@testset "4.2.2.1 (a+b cos)^m (c+d cos)^n" begin
    (A, B, a, b, c, d, e, f, m, n, x, ) = @variables A B a b c d e f m n x

    #= ::Package:: =#

    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^n*(a+b*cos(e+f*x))^m=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^n*(a+a*cos(e+f*x))^m=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^m*(a+a*cos(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^5*(a + a*cos(c + d*x)), x, 7, (5*a*x)/16 + (a*sin(c + d*x))/d + (5*a*cos(c + d*x)*sin(c + d*x))/(16*d) + (5*a*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (2*a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^4*(a + a*cos(c + d*x)), x, 6, (3*a*x)/8 + (a*sin(c + d*x))/d + (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (2*a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^3*(a + a*cos(c + d*x)), x, 6, (3*a*x)/8 + (a*sin(c + d*x))/d + (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^2*(a + a*cos(c + d*x)), x, 5, (a*x)/2 + (a*sin(c + d*x))/d + (a*cos(c + d*x)*sin(c + d*x))/(2*d) - (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^1*(a + a*cos(c + d*x)), x, 1, (a*x)/2 + (a*sin(c + d*x))/d + (a*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x)), x, 2, a*x + (a*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^1*(a + a*cos(c + d*x)), x, 2, a*x + (a*atanh(sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^2*(a + a*cos(c + d*x)), x, 4, (a*atanh(sin(c + d*x)))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + a*cos(c + d*x)), x, 5, (a*atanh(sin(c + d*x)))/(2*d) + (a*tan(c + d*x))/d + (a*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + a*cos(c + d*x)), x, 5, (a*atanh(sin(c + d*x)))/(2*d) + (a*tan(c + d*x))/d + (a*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*cos(c + d*x)), x, 6, (3*a*atanh(sin(c + d*x)))/(8*d) + (a*tan(c + d*x))/d + (3*a*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^6*(a + a*cos(c + d*x)), x, 6, (3*a*atanh(sin(c + d*x)))/(8*d) + (a*tan(c + d*x))/d + (3*a*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (2*a*tan(c + d*x)^3)/(3*d) + (a*tan(c + d*x)^5)/(5*d)]


    @test_int [cos(c + d*x)^4*(a + a*cos(c + d*x))^2, x, 11, (11*a^2*x)/16 + (2*a^2*sin(c + d*x))/d + (11*a^2*cos(c + d*x)*sin(c + d*x))/(16*d) + (11*a^2*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^2*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (4*a^2*sin(c + d*x)^3)/(3*d) + (2*a^2*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^3*(a + a*cos(c + d*x))^2, x, 9, (3*a^2*x)/4 + (2*a^2*sin(c + d*x))/d + (3*a^2*cos(c + d*x)*sin(c + d*x))/(4*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*d) - (a^2*sin(c + d*x)^3)/d + (a^2*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^2*(a + a*cos(c + d*x))^2, x, 9, (7*a^2*x)/8 + (2*a^2*sin(c + d*x))/d + (7*a^2*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^2*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (2*a^2*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^1*(a + a*cos(c + d*x))^2, x, 2, a^2*x + (2*a^2*sin(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/d - (a^2*sin(c + d*x)^3)/(3*d), a^2*x + (4*a^2*sin(c + d*x))/(3*d) + (a^2*cos(c + d*x)*sin(c + d*x))/(3*d) + ((a + a*cos(c + d*x))^2*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x))^2, x, 1, (3*a^2*x)/2 + (2*a^2*sin(c + d*x))/d + (a^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^1*(a + a*cos(c + d*x))^2, x, 3, 2*a^2*x + (a^2*atanh(sin(c + d*x)))/d + (a^2*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^2*(a + a*cos(c + d*x))^2, x, 5, a^2*x + (2*a^2*atanh(sin(c + d*x)))/d + (a^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + a*cos(c + d*x))^2, x, 7, (3*a^2*atanh(sin(c + d*x)))/(2*d) + (2*a^2*tan(c + d*x))/d + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + a*cos(c + d*x))^2, x, 8, (a^2*atanh(sin(c + d*x)))/d + (2*a^2*tan(c + d*x))/d + (a^2*sec(c + d*x)*tan(c + d*x))/d + (a^2*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*cos(c + d*x))^2, x, 9, (7*a^2*atanh(sin(c + d*x)))/(8*d) + (2*a^2*tan(c + d*x))/d + (7*a^2*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (2*a^2*tan(c + d*x)^3)/(3*d)]


    @test_int [cos(c + d*x)^3*(a + a*cos(c + d*x))^3, x, 13, (23*a^3*x)/16 + (4*a^3*sin(c + d*x))/d + (23*a^3*cos(c + d*x)*sin(c + d*x))/(16*d) + (23*a^3*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^3*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (7*a^3*sin(c + d*x)^3)/(3*d) + (3*a^3*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^2*(a + a*cos(c + d*x))^3, x, 11, (13*a^3*x)/8 + (4*a^3*sin(c + d*x))/d + (13*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (3*a^3*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (5*a^3*sin(c + d*x)^3)/(3*d) + (a^3*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^1*(a + a*cos(c + d*x))^3, x, 8, (15*a^3*x)/8 + (4*a^3*sin(c + d*x))/d + (15*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^3*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (a^3*sin(c + d*x)^3)/d, (15*a^3*x)/8 + (3*a^3*sin(c + d*x))/d + (9*a^3*cos(c + d*x)*sin(c + d*x))/(8*d) + ((a + a*cos(c + d*x))^3*sin(c + d*x))/(4*d) - (a^3*sin(c + d*x)^3)/(4*d)]
    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x))^3, x, 7, (5*a^3*x)/2 + (4*a^3*sin(c + d*x))/d + (3*a^3*cos(c + d*x)*sin(c + d*x))/(2*d) - (a^3*sin(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^1*(a + a*cos(c + d*x))^3, x, 6, (7*a^3*x)/2 + (a^3*atanh(sin(c + d*x)))/d + (3*a^3*sin(c + d*x))/d + (a^3*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*(a + a*cos(c + d*x))^3, x, 6, 3*a^3*x + (3*a^3*atanh(sin(c + d*x)))/d + (a^3*sin(c + d*x))/d + (a^3*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + a*cos(c + d*x))^3, x, 7, a^3*x + (7*a^3*atanh(sin(c + d*x)))/(2*d) + (3*a^3*tan(c + d*x))/d + (a^3*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + a*cos(c + d*x))^3, x, 9, (5*a^3*atanh(sin(c + d*x)))/(2*d) + (4*a^3*tan(c + d*x))/d + (3*a^3*sec(c + d*x)*tan(c + d*x))/(2*d) + (a^3*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*cos(c + d*x))^3, x, 11, (15*a^3*atanh(sin(c + d*x)))/(8*d) + (4*a^3*tan(c + d*x))/d + (15*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^3*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (a^3*tan(c + d*x)^3)/d]
    @test_int [sec(c + d*x)^6*(a + a*cos(c + d*x))^3, x, 11, (13*a^3*atanh(sin(c + d*x)))/(8*d) + (4*a^3*tan(c + d*x))/d + (13*a^3*sec(c + d*x)*tan(c + d*x))/(8*d) + (3*a^3*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (5*a^3*tan(c + d*x)^3)/(3*d) + (a^3*tan(c + d*x)^5)/(5*d)]


    @test_int [cos(c + d*x)^2*(a + a*cos(c + d*x))^4, x, 15, (49*a^4*x)/16 + (8*a^4*sin(c + d*x))/d + (49*a^4*cos(c + d*x)*sin(c + d*x))/(16*d) + (41*a^4*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^4*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (4*a^4*sin(c + d*x)^3)/d + (4*a^4*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^1*(a + a*cos(c + d*x))^4, x, 11, (7*a^4*x)/2 + (8*a^4*sin(c + d*x))/d + (7*a^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (a^4*cos(c + d*x)^3*sin(c + d*x))/d - (8*a^4*sin(c + d*x)^3)/(3*d) + (a^4*sin(c + d*x)^5)/(5*d), (7*a^4*x)/2 + (32*a^4*sin(c + d*x))/(5*d) + (27*a^4*cos(c + d*x)*sin(c + d*x))/(10*d) + (a^4*cos(c + d*x)^3*sin(c + d*x))/(5*d) + ((a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d) - (16*a^4*sin(c + d*x)^3)/(15*d)]
    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x))^4, x, 10, (35*a^4*x)/8 + (8*a^4*sin(c + d*x))/d + (27*a^4*cos(c + d*x)*sin(c + d*x))/(8*d) + (a^4*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (4*a^4*sin(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^1*(a + a*cos(c + d*x))^4, x, 8, 6*a^4*x + (a^4*atanh(sin(c + d*x)))/d + (7*a^4*sin(c + d*x))/d + (2*a^4*cos(c + d*x)*sin(c + d*x))/d - (a^4*sin(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^2*(a + a*cos(c + d*x))^4, x, 8, (13*a^4*x)/2 + (4*a^4*atanh(sin(c + d*x)))/d + (4*a^4*sin(c + d*x))/d + (a^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (a^4*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + a*cos(c + d*x))^4, x, 8, 4*a^4*x + (13*a^4*atanh(sin(c + d*x)))/(2*d) + (a^4*sin(c + d*x))/d + (4*a^4*tan(c + d*x))/d + (a^4*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + a*cos(c + d*x))^4, x, 9, a^4*x + (6*a^4*atanh(sin(c + d*x)))/d + (7*a^4*tan(c + d*x))/d + (2*a^4*sec(c + d*x)*tan(c + d*x))/d + (a^4*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*cos(c + d*x))^4, x, 12, (35*a^4*atanh(sin(c + d*x)))/(8*d) + (8*a^4*tan(c + d*x))/d + (27*a^4*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^4*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (4*a^4*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^6*(a + a*cos(c + d*x))^4, x, 13, (7*a^4*atanh(sin(c + d*x)))/(2*d) + (8*a^4*tan(c + d*x))/d + (7*a^4*sec(c + d*x)*tan(c + d*x))/(2*d) + (a^4*sec(c + d*x)^3*tan(c + d*x))/d + (8*a^4*tan(c + d*x)^3)/(3*d) + (a^4*tan(c + d*x)^5)/(5*d)]
    @test_int [sec(c + d*x)^7*(a + a*cos(c + d*x))^4, x, 15, (49*a^4*atanh(sin(c + d*x)))/(16*d) + (8*a^4*tan(c + d*x))/d + (49*a^4*sec(c + d*x)*tan(c + d*x))/(16*d) + (41*a^4*sec(c + d*x)^3*tan(c + d*x))/(24*d) + (a^4*sec(c + d*x)^5*tan(c + d*x))/(6*d) + (4*a^4*tan(c + d*x)^3)/d + (4*a^4*tan(c + d*x)^5)/(5*d)]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^5/(a + a*cos(c + d*x)), x, 7, (15*x)/(8*a) - (4*sin(c + d*x))/(a*d) + (15*cos(c + d*x)*sin(c + d*x))/(8*a*d) + (5*cos(c + d*x)^3*sin(c + d*x))/(4*a*d) - (cos(c + d*x)^4*sin(c + d*x))/(d*(a + a*cos(c + d*x))) + (4*sin(c + d*x)^3)/(3*a*d)]
    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x)), x, 6, -((3*x)/(2*a)) + (4*sin(c + d*x))/(a*d) - (3*cos(c + d*x)*sin(c + d*x))/(2*a*d) - (cos(c + d*x)^3*sin(c + d*x))/(d*(a + a*cos(c + d*x))) - (4*sin(c + d*x)^3)/(3*a*d)]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x)), x, 2, (3*x)/(2*a) - (2*sin(c + d*x))/(a*d) + (3*cos(c + d*x)*sin(c + d*x))/(2*a*d) - (cos(c + d*x)^2*sin(c + d*x))/(d*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x)), x, 4, -(x/a) + sin(c + d*x)/(a*d) + sin(c + d*x)/(a*d*(1 + cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x)), x, 2, x/a - sin(c + d*x)/(d*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x)), x, 1, sin(c + d*x)/(d*(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*cos(c + d*x)), x, 3, atanh(sin(c + d*x))/(a*d) - sin(c + d*x)/(d*(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*cos(c + d*x)), x, 5, -(atanh(sin(c + d*x))/(a*d)) + (2*tan(c + d*x))/(a*d) - tan(c + d*x)/(d*(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*cos(c + d*x)), x, 6, (3*atanh(sin(c + d*x)))/(2*a*d) - (2*tan(c + d*x))/(a*d) + (3*sec(c + d*x)*tan(c + d*x))/(2*a*d) - (sec(c + d*x)*tan(c + d*x))/(d*(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + a*cos(c + d*x)), x, 6, -((3*atanh(sin(c + d*x)))/(2*a*d)) + (4*tan(c + d*x))/(a*d) - (3*sec(c + d*x)*tan(c + d*x))/(2*a*d) - (sec(c + d*x)^2*tan(c + d*x))/(d*(a + a*cos(c + d*x))) + (4*tan(c + d*x)^3)/(3*a*d)]


    @test_int [cos(c + d*x)^5/(a + a*cos(c + d*x))^2, x, 7, -((5*x)/a^2) + (12*sin(c + d*x))/(a^2*d) - (5*cos(c + d*x)*sin(c + d*x))/(a^2*d) - (10*cos(c + d*x)^3*sin(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) - (cos(c + d*x)^4*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) - (4*sin(c + d*x)^3)/(a^2*d)]
    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^2, x, 3, (7*x)/(2*a^2) - (16*sin(c + d*x))/(3*a^2*d) + (7*cos(c + d*x)*sin(c + d*x))/(2*a^2*d) - (8*cos(c + d*x)^2*sin(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) - (cos(c + d*x)^3*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x))^2, x, 6, -((2*x)/a^2) + (4*sin(c + d*x))/(3*a^2*d) + (2*sin(c + d*x))/(a^2*d*(1 + cos(c + d*x))) - (cos(c + d*x)^2*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x))^2, x, 3, x/a^2 - (5*sin(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) + sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x))^2, x, 2, -(sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2)) + (2*sin(c + d*x))/(3*d*(a^2 + a^2*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x))^2, x, 2, sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2) + sin(c + d*x)/(3*d*(a^2 + a^2*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*cos(c + d*x))^2, x, 4, atanh(sin(c + d*x))/(a^2*d) - (4*sin(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) - sin(c + d*x)/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [sec(c + d*x)^2/(a + a*cos(c + d*x))^2, x, 6, -((2*atanh(sin(c + d*x)))/(a^2*d)) + (10*tan(c + d*x))/(3*a^2*d) - (2*tan(c + d*x))/(a^2*d*(1 + cos(c + d*x))) - tan(c + d*x)/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [sec(c + d*x)^3/(a + a*cos(c + d*x))^2, x, 7, (7*atanh(sin(c + d*x)))/(2*a^2*d) - (16*tan(c + d*x))/(3*a^2*d) + (7*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) - (8*sec(c + d*x)*tan(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) - (sec(c + d*x)*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [sec(c + d*x)^4/(a + a*cos(c + d*x))^2, x, 7, -((5*atanh(sin(c + d*x)))/(a^2*d)) + (12*tan(c + d*x))/(a^2*d) - (5*sec(c + d*x)*tan(c + d*x))/(a^2*d) - (10*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) - (sec(c + d*x)^2*tan(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + (4*tan(c + d*x)^3)/(a^2*d)]


    @test_int [cos(c + d*x)^5/(a + a*cos(c + d*x))^3, x, 4, (13*x)/(2*a^3) - (152*sin(c + d*x))/(15*a^3*d) + (13*cos(c + d*x)*sin(c + d*x))/(2*a^3*d) - (cos(c + d*x)^4*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (11*cos(c + d*x)^3*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (76*cos(c + d*x)^2*sin(c + d*x))/(15*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^3, x, 7, -((3*x)/a^3) + (9*sin(c + d*x))/(5*a^3*d) - (cos(c + d*x)^3*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (3*cos(c + d*x)^2*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) + (3*sin(c + d*x))/(d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x))^3, x, 5, x/a^3 - (cos(c + d*x)^2*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (7*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (29*sin(c + d*x))/(15*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x))^3, x, 3, sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (7*sin(c + d*x))/(15*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x))^3, x, 3, -(sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3)) + sin(c + d*x)/(5*a*d*(a + a*cos(c + d*x))^2) + sin(c + d*x)/(5*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x))^3, x, 3, sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (2*sin(c + d*x))/(15*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*cos(c + d*x))^3, x, 5, atanh(sin(c + d*x))/(a^3*d) - sin(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) - (7*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (22*sin(c + d*x))/(15*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*cos(c + d*x))^3, x, 7, -((3*atanh(sin(c + d*x)))/(a^3*d)) + (24*tan(c + d*x))/(5*a^3*d) - tan(c + d*x)/(5*d*(a + a*cos(c + d*x))^3) - (3*tan(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) - (3*tan(c + d*x))/(d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*cos(c + d*x))^3, x, 8, (13*atanh(sin(c + d*x)))/(2*a^3*d) - (152*tan(c + d*x))/(15*a^3*d) + (13*sec(c + d*x)*tan(c + d*x))/(2*a^3*d) - (sec(c + d*x)*tan(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (11*sec(c + d*x)*tan(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (76*sec(c + d*x)*tan(c + d*x))/(15*d*(a^3 + a^3*cos(c + d*x)))]


    @test_int [cos(c + d*x)^6/(a + a*cos(c + d*x))^4, x, 5, (21*x)/(2*a^4) - (576*sin(c + d*x))/(35*a^4*d) + (21*cos(c + d*x)*sin(c + d*x))/(2*a^4*d) - (43*cos(c + d*x)^3*sin(c + d*x))/(35*a^4*d*(1 + cos(c + d*x))^2) - (288*cos(c + d*x)^2*sin(c + d*x))/(35*a^4*d*(1 + cos(c + d*x))) - (cos(c + d*x)^5*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*cos(c + d*x)^4*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3)]
    @test_int [cos(c + d*x)^5/(a + a*cos(c + d*x))^4, x, 8, -((4*x)/a^4) + (244*sin(c + d*x))/(105*a^4*d) - (88*cos(c + d*x)^2*sin(c + d*x))/(105*a^4*d*(1 + cos(c + d*x))^2) + (4*sin(c + d*x))/(a^4*d*(1 + cos(c + d*x))) - (cos(c + d*x)^4*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (12*cos(c + d*x)^3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3)]
    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^4, x, 6, x/a^4 + (11*sin(c + d*x))/(21*a^4*d*(1 + cos(c + d*x))^2) - (43*sin(c + d*x))/(21*a^4*d*(1 + cos(c + d*x))) - (cos(c + d*x)^3*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*cos(c + d*x)^2*sin(c + d*x))/(7*a*d*(a + a*cos(c + d*x))^3)]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x))^4, x, 5, -((18*sin(c + d*x))/(35*a^4*d*(1 + cos(c + d*x))^2)) + (12*sin(c + d*x))/(35*a^4*d*(1 + cos(c + d*x))) - (cos(c + d*x)^2*sin(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) + (8*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3)]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x))^4, x, 4, sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) - (11*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + (13*sin(c + d*x))/(105*d*(a^2 + a^2*cos(c + d*x))^2) + (13*sin(c + d*x))/(105*d*(a^4 + a^4*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x))^4, x, 4, -(sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4)) + (4*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + (8*sin(c + d*x))/(105*d*(a^2 + a^2*cos(c + d*x))^2) + (8*sin(c + d*x))/(105*d*(a^4 + a^4*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x))^4, x, 4, sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) + (3*sin(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(35*d*(a^2 + a^2*cos(c + d*x))^2) + (2*sin(c + d*x))/(35*d*(a^4 + a^4*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*cos(c + d*x))^4, x, 6, atanh(sin(c + d*x))/(a^4*d) - (11*sin(c + d*x))/(21*a^4*d*(1 + cos(c + d*x))^2) - (32*sin(c + d*x))/(21*a^4*d*(1 + cos(c + d*x))) - sin(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) - (2*sin(c + d*x))/(7*a*d*(a + a*cos(c + d*x))^3)]
    @test_int [sec(c + d*x)^2/(a + a*cos(c + d*x))^4, x, 8, -((4*atanh(sin(c + d*x)))/(a^4*d)) + (664*tan(c + d*x))/(105*a^4*d) - (88*tan(c + d*x))/(105*a^4*d*(1 + cos(c + d*x))^2) - (4*tan(c + d*x))/(a^4*d*(1 + cos(c + d*x))) - tan(c + d*x)/(7*d*(a + a*cos(c + d*x))^4) - (12*tan(c + d*x))/(35*a*d*(a + a*cos(c + d*x))^3)]
    @test_int [sec(c + d*x)^3/(a + a*cos(c + d*x))^4, x, 9, (21*atanh(sin(c + d*x)))/(2*a^4*d) - (576*tan(c + d*x))/(35*a^4*d) + (21*sec(c + d*x)*tan(c + d*x))/(2*a^4*d) - (43*sec(c + d*x)*tan(c + d*x))/(35*a^4*d*(1 + cos(c + d*x))^2) - (288*sec(c + d*x)*tan(c + d*x))/(35*a^4*d*(1 + cos(c + d*x))) - (sec(c + d*x)*tan(c + d*x))/(7*d*(a + a*cos(c + d*x))^4) - (2*sec(c + d*x)*tan(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^3)]


    @test_int [cos(c + d*x)^7/(a + a*cos(c + d*x))^5, x, 6, (31*x)/(2*a^5) - (7664*sin(c + d*x))/(315*a^5*d) + (31*cos(c + d*x)*sin(c + d*x))/(2*a^5*d) - (cos(c + d*x)^6*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (17*cos(c + d*x)^5*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (28*cos(c + d*x)^4*sin(c + d*x))/(45*a^2*d*(a + a*cos(c + d*x))^3) - (577*cos(c + d*x)^3*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (3832*cos(c + d*x)^2*sin(c + d*x))/(315*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [cos(c + d*x)^6/(a + a*cos(c + d*x))^5, x, 9, -((5*x)/a^5) + (181*sin(c + d*x))/(63*a^5*d) - (cos(c + d*x)^5*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (5*cos(c + d*x)^4*sin(c + d*x))/(21*a*d*(a + a*cos(c + d*x))^4) - (29*cos(c + d*x)^3*sin(c + d*x))/(63*a^2*d*(a + a*cos(c + d*x))^3) - (67*cos(c + d*x)^2*sin(c + d*x))/(63*a^3*d*(a + a*cos(c + d*x))^2) + (5*sin(c + d*x))/(d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [cos(c + d*x)^5/(a + a*cos(c + d*x))^5, x, 7, x/a^5 - (cos(c + d*x)^4*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (13*cos(c + d*x)^3*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (34*cos(c + d*x)^2*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^3) + (173*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (661*sin(c + d*x))/(315*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^5, x, 6, -((cos(c + d*x)^3*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5)) - (11*cos(c + d*x)^2*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) + (67*sin(c + d*x))/(315*a^2*d*(a + a*cos(c + d*x))^3) - (142*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) + (83*sin(c + d*x))/(315*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x))^5, x, 6, -((cos(c + d*x)^2*sin(c + d*x))/(9*d*(a + a*cos(c + d*x))^5)) + sin(c + d*x)/(7*a*d*(a + a*cos(c + d*x))^4) - (17*sin(c + d*x))/(63*a^2*d*(a + a*cos(c + d*x))^3) + (5*sin(c + d*x))/(63*a^3*d*(a + a*cos(c + d*x))^2) + (5*sin(c + d*x))/(63*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x))^5, x, 5, sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) - (2*sin(c + d*x))/(9*a*d*(a + a*cos(c + d*x))^4) + sin(c + d*x)/(15*a^2*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(45*a^3*d*(a + a*cos(c + d*x))^2) + (2*sin(c + d*x))/(45*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x))^5, x, 5, -(sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5)) + (5*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) + sin(c + d*x)/(21*a^2*d*(a + a*cos(c + d*x))^3) + (2*sin(c + d*x))/(63*a*d*(a^2 + a^2*cos(c + d*x))^2) + (2*sin(c + d*x))/(63*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x))^5, x, 5, sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) + (4*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) + (4*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^3) + (8*sin(c + d*x))/(315*a*d*(a^2 + a^2*cos(c + d*x))^2) + (8*sin(c + d*x))/(315*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + a*cos(c + d*x))^5, x, 7, atanh(sin(c + d*x))/(a^5*d) - sin(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) - (13*sin(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (34*sin(c + d*x))/(105*a^2*d*(a + a*cos(c + d*x))^3) - (173*sin(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (488*sin(c + d*x))/(315*d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + a*cos(c + d*x))^5, x, 9, -((5*atanh(sin(c + d*x)))/(a^5*d)) + (496*tan(c + d*x))/(63*a^5*d) - tan(c + d*x)/(9*d*(a + a*cos(c + d*x))^5) - (5*tan(c + d*x))/(21*a*d*(a + a*cos(c + d*x))^4) - (29*tan(c + d*x))/(63*a^2*d*(a + a*cos(c + d*x))^3) - (67*tan(c + d*x))/(63*a^3*d*(a + a*cos(c + d*x))^2) - (5*tan(c + d*x))/(d*(a^5 + a^5*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*cos(c + d*x))^5, x, 10, (31*atanh(sin(c + d*x)))/(2*a^5*d) - (7664*tan(c + d*x))/(315*a^5*d) + (31*sec(c + d*x)*tan(c + d*x))/(2*a^5*d) - (sec(c + d*x)*tan(c + d*x))/(9*d*(a + a*cos(c + d*x))^5) - (17*sec(c + d*x)*tan(c + d*x))/(63*a*d*(a + a*cos(c + d*x))^4) - (28*sec(c + d*x)*tan(c + d*x))/(45*a^2*d*(a + a*cos(c + d*x))^3) - (577*sec(c + d*x)*tan(c + d*x))/(315*a^3*d*(a + a*cos(c + d*x))^2) - (3832*sec(c + d*x)*tan(c + d*x))/(315*d*(a^5 + a^5*cos(c + d*x)))]


    @test_int [cos(c + d*x)^5/(a + a*cos(c + d*x))^6, x, 7, (130*sin(c + d*x))/(693*a^6*d*(1 + cos(c + d*x))^3) - (268*sin(c + d*x))/(693*a^6*d*(1 + cos(c + d*x))^2) + (146*sin(c + d*x))/(693*a^6*d*(1 + cos(c + d*x))) - (cos(c + d*x)^4*sin(c + d*x))/(11*d*(a + a*cos(c + d*x))^6) - (14*cos(c + d*x)^3*sin(c + d*x))/(99*a*d*(a + a*cos(c + d*x))^5) - (118*cos(c + d*x)^2*sin(c + d*x))/(693*a^2*d*(a + a*cos(c + d*x))^4)]
    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^6, x, 7, -((241*sin(c + d*x))/(1155*a^6*d*(1 + cos(c + d*x))^3)) + (61*sin(c + d*x))/(1155*a^6*d*(1 + cos(c + d*x))^2) + (61*sin(c + d*x))/(1155*a^6*d*(1 + cos(c + d*x))) - (cos(c + d*x)^3*sin(c + d*x))/(11*d*(a + a*cos(c + d*x))^6) - (4*cos(c + d*x)^2*sin(c + d*x))/(33*a*d*(a + a*cos(c + d*x))^5) + (9*sin(c + d*x))/(77*a^2*d*(a + a*cos(c + d*x))^4)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^m*(a+a*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^4*(a + a*cos(c + d*x))^(1/2), x, 5, (32*a*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (16*a*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a*cos(c + d*x)^4*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x))) - (64*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (32*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*a*d)]
    @test_int [cos(c + d*x)^3*(a + a*cos(c + d*x))^(1/2), x, 4, (4*a*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) - (8*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (12*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*a*d)]
    @test_int [cos(c + d*x)^2*(a + a*cos(c + d*x))^(1/2), x, 3, (14*a*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (4*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*a*d)]
    @test_int [cos(c + d*x)^1*(a + a*cos(c + d*x))^(1/2), x, 2, (2*a*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x))^(1/2), x, 1, (2*a*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1*(a + a*cos(c + d*x))^(1/2), x, 2, (2*sqrt(a)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d]
    @test_int [sec(c + d*x)^2*(a + a*cos(c + d*x))^(1/2), x, 3, (sqrt(a)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (a*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3*(a + a*cos(c + d*x))^(1/2), x, 4, (3*sqrt(a)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*d) + (3*a*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + a*cos(c + d*x))^(1/2), x, 5, (5*sqrt(a)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(8*d) + (5*a*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (5*a*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^3*(a + a*cos(c + d*x))^(3/2), x, 6, (68*a^2*sin(c + d*x))/(45*d*sqrt(a + a*cos(c + d*x))) + (34*a^2*cos(c + d*x)^3*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*cos(c + d*x)^4*sin(c + d*x))/(9*d*sqrt(a + a*cos(c + d*x))) - (136*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (68*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d)]
    @test_int [cos(c + d*x)^2*(a + a*cos(c + d*x))^(3/2), x, 4, (152*a^2*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (38*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*d) - (4*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*a*d)]
    @test_int [cos(c + d*x)^1*(a + a*cos(c + d*x))^(3/2), x, 3, (8*a^2*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x))^(3/2), x, 2, (8*a^2*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^1*(a + a*cos(c + d*x))^(3/2), x, 4, (2*a^(3/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (2*a^2*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2*(a + a*cos(c + d*x))^(3/2), x, 4, (3*a^(3/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (a^2*tan(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3*(a + a*cos(c + d*x))^(3/2), x, 5, (7*a^(3/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*d) + (7*a^2*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4*(a + a*cos(c + d*x))^(3/2), x, 6, (11*a^(3/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(8*d) + (11*a^2*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (11*a^2*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^3*(a + a*cos(c + d*x))^(5/2), x, 6, (284*a^3*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) + (710*a^3*cos(c + d*x)^3*sin(c + d*x))/(693*d*sqrt(a + a*cos(c + d*x))) + (46*a^3*cos(c + d*x)^4*sin(c + d*x))/(99*d*sqrt(a + a*cos(c + d*x))) - (568*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(693*d) + (2*a^2*cos(c + d*x)^4*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(11*d) + (284*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(231*d)]
    @test_int [cos(c + d*x)^2*(a + a*cos(c + d*x))^(5/2), x, 5, (832*a^3*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (208*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(315*d) + (26*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(105*d) - (4*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*d) + (2*(a + a*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*a*d)]
    @test_int [cos(c + d*x)^1*(a + a*cos(c + d*x))^(5/2), x, 4, (64*a^3*sin(c + d*x))/(21*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x))^(5/2), x, 3, (64*a^3*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (16*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*a*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^1*(a + a*cos(c + d*x))^(5/2), x, 4, (2*a^(5/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (14*a^3*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^2*(a + a*cos(c + d*x))^(5/2), x, 4, (5*a^(5/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (a^3*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + a*cos(c + d*x))^(5/2), x, 4, (19*a^(5/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*d) + (9*a^3*tan(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + a*cos(c + d*x))^(5/2), x, 5, (25*a^(5/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(8*d) + (25*a^3*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (13*a^3*sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^5*(a + a*cos(c + d*x))^(5/2), x, 6, (163*a^(5/2)*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(64*d) + (163*a^3*tan(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (163*a^3*sec(c + d*x)*tan(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (17*a^3*sec(c + d*x)^2*tan(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)]


    @test_int [cos(c + d*x)^0*(a + a*cos(c + d*x))^(7/2), x, 4, (256*a^4*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (64*a^3*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(35*d) + (24*a^2*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*a*(a + a*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^(1/2), x, 7, (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (148*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) - (2*cos(c + d*x)^2*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*cos(c + d*x)^3*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) + (62*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(105*a*d)]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x))^(1/2), x, 6, -((sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (28*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*cos(c + d*x)^2*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) - (2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(15*a*d)]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x))^(1/2), x, 4, (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (4*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*a*d)]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x))^(1/2), x, 3, -((sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x))^(1/2), x, 2, (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)]
    @test_int [sec(c + d*x)^1/sqrt(a + a*cos(c + d*x)), x, 5, (2*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)]
    @test_int [sec(c + d*x)^2/sqrt(a + a*cos(c + d*x)), x, 6, -(atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + tan(c + d*x)/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/sqrt(a + a*cos(c + d*x)), x, 7, (7*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - tan(c + d*x)/(4*d*sqrt(a + a*cos(c + d*x))) + (sec(c + d*x)*tan(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4/sqrt(a + a*cos(c + d*x)), x, 8, -((9*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(8*sqrt(a)*d)) + (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (7*tan(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)*tan(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (sec(c + d*x)^2*tan(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^(3/2), x, 7, -((15*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - (cos(c + d*x)^3*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (31*sin(c + d*x))/(5*a*d*sqrt(a + a*cos(c + d*x))) + (9*cos(c + d*x)^2*sin(c + d*x))/(10*a*d*sqrt(a + a*cos(c + d*x))) - (13*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(10*a^2*d)]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x))^(3/2), x, 6, (11*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (cos(c + d*x)^2*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) - (13*sin(c + d*x))/(3*a*d*sqrt(a + a*cos(c + d*x))) + (7*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(6*a^2*d)]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x))^(3/2), x, 4, -((7*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) + sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)) + (2*sin(c + d*x))/(a*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x))^(3/2), x, 3, (3*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x))^(3/2), x, 3, atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x))))/(2*sqrt(2)*a^(3/2)*d) + sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^1/(a + a*cos(c + d*x))^(3/2), x, 6, (2*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(3/2)*d) - (5*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^2/(a + a*cos(c + d*x))^(3/2), x, 7, -((3*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(3/2)*d)) + (9*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - tan(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)) + (3*tan(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + a*cos(c + d*x))^(3/2), x, 8, (19*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*a^(3/2)*d) - (13*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (7*tan(c + d*x))/(4*a*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)*tan(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (sec(c + d*x)*tan(c + d*x))/(a*d*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^4/(a + a*cos(c + d*x))^(5/2), x, 7, (163*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (cos(c + d*x)^3*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (17*cos(c + d*x)^2*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) - (197*sin(c + d*x))/(24*a^2*d*sqrt(a + a*cos(c + d*x))) + (95*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(48*a^3*d)]
    @test_int [cos(c + d*x)^3/(a + a*cos(c + d*x))^(5/2), x, 6, -((75*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - (cos(c + d*x)^2*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + (13*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (9*sin(c + d*x))/(4*a^2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + a*cos(c + d*x))^(5/2), x, 4, (19*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) - (13*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^1/(a + a*cos(c + d*x))^(5/2), x, 4, (5*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) + (5*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^0/(a + a*cos(c + d*x))^(5/2), x, 4, (3*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) + (3*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^1/(a + a*cos(c + d*x))^(5/2), x, 7, (2*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(5/2)*d) - (43*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) - (11*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [sec(c + d*x)^2/(a + a*cos(c + d*x))^(5/2), x, 8, -((5*atanh((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(5/2)*d)) + (115*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - tan(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)) - (15*tan(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (35*tan(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/2)*(a+a*cos(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^(5/2)*(a + a*cos(c + d*x)), x, 6, (6*a*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (10*a*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (10*a*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)), x, 5, (6*a*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*cos(c + d*x)), x, 4, (2*a*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x)), x, 3, (2*a*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/d]
    @test_int [cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x)), x, 4, -((2*a*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x)), x, 5, -((2*a*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x)), x, 6, -((6*a*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (2*a*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*a*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^2, x, 10, (32*a^2*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (20*a^2*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (20*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^2*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2, x, 9, (12*a^2*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (8*a^2*Elliptic.F((1/2)*(c + d*x), 2))/(7*d) + (8*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(7*d) + (4*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^2, x, 7, (16*a^2*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (4*a^2*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (4*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^2, x, 6, (4*a^2*Elliptic.E((1/2)*(c + d*x), 2))/d + (8*a^2*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^2, x, 6, (4*a^2*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^2, x, 7, -((4*a^2*Elliptic.E((1/2)*(c + d*x), 2))/d) + (8*a^2*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (4*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^2, x, 9, -((16*a^2*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (4*a^2*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (16*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3, x, 12, (68*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (44*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (44*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (68*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (6*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^3*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^3, x, 10, (28*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (52*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (52*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (6*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^3, x, 8, (36*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (4*a^3*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/d + (2*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^3, x, 8, (4*a^3*Elliptic.E((1/2)*(c + d*x), 2))/d + (20*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^3*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^3, x, 8, -((4*a^3*Elliptic.E((1/2)*(c + d*x), 2))/d) + (20*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^3*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*a^3*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^3, x, 10, -((36*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (4*a^3*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*a^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*a^3*sin(c + d*x))/(d*cos(c + d*x)^(3/2)) + (36*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^3, x, 12, -((28*a^3*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (52*a^3*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*a^3*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (6*a^3*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (52*a^3*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (28*a^3*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^4, x, 16, (128*a^4*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (904*a^4*Elliptic.F((1/2)*(c + d*x), 2))/(231*d) + (904*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(231*d) + (128*a^4*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (150*a^4*cos(c + d*x)^(5/2)*sin(c + d*x))/(77*d) + (8*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d) + (2*a^4*cos(c + d*x)^(9/2)*sin(c + d*x))/(11*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^4, x, 13, (152*a^4*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (32*a^4*Elliptic.F((1/2)*(c + d*x), 2))/(7*d) + (32*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(7*d) + (122*a^4*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (8*a^4*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^4*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^4, x, 11, (64*a^4*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (136*a^4*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (94*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (8*a^4*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^4*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^4, x, 10, (56*a^4*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (32*a^4*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^4*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (8*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^4*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^4, x, 10, (40*a^4*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^4*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (8*a^4*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (2*a^4*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^4, x, 11, -((56*a^4*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (32*a^4*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^4*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (8*a^4*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (66*a^4*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^4, x, 13, -((64*a^4*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (136*a^4*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*a^4*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (8*a^4*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (94*a^4*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (64*a^4*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(7/2)/(a + a*cos(c + d*x)), x, 6, (21*Elliptic.E((1/2)*(c + d*x), 2))/(5*a*d) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) + (7*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x)), x, 5, -((3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d)) + (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a*d) + (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*cos(c + d*x)), x, 4, (3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d) - Elliptic.F((1/2)*(c + d*x), 2)/(a*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*cos(c + d*x)), x, 4, -(Elliptic.E((1/2)*(c + d*x), 2)/(a*d)) + Elliptic.F((1/2)*(c + d*x), 2)/(a*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x)), x, 4, Elliptic.E((1/2)*(c + d*x), 2)/(a*d) + Elliptic.F((1/2)*(c + d*x), 2)/(a*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(d*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x)), x, 5, -((3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d)) - Elliptic.F((1/2)*(c + d*x), 2)/(a*d) + (3*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x)), x, 6, (3*Elliptic.E((1/2)*(c + d*x), 2))/(a*d) + (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a*d) + (5*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (3*sin(c + d*x))/(a*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)/(a + a*cos(c + d*x))^2, x, 7, (56*Elliptic.E((1/2)*(c + d*x), 2))/(5*a^2*d) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(a^2*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(a^2*d) + (56*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a^2*d) - (3*cos(c + d*x)^(5/2)*sin(c + d*x))/(a^2*d*(1 + cos(c + d*x))) - (cos(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^2, x, 6, -((7*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d)) + (10*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d) - (7*cos(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^2, x, 5, (4*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a^2*d*(1 + cos(c + d*x))) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^2, x, 5, -(Elliptic.E((1/2)*(c + d*x), 2)/(a^2*d)) + (2*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(a^2*d*(1 + cos(c + d*x))) - (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^2, x, 3, Elliptic.F((1/2)*(c + d*x), 2)/(3*a^2*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^2, x, 5, Elliptic.E((1/2)*(c + d*x), 2)/(a^2*d) + (2*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(a^2*d*(1 + cos(c + d*x))) - (sqrt(cos(c + d*x))*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^2, x, 6, -((4*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d)) - (5*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + (4*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - (5*sin(c + d*x))/(3*a^2*d*sqrt(cos(c + d*x))*(1 + cos(c + d*x))) - sin(c + d*x)/(3*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2)]
    @test_int [cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^2, x, 7, (7*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d) + (10*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*d) + (10*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)) - (7*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x))) - (7*sin(c + d*x))/(3*a^2*d*cos(c + d*x)^(3/2)*(1 + cos(c + d*x))) - sin(c + d*x)/(3*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2)]


    @test_int [cos(c + d*x)^(11/2)/(a + a*cos(c + d*x))^3, x, 8, (231*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) - (21*Elliptic.F((1/2)*(c + d*x), 2))/(2*a^3*d) - (21*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) + (77*cos(c + d*x)^(3/2)*sin(c + d*x))/(10*a^3*d) - (cos(c + d*x)^(9/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (4*cos(c + d*x)^(7/2)*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) - (63*cos(c + d*x)^(5/2)*sin(c + d*x))/(10*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(9/2)/(a + a*cos(c + d*x))^3, x, 7, -((119*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d)) + (11*Elliptic.F((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a^3*d) - (cos(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*a*d*(a + a*cos(c + d*x))^2) - (119*cos(c + d*x)^(3/2)*sin(c + d*x))/(30*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^3, x, 6, (49*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) - (13*Elliptic.F((1/2)*(c + d*x), 2))/(6*a^3*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (8*cos(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (13*sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^3, x, 6, -((9*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d)) + Elliptic.F((1/2)*(c + d*x), 2)/(2*a^3*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) + (9*sqrt(cos(c + d*x))*sin(c + d*x))/(10*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^3, x, 6, -(Elliptic.E((1/2)*(c + d*x), 2)/(10*a^3*d)) + Elliptic.F((1/2)*(c + d*x), 2)/(6*a^3*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (4*sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) + (sqrt(cos(c + d*x))*sin(c + d*x))/(10*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^3, x, 6, Elliptic.E((1/2)*(c + d*x), 2)/(10*a^3*d) + Elliptic.F((1/2)*(c + d*x), 2)/(6*a^3*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) + (sqrt(cos(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*cos(c + d*x))^2) - (sqrt(cos(c + d*x))*sin(c + d*x))/(10*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^3, x, 6, (9*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) + Elliptic.F((1/2)*(c + d*x), 2)/(2*a^3*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(5*d*(a + a*cos(c + d*x))^3) - (2*sqrt(cos(c + d*x))*sin(c + d*x))/(5*a*d*(a + a*cos(c + d*x))^2) - (9*sqrt(cos(c + d*x))*sin(c + d*x))/(10*d*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^3, x, 7, -((49*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d)) - (13*Elliptic.F((1/2)*(c + d*x), 2))/(6*a^3*d) + (49*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^3) - (8*sin(c + d*x))/(15*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^2) - (13*sin(c + d*x))/(6*d*sqrt(cos(c + d*x))*(a^3 + a^3*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^3, x, 8, (119*Elliptic.E((1/2)*(c + d*x), 2))/(10*a^3*d) + (11*Elliptic.F((1/2)*(c + d*x), 2))/(2*a^3*d) + (11*sin(c + d*x))/(2*a^3*d*cos(c + d*x)^(3/2)) - (119*sin(c + d*x))/(10*a^3*d*sqrt(cos(c + d*x))) - sin(c + d*x)/(5*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^3) - (2*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^2) - (119*sin(c + d*x))/(30*d*cos(c + d*x)^(3/2)*(a^3 + a^3*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/2)*(a+a*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^(5/2)*(a + a*cos(c + d*x))^(1/2), x, 5, (5*sqrt(a)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(8*d) + (5*a*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (5*a*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(1/2), x, 4, (3*sqrt(a)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*d) + (3*a*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(1/2), x, 3, (sqrt(a)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (a*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^(1/2), x, 2, (2*sqrt(a)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d]
    @test_int [cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^(1/2), x, 1, (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^(1/2), x, 2, (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (4*a*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^(1/2), x, 3, (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (8*a*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (16*a*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^(1/2), x, 4, (2*a*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (12*a*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (16*a*sin(c + d*x))/(35*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (32*a*sin(c + d*x))/(35*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2), x, 6, (11*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(8*d) + (11*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (11*a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(3/2), x, 5, (7*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*d) + (7*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^(3/2), x, 4, (3*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^(3/2), x, 4, (2*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^(3/2), x, 3, (2*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (10*a^2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^(3/2), x, 4, (2*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (6*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (12*a^2*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^(3/2), x, 5, (2*a^2*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (26*a^2*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (104*a^2*sin(c + d*x))/(105*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (208*a^2*sin(c + d*x))/(105*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2), x, 6, (163*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(64*d) + (163*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))) + (163*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))) + (17*a^3*cos(c + d*x)^(5/2)*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + a*cos(c + d*x))^(5/2), x, 5, (25*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(8*d) + (25*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))) + (13*a^3*cos(c + d*x)^(3/2)*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))) + (a^2*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + a*cos(c + d*x))^(5/2), x, 4, (19*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*d) + (9*a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (a^2*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^(-3/2)*(a + a*cos(c + d*x))^(5/2), x, 4, (5*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d - (a^3*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + a*cos(c + d*x))^(5/2), x, 4, (2*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/d + (14*a^3*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-7/2)*(a + a*cos(c + d*x))^(5/2), x, 3, (22*a^3*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (86*a^3*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))]
    @test_int [cos(c + d*x)^(-9/2)*(a + a*cos(c + d*x))^(5/2), x, 4, (6*a^3*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (46*a^3*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (92*a^3*sin(c + d*x))/(21*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))]
    @test_int [cos(c + d*x)^(-11/2)*(a + a*cos(c + d*x))^(5/2), x, 5, (38*a^3*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)*sqrt(a + a*cos(c + d*x))) + (146*a^3*sin(c + d*x))/(105*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) + (584*a^3*sin(c + d*x))/(315*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (1168*a^3*sin(c + d*x))/(315*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2))]


    @test_int [cos(c + d*x)^(-5/4)*(a + a*cos(c + d*x))^(3/2), x, 2, (4*a^2*sin(c + d*x))/(d*cos(c + d*x)^(1/4)*sqrt(a + a*cos(c + d*x)))]


    @test_int [sqrt(a + a*cos(e + f*x))/sqrt(cos(e + f*x)), x, 2, (2*sqrt(a)*asin((sqrt(a)*sin(e + f*x))/sqrt(a + a*cos(e + f*x))))/f]
    @test_int [sqrt(a - a*cos(e + f*x))/sqrt(-cos(e + f*x)), x, 2, -((2*sqrt(a)*asin((sqrt(a)*sin(e + f*x))/sqrt(a - a*cos(e + f*x))))/f)]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(1/2), x, 7, (7*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(4*sqrt(a)*d) - (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(1/2), x, 6, -(asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(1/2), x, 5, (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)]
    @test_int [cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(1/2), x, 2, (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)]
    @test_int [cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(1/2), x, 4, -((sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(1/2), x, 5, (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d) + (2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)/(a + a*cos(c + d*x))^(1/2), x, 6, -((sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a + a*cos(c + d*x))) - (2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)/(1 + cos(c + d*x))^(1/2), x, 7, -((sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x))))/d) + (7*asin(sin(c + d*x)/sqrt(1 + cos(c + d*x))))/(4*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(1 + cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(1 + cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(1 + cos(c + d*x))^(1/2), x, 6, (sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x))))/d - asin(sin(c + d*x)/sqrt(1 + cos(c + d*x)))/d + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(1 + cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(1 + cos(c + d*x))^(1/2), x, 5, -((sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x))))/d) + (2*asin(sin(c + d*x)/sqrt(1 + cos(c + d*x))))/d]
    @test_int [cos(c + d*x)^(-1/2)/(1 + cos(c + d*x))^(1/2), x, 2, (sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x))))/d]
    @test_int [cos(c + d*x)^(-3/2)/(1 + cos(c + d*x))^(1/2), x, 3, -((sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x))))/d) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(1 + cos(c + d*x))^(1/2), x, 5, (sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x))))/d + (2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(1 + cos(c + d*x))) - (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)/(1 + cos(c + d*x))^(1/2), x, 6, -((sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x))))/d) + (2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(1 + cos(c + d*x))) - (2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(1 + cos(c + d*x))) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(1 + cos(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(3/2), x, 7, -((3*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(3/2)*d)) + (9*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (3*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(3/2), x, 6, (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(3/2)*d) - (5*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(3/2), x, 4, atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))/(2*sqrt(2)*a^(3/2)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(3/2), x, 4, (3*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(3/2), x, 5, -((7*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d)) - sin(c + d*x)/(2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + (5*sin(c + d*x))/(2*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(3/2), x, 6, (11*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + (7*sin(c + d*x))/(6*a*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (19*sin(c + d*x))/(6*a*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(5/2), x, 8, -((5*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(5/2)*d)) + (115*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (15*cos(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (35*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(5/2), x, 7, (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(5/2)*d) - (43*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (11*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(5/2), x, 5, (3*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + (7*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(5/2), x, 5, (5*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) + (sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(5/2), x, 5, (19*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (9*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(5/2), x, 6, -((75*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d)) - sin(c + d*x)/(4*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - (13*sin(c + d*x))/(16*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + (49*sin(c + d*x))/(16*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(5/2), x, 7, (163*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - (17*sin(c + d*x))/(16*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + (95*sin(c + d*x))/(48*a^2*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (299*sin(c + d*x))/(48*a^2*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)/(a + a*cos(c + d*x))^(7/2), x, 9, -((7*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(7/2)*d)) + (637*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (cos(c + d*x)^(7/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (7*cos(c + d*x)^(5/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - (259*cos(c + d*x)^(3/2)*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)) + (189*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(7/2), x, 8, (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x))))/(a^(7/2)*d) - (177*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (17*cos(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - (49*sqrt(cos(c + d*x))*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(7/2), x, 6, (5*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (13*sqrt(cos(c + d*x))*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) + (67*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(7/2), x, 6, (7*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + (3*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) + (17*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(7/2), x, 6, (13*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) + (sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(-1/2)/(a + a*cos(c + d*x))^(7/2), x, 6, (63*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - (sqrt(cos(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)) - (103*sqrt(cos(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(-3/2)/(a + a*cos(c + d*x))^(7/2), x, 7, -((363*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d)) - sin(c + d*x)/(6*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(7/2)) - (19*sin(c + d*x))/(48*a*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(5/2)) - (199*sin(c + d*x))/(192*a^2*d*sqrt(cos(c + d*x))*(a + a*cos(c + d*x))^(3/2)) + (691*sin(c + d*x))/(192*a^3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + a*cos(c + d*x))^(7/2), x, 8, (1015*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(7/2)) - (23*sin(c + d*x))/(48*a*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(5/2)) - (109*sin(c + d*x))/(64*a^2*d*cos(c + d*x)^(3/2)*(a + a*cos(c + d*x))^(3/2)) + (193*sin(c + d*x))/(64*a^3*d*cos(c + d*x)^(3/2)*sqrt(a + a*cos(c + d*x))) - (629*sin(c + d*x))/(64*a^3*d*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(7/2)/(a + a*cos(c + d*x))^(9/2), x, 7, (35*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(1024*sqrt(2)*a^(9/2)*d) - (cos(c + d*x)^(5/2)*sin(c + d*x))/(8*d*(a + a*cos(c + d*x))^(9/2)) - (19*cos(c + d*x)^(3/2)*sin(c + d*x))/(96*a*d*(a + a*cos(c + d*x))^(7/2)) - (187*sqrt(cos(c + d*x))*sin(c + d*x))/(768*a^2*d*(a + a*cos(c + d*x))^(5/2)) + (853*sqrt(cos(c + d*x))*sin(c + d*x))/(3072*a^3*d*(a + a*cos(c + d*x))^(3/2))]
    @test_int [cos(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(9/2), x, 7, (45*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x)))))/(1024*sqrt(2)*a^(9/2)*d) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(8*d*(a + a*cos(c + d*x))^(9/2)) - (5*sqrt(cos(c + d*x))*sin(c + d*x))/(32*a*d*(a + a*cos(c + d*x))^(7/2)) + (33*sqrt(cos(c + d*x))*sin(c + d*x))/(256*a^2*d*(a + a*cos(c + d*x))^(5/2)) + (73*sqrt(cos(c + d*x))*sin(c + d*x))/(1024*a^3*d*(a + a*cos(c + d*x))^(3/2))]


    @test_int [1/(sqrt(1 + cos(x))*sqrt(cos(x))), x, 2, sqrt(2)*asin(sin(x)/(1 + cos(x)))]
    @test_int [1/(sqrt(a + a*cos(x))*sqrt(cos(x))), x, 2, (sqrt(2)*atan((sqrt(a)*sin(x))/(sqrt(2)*sqrt(cos(x))*sqrt(a + a*cos(x)))))/sqrt(a)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/2)*(a - a*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^(3/2)*(a - a*cos(c + d*x))^(1/2), x, 4, -((3*sqrt(a)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(4*d)) + (3*a*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a - a*cos(c + d*x))) - (a*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(a - a*cos(c + d*x))^(1/2), x, 3, (sqrt(a)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/d - (a*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(a - a*cos(c + d*x))^(1/2), x, 2, -((2*sqrt(a)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/d)]
    @test_int [cos(c + d*x)^(-3/2)*(a - a*cos(c + d*x))^(1/2), x, 1, (2*a*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a - a*cos(c + d*x))^(1/2), x, 2, (2*a*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a - a*cos(c + d*x))) - (4*a*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(a - a*cos(c + d*x))^(1/2), x, 3, (2*a*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a - a*cos(c + d*x))) - (8*a*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a - a*cos(c + d*x))) + (16*a*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)*(1 - cos(c + d*x))^(1/2), x, 4, -((3*atanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/(4*d)) + (3*sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(1 - cos(c + d*x))) - (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(1 - cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(1 - cos(c + d*x))^(1/2), x, 3, atanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x))))/d - (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(1 - cos(c + d*x))^(1/2), x, 2, -((2*atanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d)]
    @test_int [cos(c + d*x)^(-3/2)*(1 - cos(c + d*x))^(1/2), x, 1, (2*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(1 - cos(c + d*x))^(1/2), x, 2, (2*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(3/2)) - (4*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(1 - cos(c + d*x))^(1/2), x, 3, (2*sin(c + d*x))/(5*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(5/2)) - (8*sin(c + d*x))/(15*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(3/2)) + (16*sin(c + d*x))/(15*d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(5/2)/(a - a*cos(c + d*x))^(1/2), x, 7, (7*atanh((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(4*sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(a - a*cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a - a*cos(c + d*x))^(1/2), x, 6, atanh((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d) + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a - a*cos(c + d*x))^(1/2), x, 5, (2*atanh((sqrt(a)*sin(c + d*x))/(sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d) - (sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d)]
    @test_int [cos(c + d*x)^(-1/2)/(a - a*cos(c + d*x))^(1/2), x, 2, -((sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d))]
    @test_int [cos(c + d*x)^(-3/2)/(a - a*cos(c + d*x))^(1/2), x, 4, -((sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a - a*cos(c + d*x))^(1/2), x, 5, -((sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)*sqrt(a - a*cos(c + d*x))) + (2*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)/(a - a*cos(c + d*x))^(1/2), x, 6, -((sqrt(2)*atanh((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))))/(sqrt(a)*d)) + (2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)*sqrt(a - a*cos(c + d*x))) + (2*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2)*sqrt(a - a*cos(c + d*x))) + (26*sin(c + d*x))/(15*d*sqrt(cos(c + d*x))*sqrt(a - a*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)/(1 - cos(c + d*x))^(1/2), x, 7, (7*atanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/(4*d) - (sqrt(2)*atanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d + (sqrt(cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(1 - cos(c + d*x))) + (cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(1 - cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(1 - cos(c + d*x))^(1/2), x, 6, atanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x))))/d - (sqrt(2)*atanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(1 - cos(c + d*x))^(1/2), x, 5, (2*atanh(sin(c + d*x)/(sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d - (sqrt(2)*atanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d]
    @test_int [cos(c + d*x)^(-1/2)/(1 - cos(c + d*x))^(1/2), x, 2, -((sqrt(2)*atanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d)]
    @test_int [cos(c + d*x)^(-3/2)/(1 - cos(c + d*x))^(1/2), x, 3, -((sqrt(2)*atanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d) + (2*sin(c + d*x))/(d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(1 - cos(c + d*x))^(1/2), x, 5, -((sqrt(2)*atanh(sin(c + d*x)/(sqrt(2)*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))))/d) + (2*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*cos(c + d*x)^(3/2)) + (2*sin(c + d*x))/(3*d*sqrt(1 - cos(c + d*x))*sqrt(cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/3)*(a+a*cos(e+f*x))^(n/3)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^(4/3)*(a + a*cos(c + d*x))^(1/3), x, 3, (2^(5/6)*AppellF1(1/2, -(4/3), 1/6, 3/2, 1 - cos(c + d*x), (1/2)*(1 - cos(c + d*x)))*(a + a*cos(c + d*x))^(1/3)*sin(c + d*x))/(d*(1 + cos(c + d*x))^(5/6))]
    @test_int [cos(c + d*x)^(4/3)*(a + a*cos(c + d*x))^(2/3), x, 3, (2*2^(1/6)*AppellF1(1/2, -(4/3), -(1/6), 3/2, 1 - cos(c + d*x), (1/2)*(1 - cos(c + d*x)))*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(d*(1 + cos(c + d*x))^(7/6))]
    @test_int [cos(c + d*x)^(5/3)*(a + a*cos(c + d*x))^(2/3), x, 3, (2*2^(1/6)*AppellF1(1/2, -(5/3), -(1/6), 3/2, 1 - cos(c + d*x), (1/2)*(1 - cos(c + d*x)))*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(d*(1 + cos(c + d*x))^(7/6))]


    #= ::Subsubsection:: =#
    #=n<0=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^(m/2)*(a+a*cos(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [(a + a*cos(c + d*x))*sec(c + d*x)^(7/2), x, 9, -((6*a*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (6*a*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*cos(c + d*x))*sec(c + d*x)^(5/2), x, 8, (-2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*cos(c + d*x))*sec(c + d*x)^(3/2), x, 7, (-2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*cos(c + d*x))*sec(c + d*x)^(1/2), x, 6, (2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d]
    @test_int [(a + a*cos(c + d*x))/sec(c + d*x)^(1/2), x, 7, (2*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))/sec(c + d*x)^(3/2), x, 8, (6*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*a*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))/sec(c + d*x)^(5/2), x, 9, (6*a*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (10*a*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*a*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*a*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [(a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2), x, 9, -((16*a^2*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2), x, 8, (-4*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2), x, 5, (4*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*cos(c + d*x))^2*sec(c + d*x)^(1/2), x, 7, (4*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (8*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^2/sec(c + d*x)^(1/2), x, 8, (16*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (4*a^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^2/sec(c + d*x)^(3/2), x, 9, (12*a^2*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (8*a^2*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(7*d) + (2*a^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (4*a^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (8*a^2*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))]


    @test_int [(a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2), x, 17, -((28*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (52*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (28*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (52*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (6*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2), x, 15, -((36*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (36*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/d + (2*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2), x, 13, -((4*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (20*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (6*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2), x, 13, (4*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (20*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^3*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*cos(c + d*x))^3*sec(c + d*x)^(1/2), x, 13, (36*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*a^3*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*a^3*sin(c + d*x))/(d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^3/sec(c + d*x)^(1/2), x, 15, (28*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (52*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^3*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (6*a^3*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (52*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^3/sec(c + d*x)^(3/2), x, 17, (68*a^3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (44*a^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^3*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (6*a^3*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (68*a^3*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (44*a^3*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [(a + a*cos(c + d*x))^4*sec(c + d*x)^(9/2), x, 19, -((64*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (136*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (64*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (94*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (8*a^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a^4*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + a*cos(c + d*x))^4*sec(c + d*x)^(7/2), x, 17, -((56*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (32*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (66*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^4*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*cos(c + d*x))^4*sec(c + d*x)^(5/2), x, 16, (40*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^4*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (8*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^4*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*cos(c + d*x))^4*sec(c + d*x)^(3/2), x, 16, (56*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (32*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a^4*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (8*a^4*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + (2*a^4*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*cos(c + d*x))^4*sec(c + d*x)^(1/2), x, 17, (64*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (136*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*a^4*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (8*a^4*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (94*a^4*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^4/sec(c + d*x)^(1/2), x, 19, (152*a^4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (32*a^4*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(7*d) + (2*a^4*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (8*a^4*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (122*a^4*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (32*a^4*sin(c + d*x))/(7*d*sqrt(sec(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [sec(c + d*x)^(5/2)/(a + a*cos(c + d*x)), x, 9, (3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) - (3*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) + (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*cos(c + d*x)), x, 8, -((3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) - (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (3*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + a*cos(c + d*x)), x, 7, (sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))*sec(c + d*x)^(1/2)), x, 7, -((sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) + (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 7, (3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(d*(a + a*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))*sec(c + d*x)^(5/2)), x, 8, -((3*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) + (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (5*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))*sec(c + d*x)^(7/2)), x, 9, (21*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a*d) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (7*sin(c + d*x))/(5*a*d*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^2, x, 10, (7*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (10*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (7*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (10*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d) - (7*sec(c + d*x)^(5/2)*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^2, x, 9, -((4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (4*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) - (5*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^2, x, 8, (sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(1/2)), x, 5, (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(3/2)), x, 8, -((sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d*(1 + sec(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(5/2)), x, 8, (4*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) - (5*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*d*(1 + sec(c + d*x))) - (sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*(a + a*sec(c + d*x))^2)]
    @test_int [1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(7/2)), x, 9, -((7*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d)) + (10*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*d) + (10*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))) - (7*sin(c + d*x))/(3*a^2*d*sqrt(sec(c + d*x))*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2)]
    @test_int [1/((a + a*cos(c + d*x))^2*sec(c + d*x)^(9/2)), x, 10, (56*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*a^2*d) - (5*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (56*sin(c + d*x))/(15*a^2*d*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(a^2*d*sqrt(sec(c + d*x))) - (3*sin(c + d*x))/(a^2*d*sec(c + d*x)^(3/2)*(1 + sec(c + d*x))) - sin(c + d*x)/(3*d*sec(c + d*x)^(3/2)*(a + a*sec(c + d*x))^2)]


    @test_int [sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^3, x, 10, -((49*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) - (13*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(10*a^3*d) - (sec(c + d*x)^(7/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (8*sec(c + d*x)^(5/2)*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (13*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^3, x, 9, (9*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) - (9*sqrt(sec(c + d*x))*sin(c + d*x))/(10*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(1/2)), x, 9, (sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (4*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(3/2)), x, 9, -((sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) + (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(5/2)), x, 9, -((9*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*a*d*(a + a*sec(c + d*x))^2) + (sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(7/2)), x, 9, (49*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d) - (13*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(6*a^3*d) - (sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*(a + a*sec(c + d*x))^3) - (8*sqrt(sec(c + d*x))*sin(c + d*x))/(15*a*d*(a + a*sec(c + d*x))^2) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a^3 + a^3*sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^3*sec(c + d*x)^(9/2)), x, 10, -((119*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(10*a^3*d)) + (11*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(2*a^3*d) + (11*sin(c + d*x))/(2*a^3*d*sqrt(sec(c + d*x))) - sin(c + d*x)/(5*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^3) - (2*sin(c + d*x))/(3*a*d*sqrt(sec(c + d*x))*(a + a*sec(c + d*x))^2) - (119*sin(c + d*x))/(30*d*sqrt(sec(c + d*x))*(a^3 + a^3*sec(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^(m/2)*(a+a*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2), x, 5, (32*a*sqrt(sec(c + d*x))*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (16*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (12*a*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*a*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2), x, 4, (16*a*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (8*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2), x, 3, (4*a*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2), x, 2, (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(1/2), x, 3, (2*sqrt(a)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d]
    @test_int [sqrt(a + a*cos(c + d*x))/sec(c + d*x)^(1/2), x, 4, (sqrt(a)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]
    @test_int [sqrt(a + a*cos(c + d*x))/sec(c + d*x)^(3/2), x, 5, (3*sqrt(a)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (3*a*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]


    @test_int [(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2), x, 6, (208*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (104*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (26*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2), x, 5, (12*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (6*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2), x, 4, (10*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2), x, 5, (2*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2), x, 5, (3*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (a^2*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(3/2)/sec(c + d*x)^(1/2), x, 6, (7*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (a^2*sin(c + d*x))/(2*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (7*a^2*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(3/2)/sec(c + d*x)^(3/2), x, 7, (11*a^(3/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (a^2*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (11*a^2*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (11*a^2*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]


    @test_int [(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(11/2), x, 6, (1168*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (584*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*d*sqrt(a + a*cos(c + d*x))) + (146*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(105*d*sqrt(a + a*cos(c + d*x))) + (38*a^3*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)]
    @test_int [(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(9/2), x, 5, (92*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(21*d*sqrt(a + a*cos(c + d*x))) + (46*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d*sqrt(a + a*cos(c + d*x))) + (6*a^3*sec(c + d*x)^(5/2)*sin(c + d*x))/(7*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2), x, 4, (86*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (22*a^3*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2), x, 5, (2*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + (14*a^3*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2), x, 5, (5*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (a^3*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2), x, 5, (19*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(4*d) + (9*a^3*sin(c + d*x))/(4*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x))) + (a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(5/2)/sec(c + d*x)^(1/2), x, 6, (25*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(8*d) + (13*a^3*sin(c + d*x))/(12*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (25*a^3*sin(c + d*x))/(8*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]
    @test_int [(a + a*cos(c + d*x))^(5/2)/sec(c + d*x)^(3/2), x, 7, (163*a^(5/2)*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*d) + (17*a^3*sin(c + d*x))/(24*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(5/2)) + (a^2*sqrt(a + a*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (163*a^3*sin(c + d*x))/(96*d*sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)) + (163*a^3*sin(c + d*x))/(64*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [sec(c + d*x)^(7/2)/sqrt(1 + cos(c + d*x)), x, 7, -((sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(1 + cos(c + d*x))) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(1 + cos(c + d*x))) + (2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(1 + cos(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/sqrt(1 + cos(c + d*x)), x, 6, (sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(1 + cos(c + d*x))) + (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(1 + cos(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/sqrt(1 + cos(c + d*x)), x, 4, -((sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(1 + cos(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/sqrt(1 + cos(c + d*x)), x, 3, (sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d]
    @test_int [1/(sqrt(1 + cos(c + d*x))*sec(c + d*x)^(1/2)), x, 6, -((sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d) + (2*asin(sin(c + d*x)/sqrt(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d]
    @test_int [1/(sqrt(1 + cos(c + d*x))*sec(c + d*x)^(3/2)), x, 7, (sqrt(2)*asin(sin(c + d*x)/(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d - (asin(sin(c + d*x)/sqrt(1 + cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/d + sin(c + d*x)/(d*sqrt(1 + cos(c + d*x))*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(7/2)/sqrt(a + a*cos(c + d*x)), x, 7, -((sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (26*sqrt(sec(c + d*x))*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) - (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d*sqrt(a + a*cos(c + d*x))) + (2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(5/2)/sqrt(a + a*cos(c + d*x)), x, 6, (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x))) + (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/sqrt(a + a*cos(c + d*x)), x, 5, -((sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/sqrt(a + a*cos(c + d*x)), x, 3, (sqrt(2)*atan((sqrt(a)*sin(c + d*x)*sqrt(sec(c + d*x)))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d), (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)]
    @test_int [1/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(1/2)), x, 6, (2*atan((sqrt(a)*sin(c + d*x)*sqrt(sec(c + d*x)))/sqrt(a + a*cos(c + d*x))))/(sqrt(a)*d) - (sqrt(2)*atan((sqrt(a)*sin(c + d*x)*sqrt(sec(c + d*x)))/(sqrt(2)*sqrt(a + a*cos(c + d*x)))))/(sqrt(a)*d), (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) - (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)]
    @test_int [1/(sqrt(a + a*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 7, -((asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d)) + (sqrt(2)*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(sqrt(a)*d) + sin(c + d*x)/(d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(3/2), x, 7, (11*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - (19*sqrt(sec(c + d*x))*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (7*sec(c + d*x)^(3/2)*sin(c + d*x))/(6*a*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(3/2), x, 6, -((7*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(3/2)) + (5*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(3/2), x, 5, (3*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(1/2)), x, 5, (atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) + sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 7, (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d) - (5*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)), x, 8, -((3*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(3/2)*d)) + (9*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(2*sqrt(2)*a^(3/2)*d) - sin(c + d*x)/(2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (3*sin(c + d*x))/(2*a*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(5/2), x, 8, (163*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - (299*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (17*sec(c + d*x)^(3/2)*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (95*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a^2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(5/2), x, 7, -((75*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(4*d*(a + a*cos(c + d*x))^(5/2)) - (13*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)) + (49*sqrt(sec(c + d*x))*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(5/2), x, 6, (19*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - (9*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(1/2)), x, 6, (5*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) + sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) + sin(c + d*x)/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 6, (3*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) + (7*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)), x, 8, (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d) - (43*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) - (11*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2)), x, 9, -((5*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(5/2)*d)) + (115*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(16*sqrt(2)*a^(5/2)*d) - sin(c + d*x)/(4*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) - (15*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (35*sin(c + d*x))/(16*a^2*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)/(a + a*cos(c + d*x))^(7/2), x, 9, (1015*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - (629*sqrt(sec(c + d*x))*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x))) - (sec(c + d*x)^(3/2)*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (23*sec(c + d*x)^(3/2)*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - (109*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)) + (193*sec(c + d*x)^(3/2)*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + a*cos(c + d*x))^(7/2), x, 8, -((363*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d)) - (sqrt(sec(c + d*x))*sin(c + d*x))/(6*d*(a + a*cos(c + d*x))^(7/2)) - (19*sqrt(sec(c + d*x))*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)) - (199*sqrt(sec(c + d*x))*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)) + (691*sqrt(sec(c + d*x))*sin(c + d*x))/(192*a^3*d*sqrt(a + a*cos(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + a*cos(c + d*x))^(7/2), x, 7, (63*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) - (5*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - (103*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(1/2)), x, 7, (13*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) + sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) + sin(c + d*x)/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) - (5*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(3/2)), x, 7, (7*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) + (3*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) + (17*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)), x, 7, (5*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(3/2)) - (13*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) + (67*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(7/2)), x, 9, (2*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(7/2)*d) - (177*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(5/2)) - (17*sin(c + d*x))/(48*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)) - (49*sin(c + d*x))/(64*a^2*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(9/2)), x, 10, -((7*asin((sqrt(a)*sin(c + d*x))/sqrt(a + a*cos(c + d*x)))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(a^(7/2)*d)) + (637*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(64*sqrt(2)*a^(7/2)*d) - sin(c + d*x)/(6*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(7/2)) - (7*sin(c + d*x))/(16*a*d*(a + a*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)) - (259*sin(c + d*x))/(192*a^2*d*(a + a*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)) + (189*sin(c + d*x))/(64*a^3*d*sqrt(a + a*cos(c + d*x))*sqrt(sec(c + d*x)))]


    @test_int [1/((a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(5/2)), x, 8, (45*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(1024*sqrt(2)*a^(9/2)*d) - sin(c + d*x)/(8*d*(a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(3/2)) - (5*sin(c + d*x))/(32*a*d*(a + a*cos(c + d*x))^(7/2)*sqrt(sec(c + d*x))) + (33*sin(c + d*x))/(256*a^2*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) + (73*sin(c + d*x))/(1024*a^3*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]
    @test_int [1/((a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(7/2)), x, 8, (35*atan((sqrt(a)*sin(c + d*x))/(sqrt(2)*sqrt(cos(c + d*x))*sqrt(a + a*cos(c + d*x))))*sqrt(cos(c + d*x))*sqrt(sec(c + d*x)))/(1024*sqrt(2)*a^(9/2)*d) - sin(c + d*x)/(8*d*(a + a*cos(c + d*x))^(9/2)*sec(c + d*x)^(5/2)) - (19*sin(c + d*x))/(96*a*d*(a + a*cos(c + d*x))^(7/2)*sec(c + d*x)^(3/2)) - (187*sin(c + d*x))/(768*a^2*d*(a + a*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))) + (853*sin(c + d*x))/(3072*a^3*d*(a + a*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^(m/4) / (a+a*cos(e+f*x))^(n/2)=#


    @test_int [sec(c + d*x)^(5/4)*(a + a*cos(c + d*x))^(3/2), x, 3, (4*a^2*sec(c + d*x)^(1/4)*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^m*(a+a*cos(e+f*x))^n*with*m*and/or*n*symbolic=#


    @test_int [cos(c + d*x)^m*(a + a*cos(c + d*x))^4, x, 7, If(13>=8, (a^4*(55 + 29*m + 4*m^2)*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m)*(3 + m)*(4 + m)) + (cos(c + d*x)^(1 + m)*(a^2 + a^2*cos(c + d*x))^2*sin(c + d*x))/(d*(4 + m)) + (2*(5 + m)*cos(c + d*x)^(1 + m)*(a^4 + a^4*cos(c + d*x))*sin(c + d*x))/(d*(3 + m)*(4 + m)) - (a^4*(35 + 40*m + 8*m^2)*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*(2 + m)*(4 + m)*sqrt(sin(c + d*x)^2)) - (4*a^4*(5 + 2*m)*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*(3 + m)*sqrt(sin(c + d*x)^2)), (a^4*(55 + 29*m + 4*m^2)*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(4 + m)*(6 + 5*m + m^2)) + (cos(c + d*x)^(1 + m)*(a^2 + a^2*cos(c + d*x))^2*sin(c + d*x))/(d*(4 + m)) + (2*(5 + m)*cos(c + d*x)^(1 + m)*(a^4 + a^4*cos(c + d*x))*sin(c + d*x))/(d*(3 + m)*(4 + m)) - (a^4*(35 + 40*m + 8*m^2)*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(8 + 14*m + 7*m^2 + m^3)*sqrt(sin(c + d*x)^2)) - (4*a^4*(5 + 2*m)*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*(3 + m)*sqrt(sin(c + d*x)^2)))]
    @test_int [cos(c + d*x)^m*(a + a*cos(c + d*x))^3, x, 6, (a^3*(7 + 2*m)*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m)*(3 + m)) + (cos(c + d*x)^(1 + m)*(a^3 + a^3*cos(c + d*x))*sin(c + d*x))/(d*(3 + m)) - (a^3*(5 + 4*m)*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*(2 + m)*sqrt(sin(c + d*x)^2)) - (a^3*(11 + 4*m)*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*(3 + m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m*(a + a*cos(c + d*x))^2, x, 4, (a^2*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m)) - (a^2*(3 + 2*m)*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*(2 + m)*sqrt(sin(c + d*x)^2)) - (2*a^2*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m*(a + a*cos(c + d*x))^1, x, 3, -((a*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*sqrt(sin(c + d*x)^2))) - (a*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m/(a + a*cos(c + d*x))^1, x, 4, (cos(c + d*x)^m*sin(c + d*x))/(d*(a + a*cos(c + d*x))) - (cos(c + d*x)^m*HypergeometricFunctions._₂F₁(1/2, m/2, (2 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(a*d*sqrt(sin(c + d*x)^2)) + (m*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(a*d*(1 + m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m/(a + a*cos(c + d*x))^2, x, 5, -((2*(1 - m)*cos(c + d*x)^(1 + m)*sin(c + d*x))/(3*a^2*d*(1 + cos(c + d*x)))) - (cos(c + d*x)^(1 + m)*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^2) + ((1 - 2*m)*m*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(3*a^2*d*(1 + m)*sqrt(sin(c + d*x)^2)) - (2*(1 - m)*(1 + m)*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(3*a^2*d*(2 + m)*sqrt(sin(c + d*x)^2))]


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(d*cos(e+f*x))^n*(a+b*cos(e+f*x))^m=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^m*(a+b*cos(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^7*(a + b*cos(c + d*x)), x, 8, (35*b*x)/128 + (a*sin(c + d*x))/d + (35*b*cos(c + d*x)*sin(c + d*x))/(128*d) + (35*b*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (7*b*cos(c + d*x)^5*sin(c + d*x))/(48*d) + (b*cos(c + d*x)^7*sin(c + d*x))/(8*d) - (a*sin(c + d*x)^3)/d + (3*a*sin(c + d*x)^5)/(5*d) - (a*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^6*(a + b*cos(c + d*x)), x, 7, (5*a*x)/16 + (b*sin(c + d*x))/d + (5*a*cos(c + d*x)*sin(c + d*x))/(16*d) + (5*a*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (b*sin(c + d*x)^3)/d + (3*b*sin(c + d*x)^5)/(5*d) - (b*sin(c + d*x)^7)/(7*d)]
    @test_int [cos(c + d*x)^5*(a + b*cos(c + d*x)), x, 7, (5*b*x)/16 + (a*sin(c + d*x))/d + (5*b*cos(c + d*x)*sin(c + d*x))/(16*d) + (5*b*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (b*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (2*a*sin(c + d*x)^3)/(3*d) + (a*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^4*(a + b*cos(c + d*x)), x, 6, (3*a*x)/8 + (b*sin(c + d*x))/d + (3*a*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (2*b*sin(c + d*x)^3)/(3*d) + (b*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^3*(a + b*cos(c + d*x)), x, 6, (3*b*x)/8 + (a*sin(c + d*x))/d + (3*b*cos(c + d*x)*sin(c + d*x))/(8*d) + (b*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (a*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^2*(a + b*cos(c + d*x)), x, 5, (a*x)/2 + (b*sin(c + d*x))/d + (a*cos(c + d*x)*sin(c + d*x))/(2*d) - (b*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^1*(a + b*cos(c + d*x)), x, 1, (b*x)/2 + (a*sin(c + d*x))/d + (b*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^0*(a + b*cos(c + d*x)), x, 2, a*x + (b*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^1*(a + b*cos(c + d*x)), x, 2, b*x + (a*atanh(sin(c + d*x)))/d]
    @test_int [sec(c + d*x)^2*(a + b*cos(c + d*x)), x, 4, (b*atanh(sin(c + d*x)))/d + (a*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + b*cos(c + d*x)), x, 5, (a*atanh(sin(c + d*x)))/(2*d) + (b*tan(c + d*x))/d + (a*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + b*cos(c + d*x)), x, 5, (b*atanh(sin(c + d*x)))/(2*d) + (a*tan(c + d*x))/d + (b*sec(c + d*x)*tan(c + d*x))/(2*d) + (a*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^5*(a + b*cos(c + d*x)), x, 6, (3*a*atanh(sin(c + d*x)))/(8*d) + (b*tan(c + d*x))/d + (3*a*sec(c + d*x)*tan(c + d*x))/(8*d) + (a*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (b*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^6*(a + b*cos(c + d*x)), x, 6, (3*b*atanh(sin(c + d*x)))/(8*d) + (a*tan(c + d*x))/d + (3*b*sec(c + d*x)*tan(c + d*x))/(8*d) + (b*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (2*a*tan(c + d*x)^3)/(3*d) + (a*tan(c + d*x)^5)/(5*d)]


    @test_int [cos(c + d*x)^4*(a + b*cos(c + d*x))^2, x, 7, (1/16)*(6*a^2 + 5*b^2)*x + (2*a*b*sin(c + d*x))/d + ((6*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + ((6*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (b^2*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (4*a*b*sin(c + d*x)^3)/(3*d) + (2*a*b*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^3*(a + b*cos(c + d*x))^2, x, 7, (3*a*b*x)/4 + ((a^2 + b^2)*sin(c + d*x))/d + (3*a*b*cos(c + d*x)*sin(c + d*x))/(4*d) + (a*b*cos(c + d*x)^3*sin(c + d*x))/(2*d) - ((a^2 + 2*b^2)*sin(c + d*x)^3)/(3*d) + (b^2*sin(c + d*x)^5)/(5*d)]
    @test_int [cos(c + d*x)^2*(a + b*cos(c + d*x))^2, x, 6, (1/8)*(4*a^2 + 3*b^2)*x + (2*a*b*sin(c + d*x))/d + ((4*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (b^2*cos(c + d*x)^3*sin(c + d*x))/(4*d) - (2*a*b*sin(c + d*x)^3)/(3*d)]
    @test_int [cos(c + d*x)^1*(a + b*cos(c + d*x))^2, x, 2, a*b*x + (2*(a^2 + b^2)*sin(c + d*x))/(3*d) + (a*b*cos(c + d*x)*sin(c + d*x))/(3*d) + ((a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^0*(a + b*cos(c + d*x))^2, x, 1, (1/2)*(2*a^2 + b^2)*x + (2*a*b*sin(c + d*x))/d + (b^2*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^1*(a + b*cos(c + d*x))^2, x, 3, 2*a*b*x + (a^2*atanh(sin(c + d*x)))/d + (b^2*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^2*(a + b*cos(c + d*x))^2, x, 4, b^2*x + (2*a*b*atanh(sin(c + d*x)))/d + (a^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + b*cos(c + d*x))^2, x, 5, ((a^2 + 2*b^2)*atanh(sin(c + d*x)))/(2*d) + (2*a*b*tan(c + d*x))/d + (a^2*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + b*cos(c + d*x))^2, x, 6, (a*b*atanh(sin(c + d*x)))/d + ((2*a^2 + 3*b^2)*tan(c + d*x))/(3*d) + (a*b*sec(c + d*x)*tan(c + d*x))/d + (a^2*sec(c + d*x)^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^5*(a + b*cos(c + d*x))^2, x, 6, ((3*a^2 + 4*b^2)*atanh(sin(c + d*x)))/(8*d) + (2*a*b*tan(c + d*x))/d + ((3*a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (a^2*sec(c + d*x)^3*tan(c + d*x))/(4*d) + (2*a*b*tan(c + d*x)^3)/(3*d)]
    @test_int [sec(c + d*x)^6*(a + b*cos(c + d*x))^2, x, 7, (3*a*b*atanh(sin(c + d*x)))/(4*d) + ((4*a^2 + 5*b^2)*tan(c + d*x))/(5*d) + (3*a*b*sec(c + d*x)*tan(c + d*x))/(4*d) + (a*b*sec(c + d*x)^3*tan(c + d*x))/(2*d) + (a^2*sec(c + d*x)^4*tan(c + d*x))/(5*d) + ((4*a^2 + 5*b^2)*tan(c + d*x)^3)/(15*d)]


    @test_int [cos(c + d*x)^3*(a + b*cos(c + d*x))^3, x, 8, (9/8)*a^2*b*x + (5*b^3*x)/16 + (a*(a^2 + 3*b^2)*sin(c + d*x))/d + (b*(18*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + (b*(18*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (b^3*cos(c + d*x)^5*sin(c + d*x))/(6*d) - (a*(a^2 + 6*b^2)*sin(c + d*x)^3)/(3*d) + (3*a*b^2*sin(c + d*x)^5)/(5*d), (1/16)*b*(18*a^2 + 5*b^2)*x + (a*(5*a^2 + 12*b^2)*sin(c + d*x))/(5*d) + (b*(18*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(16*d) + (b*(18*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (13*a*b^2*cos(c + d*x)^4*sin(c + d*x))/(30*d) + (b^2*cos(c + d*x)^4*(a + b*cos(c + d*x))*sin(c + d*x))/(6*d) - (a*(5*a^2 + 12*b^2)*sin(c + d*x)^3)/(15*d)]
    @test_int [cos(c + d*x)^2*(a + b*cos(c + d*x))^3, x, 4, (1/8)*a*(4*a^2 + 9*b^2)*x - ((3*a^4 - 52*a^2*b^2 - 16*b^4)*sin(c + d*x))/(30*b*d) - (a*(6*a^2 - 71*b^2)*cos(c + d*x)*sin(c + d*x))/(120*d) - ((3*a^2 - 16*b^2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(60*b*d) - (a*(a + b*cos(c + d*x))^3*sin(c + d*x))/(20*b*d) + ((a + b*cos(c + d*x))^4*sin(c + d*x))/(5*b*d)]
    @test_int [cos(c + d*x)^1*(a + b*cos(c + d*x))^3, x, 3, (3/8)*b*(4*a^2 + b^2)*x + (a*(a^2 + 4*b^2)*sin(c + d*x))/(2*d) + (b*(2*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(8*d) + (a*(a + b*cos(c + d*x))^2*sin(c + d*x))/(4*d) + ((a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)]
    @test_int [cos(c + d*x)^0*(a + b*cos(c + d*x))^3, x, 2, a^3*x + (3/2)*a*b^2*x + (b*(3*a^2 + b^2)*sin(c + d*x))/d + (3*a*b^2*cos(c + d*x)*sin(c + d*x))/(2*d) - (b^3*sin(c + d*x)^3)/(3*d), (1/2)*a*(2*a^2 + 3*b^2)*x + (2*b*(4*a^2 + b^2)*sin(c + d*x))/(3*d) + (5*a*b^2*cos(c + d*x)*sin(c + d*x))/(6*d) + (b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^1*(a + b*cos(c + d*x))^3, x, 4, (1/2)*b*(6*a^2 + b^2)*x + (a^3*atanh(sin(c + d*x)))/d + (5*a*b^2*sin(c + d*x))/(2*d) + (b^2*(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^2*(a + b*cos(c + d*x))^3, x, 4, 3*a*b^2*x + (3*a^2*b*atanh(sin(c + d*x)))/d - (b*(a^2 - b^2)*sin(c + d*x))/d + (a^2*(a + b*cos(c + d*x))*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + b*cos(c + d*x))^3, x, 4, b^3*x + (a*(a^2 + 6*b^2)*atanh(sin(c + d*x)))/(2*d) + (5*a^2*b*tan(c + d*x))/(2*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + b*cos(c + d*x))^3, x, 6, (b*(3*a^2 + 2*b^2)*atanh(sin(c + d*x)))/(2*d) + (a*(2*a^2 + 9*b^2)*tan(c + d*x))/(3*d) + (7*a^2*b*sec(c + d*x)*tan(c + d*x))/(6*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^5*(a + b*cos(c + d*x))^3, x, 7, (3*a*(a^2 + 4*b^2)*atanh(sin(c + d*x)))/(8*d) + (b*(2*a^2 + b^2)*tan(c + d*x))/d + (3*a*(a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (3*a^2*b*sec(c + d*x)^2*tan(c + d*x))/(4*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)^3*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^6*(a + b*cos(c + d*x))^3, x, 7, (b*(9*a^2 + 4*b^2)*atanh(sin(c + d*x)))/(8*d) + (a*(4*a^2 + 15*b^2)*tan(c + d*x))/(5*d) + (b*(9*a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (11*a^2*b*sec(c + d*x)^3*tan(c + d*x))/(20*d) + (a^2*(a + b*cos(c + d*x))*sec(c + d*x)^4*tan(c + d*x))/(5*d) + (a*(4*a^2 + 15*b^2)*tan(c + d*x)^3)/(15*d)]


    @test_int [cos(c + d*x)^3*(a + b*cos(c + d*x))^4, x, 9, (1/4)*a*b*(6*a^2 + 5*b^2)*x + ((35*a^4 + 168*a^2*b^2 + 24*b^4)*sin(c + d*x))/(35*d) + (a*b*(6*a^2 + 5*b^2)*cos(c + d*x)*sin(c + d*x))/(4*d) + (a*b*(6*a^2 + 5*b^2)*cos(c + d*x)^3*sin(c + d*x))/(6*d) + (b^2*(37*a^2 + 6*b^2)*cos(c + d*x)^4*sin(c + d*x))/(35*d) + (8*a*b^3*cos(c + d*x)^5*sin(c + d*x))/(21*d) + (b^2*cos(c + d*x)^4*(a + b*cos(c + d*x))^2*sin(c + d*x))/(7*d) - ((35*a^4 + 168*a^2*b^2 + 24*b^4)*sin(c + d*x)^3)/(105*d)]
    @test_int [cos(c + d*x)^2*(a + b*cos(c + d*x))^4, x, 5, (1/16)*(8*a^4 + 36*a^2*b^2 + 5*b^4)*x - (a*(4*a^4 - 121*a^2*b^2 - 128*b^4)*sin(c + d*x))/(60*b*d) - ((8*a^4 - 178*a^2*b^2 - 75*b^4)*cos(c + d*x)*sin(c + d*x))/(240*d) - (a*(4*a^2 - 53*b^2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(120*b*d) - ((4*a^2 - 25*b^2)*(a + b*cos(c + d*x))^3*sin(c + d*x))/(120*b*d) - (a*(a + b*cos(c + d*x))^4*sin(c + d*x))/(30*b*d) + ((a + b*cos(c + d*x))^5*sin(c + d*x))/(6*b*d)]
    @test_int [cos(c + d*x)^1*(a + b*cos(c + d*x))^4, x, 4, (1/2)*a*b*(4*a^2 + 3*b^2)*x + (2*(3*a^4 + 28*a^2*b^2 + 4*b^4)*sin(c + d*x))/(15*d) + (a*b*(6*a^2 + 29*b^2)*cos(c + d*x)*sin(c + d*x))/(30*d) + ((3*a^2 + 4*b^2)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(15*d) + (a*(a + b*cos(c + d*x))^3*sin(c + d*x))/(5*d) + ((a + b*cos(c + d*x))^4*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^0*(a + b*cos(c + d*x))^4, x, 3, (1/8)*(8*a^4 + 24*a^2*b^2 + 3*b^4)*x + (a*b*(19*a^2 + 16*b^2)*sin(c + d*x))/(6*d) + (b^2*(26*a^2 + 9*b^2)*cos(c + d*x)*sin(c + d*x))/(24*d) + (7*a*b*(a + b*cos(c + d*x))^2*sin(c + d*x))/(12*d) + (b*(a + b*cos(c + d*x))^3*sin(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^1*(a + b*cos(c + d*x))^4, x, 5, 2*a*b*(2*a^2 + b^2)*x + (a^4*atanh(sin(c + d*x)))/d + (b^2*(17*a^2 + 2*b^2)*sin(c + d*x))/(3*d) + (4*a*b^3*cos(c + d*x)*sin(c + d*x))/(3*d) + (b^2*(a + b*cos(c + d*x))^2*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^2*(a + b*cos(c + d*x))^4, x, 5, (1/2)*b^2*(12*a^2 + b^2)*x + (4*a^3*b*atanh(sin(c + d*x)))/d - (2*a*b*(a^2 - 2*b^2)*sin(c + d*x))/d - (b^2*(2*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(2*d) + (a^2*(a + b*cos(c + d*x))^2*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*(a + b*cos(c + d*x))^4, x, 5, 4*a*b^3*x + (a^2*(a^2 + 12*b^2)*atanh(sin(c + d*x)))/(2*d) - (b^2*(a^2 - 2*b^2)*sin(c + d*x))/(2*d) + (3*a^3*b*tan(c + d*x))/d + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [sec(c + d*x)^4*(a + b*cos(c + d*x))^4, x, 5, b^4*x + (2*a*b*(a^2 + 2*b^2)*atanh(sin(c + d*x)))/d + (a^2*(2*a^2 + 17*b^2)*tan(c + d*x))/(3*d) + (4*a^3*b*sec(c + d*x)*tan(c + d*x))/(3*d) + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)^2*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^5*(a + b*cos(c + d*x))^4, x, 7, ((3*a^4 + 24*a^2*b^2 + 8*b^4)*atanh(sin(c + d*x)))/(8*d) + (4*a*b*(2*a^2 + 3*b^2)*tan(c + d*x))/(3*d) + (a^2*(3*a^2 + 22*b^2)*sec(c + d*x)*tan(c + d*x))/(8*d) + (5*a^3*b*sec(c + d*x)^2*tan(c + d*x))/(6*d) + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)^3*tan(c + d*x))/(4*d)]
    @test_int [sec(c + d*x)^6*(a + b*cos(c + d*x))^4, x, 8, (a*b*(3*a^2 + 4*b^2)*atanh(sin(c + d*x)))/(2*d) + ((8*a^4 + 60*a^2*b^2 + 15*b^4)*tan(c + d*x))/(15*d) + (a*b*(3*a^2 + 4*b^2)*sec(c + d*x)*tan(c + d*x))/(2*d) + (a^2*(4*a^2 + 27*b^2)*sec(c + d*x)^2*tan(c + d*x))/(15*d) + (3*a^3*b*sec(c + d*x)^3*tan(c + d*x))/(5*d) + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)^4*tan(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^7*(a + b*cos(c + d*x))^4, x, 8, ((5*a^4 + 36*a^2*b^2 + 8*b^4)*atanh(sin(c + d*x)))/(16*d) + (4*a*b*(4*a^2 + 5*b^2)*tan(c + d*x))/(5*d) + ((5*a^4 + 36*a^2*b^2 + 8*b^4)*sec(c + d*x)*tan(c + d*x))/(16*d) + (a^2*(5*a^2 + 32*b^2)*sec(c + d*x)^3*tan(c + d*x))/(24*d) + (7*a^3*b*sec(c + d*x)^4*tan(c + d*x))/(15*d) + (a^2*(a + b*cos(c + d*x))^2*sec(c + d*x)^5*tan(c + d*x))/(6*d) + (4*a*b*(4*a^2 + 5*b^2)*tan(c + d*x)^3)/(15*d)]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^5/(a + b*cos(c + d*x)), x, 7, ((8*a^4 + 4*a^2*b^2 + 3*b^4)*x)/(8*b^5) - (2*a^5*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(sqrt(a - b)*b^5*sqrt(a + b)*d) - (a*(3*a^2 + 2*b^2)*sin(c + d*x))/(3*b^4*d) + ((4*a^2 + 3*b^2)*cos(c + d*x)*sin(c + d*x))/(8*b^3*d) - (a*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*d) + (cos(c + d*x)^3*sin(c + d*x))/(4*b*d)]
    @test_int [cos(c + d*x)^4/(a + b*cos(c + d*x)), x, 6, -((a*(2*a^2 + b^2)*x)/(2*b^4)) + (2*a^4*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(sqrt(a - b)*b^4*sqrt(a + b)*d) + ((3*a^2 + 2*b^2)*sin(c + d*x))/(3*b^3*d) - (a*cos(c + d*x)*sin(c + d*x))/(2*b^2*d) + (cos(c + d*x)^2*sin(c + d*x))/(3*b*d)]
    @test_int [cos(c + d*x)^3/(a + b*cos(c + d*x)), x, 5, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(sqrt(a - b)*b^3*sqrt(a + b)*d) - (a*sin(c + d*x))/(b^2*d) + (cos(c + d*x)*sin(c + d*x))/(2*b*d)]
    @test_int [cos(c + d*x)^2/(a + b*cos(c + d*x)), x, 5, -((a*x)/b^2) + (2*a^2*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(sqrt(a - b)*b^2*sqrt(a + b)*d) + sin(c + d*x)/(b*d)]
    @test_int [cos(c + d*x)^1/(a + b*cos(c + d*x)), x, 3, x/b - (2*a*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(sqrt(a - b)*b*sqrt(a + b)*d)]
    @test_int [cos(c + d*x)^0/(a + b*cos(c + d*x)), x, 2, (2*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(sqrt(a - b)*sqrt(a + b)*d)]
    @test_int [sec(c + d*x)^1/(a + b*cos(c + d*x)), x, 4, -((2*b*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a*sqrt(a - b)*sqrt(a + b)*d)) + atanh(sin(c + d*x))/(a*d)]
    @test_int [sec(c + d*x)^2/(a + b*cos(c + d*x)), x, 6, (2*b^2*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^2*sqrt(a - b)*sqrt(a + b)*d) - (b*atanh(sin(c + d*x)))/(a^2*d) + tan(c + d*x)/(a*d)]
    @test_int [sec(c + d*x)^3/(a + b*cos(c + d*x)), x, 6, -((2*b^3*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^3*sqrt(a - b)*sqrt(a + b)*d)) + ((a^2 + 2*b^2)*atanh(sin(c + d*x)))/(2*a^3*d) - (b*tan(c + d*x))/(a^2*d) + (sec(c + d*x)*tan(c + d*x))/(2*a*d)]
    @test_int [sec(c + d*x)^4/(a + b*cos(c + d*x)), x, 7, (2*b^4*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*sqrt(a - b)*sqrt(a + b)*d) - (b*(a^2 + 2*b^2)*atanh(sin(c + d*x)))/(2*a^4*d) + ((2*a^2 + 3*b^2)*tan(c + d*x))/(3*a^3*d) - (b*sec(c + d*x)*tan(c + d*x))/(2*a^2*d) + (sec(c + d*x)^2*tan(c + d*x))/(3*a*d)]


    @test_int [cos(c + d*x)^5/(a + b*cos(c + d*x))^2, x, 7, -((a*(4*a^2 + b^2)*x)/b^5) + (2*a^4*(4*a^2 - 5*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) + ((12*a^4 - 7*a^2*b^2 - 2*b^4)*sin(c + d*x))/(3*b^4*(a^2 - b^2)*d) - (a*(2*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((4*a^2 - b^2)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^4/(a + b*cos(c + d*x))^2, x, 6, ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*atanh(((a - b)*sin(c + d*x)/(1 + cos(c + d*x)))/sqrt(-a^2 + b^2)))/(b^4*(-a^2 + b^2)^(3/2)*d) - (2*a*sin(c + d*x))/(b^3*d) + (cos(c + d*x)*sin(c + d*x))/(2*b^2*d) - (a^4*sin(c + d*x))/(b^3*(a^2 - b^2)*d*(a + b*cos(c + d*x))), ((6*a^2 + b^2)*x)/(2*b^4) - (2*a^3*(3*a^2 - 4*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(3*a^2 - 2*b^2)*sin(c + d*x))/(b^3*(a^2 - b^2)*d) + ((3*a^2 - b^2)*cos(c + d*x)*sin(c + d*x))/(2*b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + b*cos(c + d*x))^2, x, 5, -((2*a*x)/b^3) + (2*a^2*(2*a^2 - 3*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + ((2*a^2 - b^2)*sin(c + d*x))/(b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*cos(c + d*x))^2, x, 4, x/b^2 - (2*a*(a^2 - 2*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - (a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*cos(c + d*x))^2, x, 4, -((2*b*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*d)) + (a*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + b*cos(c + d*x))^2, x, 4, (2*a*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*d) - (b*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*cos(c + d*x))^2, x, 5, -((2*b*(2*a^2 - b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d)) + atanh(sin(c + d*x))/(a^2*d) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*cos(c + d*x))^2, x, 6, (2*b^2*(3*a^2 - 2*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - (2*b*atanh(sin(c + d*x)))/(a^3*d) + ((a^2 - 2*b^2)*tan(c + d*x))/(a^2*(a^2 - b^2)*d) + (b^2*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*cos(c + d*x))^2, x, 7, -((2*b^3*(4*a^2 - 3*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d)) + ((a^2 + 6*b^2)*atanh(sin(c + d*x)))/(2*a^4*d) - (b*(2*a^2 - 3*b^2)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2 - 3*b^2)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4/(a + b*cos(c + d*x))^2, x, 8, (2*b^4*(5*a^2 - 4*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - (b*(a^2 + 4*b^2)*atanh(sin(c + d*x)))/(a^5*d) + ((2*a^4 + 7*a^2*b^2 - 12*b^4)*tan(c + d*x))/(3*a^4*(a^2 - b^2)*d) - (b*(a^2 - 2*b^2)*sec(c + d*x)*tan(c + d*x))/(a^3*(a^2 - b^2)*d) + ((a^2 - 4*b^2)*sec(c + d*x)^2*tan(c + d*x))/(3*a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)^2*tan(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^5/(a + b*cos(c + d*x))^3, x, 7, ((12*a^2 + b^2)*x)/(2*b^5) - (a^3*(12*a^4 - 29*a^2*b^2 + 20*b^4)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (3*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^2*d) + ((6*a^4 - 10*a^2*b^2 + b^4)*cos(c + d*x)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(4*a^2 - 7*b^2)*cos(c + d*x)^2*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^4/(a + b*cos(c + d*x))^3, x, 6, -((3*a*x)/b^4) + (3*a^2*(2*a^4 - 5*a^2*b^2 + 4*b^4)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((3*a^2 - 2*b^2)*sin(c + d*x))/(2*b^3*(a^2 - b^2)*d) - (a^2*cos(c + d*x)^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (3*a^3*(a^2 - 2*b^2)*sin(c + d*x))/(2*b^3*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + b*cos(c + d*x))^3, x, 5, x/b^3 - (a*(2*a^4 - 5*a^2*b^2 + 6*b^4)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*cos(c + d*x)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(2*a^2 - 5*b^2)*sin(c + d*x))/(2*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*cos(c + d*x))^3, x, 5, ((a^2 + 2*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d) - (a^2*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (a*(a^2 - 4*b^2)*sin(c + d*x))/(2*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*cos(c + d*x))^3, x, 5, -((3*a*b*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2 + 2*b^2)*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + b*cos(c + d*x))^3, x, 5, ((2*a^2 + b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*d) - (b*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (3*a*b*sin(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*cos(c + d*x))^3, x, 6, -((b*(6*a^4 - 5*a^2*b^2 + 2*b^4)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d)) + atanh(sin(c + d*x))/(a^3*d) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b^2*(5*a^2 - 2*b^2)*sin(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*cos(c + d*x))^3, x, 7, (3*b^2*(4*a^4 - 5*a^2*b^2 + 2*b^4)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - (3*b*atanh(sin(c + d*x)))/(a^4*d) + ((2*a^4 - 11*a^2*b^2 + 6*b^4)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b^2*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (3*b^2*(2*a^2 - b^2)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*cos(c + d*x))^3, x, 8, -((b^3*(20*a^4 - 29*a^2*b^2 + 12*b^4)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d)) + ((a^2 + 12*b^2)*atanh(sin(c + d*x)))/(2*a^5*d) - (3*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*tan(c + d*x))/(2*a^4*(a^2 - b^2)^2*d) + ((a^4 - 10*a^2*b^2 + 6*b^4)*sec(c + d*x)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^2*d) + (b^2*sec(c + d*x)*tan(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (b^2*(7*a^2 - 4*b^2)*sec(c + d*x)*tan(c + d*x))/(2*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^5/(a + b*cos(c + d*x))^4, x, 7, -((4*a*x)/b^5) + (a^2*(8*a^6 - 28*a^4*b^2 + 35*a^2*b^4 - 20*b^6)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) + ((12*a^4 - 23*a^2*b^2 + 6*b^4)*sin(c + d*x))/(6*b^4*(a^2 - b^2)^2*d) - (a^2*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a^2*(4*a^2 - 9*b^2)*cos(c + d*x)^2*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a^3*(4*a^4 - 11*a^2*b^2 + 12*b^4)*sin(c + d*x))/(2*b^4*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^4/(a + b*cos(c + d*x))^4, x, 6, x/b^4 - (a*(2*a^6 - 7*a^4*b^2 + 8*a^2*b^4 - 8*b^6)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - (a^2*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (a^3*(3*a^2 - 8*b^2)*sin(c + d*x))/(6*b^3*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - (a^2*(9*a^4 - 28*a^2*b^2 + 34*b^4)*sin(c + d*x))/(6*b^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^3/(a + b*cos(c + d*x))^4, x, 6, -((b*(3*a^2 + 2*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d)) - (a^2*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (a^2*(2*a^2 - 7*b^2)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a*(2*a^4 - 5*a^2*b^2 + 18*b^4)*sin(c + d*x))/(6*b^2*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*cos(c + d*x))^4, x, 6, (a*(a^2 + 4*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d) - (a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (a*(a^2 - 6*b^2)*sin(c + d*x))/(6*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + ((a^4 - 10*a^2*b^2 - 6*b^4)*sin(c + d*x))/(6*b*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*cos(c + d*x))^4, x, 6, -((b*(4*a^2 + b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + ((2*a^2 + 3*b^2)*sin(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (a*(2*a^2 + 13*b^2)*sin(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + b*cos(c + d*x))^4, x, 6, (a*(2*a^2 + 3*b^2)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/((a - b)^(7/2)*(a + b)^(7/2)*d) - (b*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) - (5*a*b*sin(c + d*x))/(6*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) - (b*(11*a^2 + 4*b^2)*sin(c + d*x))/(6*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*cos(c + d*x))^4, x, 7, -((b*(8*a^6 - 8*a^4*b^2 + 7*a^2*b^4 - 2*b^6)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d)) + atanh(sin(c + d*x))/(a^4*d) + (b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (b^2*(8*a^2 - 3*b^2)*sin(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (b^2*(26*a^4 - 17*a^2*b^2 + 6*b^4)*sin(c + d*x))/(6*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*cos(c + d*x))^4, x, 8, (b^2*(20*a^6 - 35*a^4*b^2 + 28*a^2*b^4 - 8*b^6)*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) - (4*b*atanh(sin(c + d*x)))/(a^5*d) + ((6*a^6 - 65*a^4*b^2 + 68*a^2*b^4 - 24*b^6)*tan(c + d*x))/(6*a^4*(a^2 - b^2)^3*d) + (b^2*tan(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^3) + (b^2*(9*a^2 - 4*b^2)*tan(c + d*x))/(6*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^2) + (b^2*(12*a^4 - 11*a^2*b^2 + 4*b^4)*tan(c + d*x))/(2*a^3*(a^2 - b^2)^3*d*(a + b*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^m*(a+b*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^3*sqrt(a + b*cos(c + d*x)), x, 8, (2*a*(8*a^2 + 19*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*(8*a^4 + 17*a^2*b^2 - 25*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(8*a^2 + 25*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b^2*d) - (8*a*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*b^2*d) + (2*cos(c + d*x)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*b*d)]
    @test_int [cos(c + d*x)^2*sqrt(a + b*cos(c + d*x)), x, 7, -((2*(2*a^2 - 9*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (4*a*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b*d) + (2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d)]
    @test_int [cos(c + d*x)^1*sqrt(a + b*cos(c + d*x)), x, 6, (2*a*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^0*sqrt(a + b*cos(c + d*x)), x, 2, (2*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt((a + b*cos(c + d*x))/(a + b)))]
    @test_int [sqrt(a + b*cos(c + d*x))*sec(c + d*x)^1, x, 5, (2*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2, x, 9, -((sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (a*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d]
    @test_int [sqrt(a + b*cos(c + d*x))*sec(c + d*x)^3, x, 10, -(b*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (3*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + (b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a*d) + (sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)]


    @test_int [cos(c + d*x)^3*(a + b*cos(c + d*x))^(3/2), x, 9, (2*(8*a^4 + 33*a^2*b^2 + 147*b^4)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*a*(8*a^4 + 31*a^2*b^2 - 39*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*a*(8*a^2 + 39*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b^2*d) + (2*(8*a^2 + 49*b^2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*b^2*d) - (8*a*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*b^2*d) + (2*cos(c + d*x)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(9*b*d)]
    @test_int [cos(c + d*x)^2*(a + b*cos(c + d*x))^(3/2), x, 8, -((4*a*(3*a^2 - 41*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (2*(6*a^4 - 31*a^2*b^2 + 25*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(105*b^2*d*sqrt(a + b*cos(c + d*x))) - (2*(6*a^2 - 25*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*b*d) - (4*a*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*b*d) + (2*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b*d)]
    @test_int [cos(c + d*x)^1*(a + b*cos(c + d*x))^(3/2), x, 7, (2*(a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*a*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b*d*sqrt(a + b*cos(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d) + (2*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^0*(a + b*cos(c + d*x))^(3/2), x, 6, (8*a*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^1, x, 8, (2*b*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt((a + b*cos(c + d*x))/(a + b))) + (2*a*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*a^2*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))]
    @test_int [(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^2, x, 9, -((a*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt((a + b*cos(c + d*x))/(a + b)))) + ((a^2 + 2*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (3*a*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d]
    @test_int [(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^3, x, 10, (-5*b*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (7*a*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 + 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (5*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)]


    @test_int [cos(c + d*x)^3*(a + b*cos(c + d*x))^(5/2), x, 10, (2*a*(8*a^4 + 51*a^2*b^2 + 741*b^4)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(693*b^3*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*(8*a^6 + 49*a^4*b^2 + 78*a^2*b^4 - 135*b^6)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(693*b^3*d*sqrt(a + b*cos(c + d*x))) + (2*(8*a^4 + 57*a^2*b^2 + 135*b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*b^2*d) + (2*a*(8*a^2 + 67*b^2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(693*b^2*d) + (2*(8*a^2 + 81*b^2)*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(693*b^2*d) - (8*a*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(99*b^2*d) + (2*cos(c + d*x)*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(11*b*d)]
    @test_int [cos(c + d*x)^2*(a + b*cos(c + d*x))^(5/2), x, 9, -((2*(10*a^4 - 279*a^2*b^2 - 147*b^4)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (4*a*(5*a^4 - 62*a^2*b^2 + 57*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(315*b^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*(5*a^2 - 57*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*b*d) - (2*(10*a^2 - 49*b^2)*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(315*b*d) - (4*a*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(63*b*d) + (2*(a + b*cos(c + d*x))^(7/2)*sin(c + d*x))/(9*b*d)]
    @test_int [cos(c + d*x)^1*(a + b*cos(c + d*x))^(5/2), x, 8, (2*a*(3*a^2 + 29*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(21*b*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*(3*a^4 + 2*a^2*b^2 - 5*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(21*b*d*sqrt(a + b*cos(c + d*x))) + (2*(3*a^2 + 5*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*a*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(7*d) + (2*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^0*(a + b*cos(c + d*x))^(5/2), x, 7, (2*(23*a^2 + 9*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (16*a*(a^2 - b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(15*d*sqrt(a + b*cos(c + d*x))) + (16*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d) + (2*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^1, x, 9, (14*a*b*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (2*b*(2*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(3*d*sqrt(a + b*cos(c + d*x))) + (2*a^3*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^2, x, 9, -(((a^2 - 2*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (a*(a^2 + 4*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (5*a^2*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) + (a^2*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/d]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^3, x, 10, (-9*a*b*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (b*(11*a^2 + 8*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (a*(4*a^2 + 15*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*d*sqrt(a + b*cos(c + d*x))) + (9*a*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*d) + (a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^4, x, 11, -((16*a^2 + 33*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (a*(16*a^2 + 59*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(24*d*sqrt(a + b*cos(c + d*x))) + (5*b*(4*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(8*d*sqrt(a + b*cos(c + d*x))) + ((16*a^2 + 33*b^2)*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(24*d) + (13*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(12*d) + (a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^2*tan(c + d*x))/(3*d)]


    @test_int [cos(c + d*x)^0*(a + b*cos(c + d*x))^(7/2), x, 8, (32*a*(11*a^2 + 13*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(105*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*(71*a^4 - 46*a^2*b^2 - 25*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(105*d*sqrt(a + b*cos(c + d*x))) + (2*b*(71*a^2 + 25*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*d) + (24*a*b*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(35*d) + (2*b*(a + b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]


    @test_int [cos(c + d*x)^3*sqrt(3 + 4*cos(c + d*x)), x, 6, (47*Elliptic.E((1/2)*(c + d*x), 8/7))/(20*sqrt(7)*d) + (59*Elliptic.F((1/2)*(c + d*x), 8/7))/(60*sqrt(7)*d) + (59*sqrt(3 + 4*cos(c + d*x))*sin(c + d*x))/(105*d) - (3*(3 + 4*cos(c + d*x))^(3/2)*sin(c + d*x))/(70*d) + (cos(c + d*x)*(3 + 4*cos(c + d*x))^(3/2)*sin(c + d*x))/(14*d)]
    @test_int [cos(c + d*x)^2*sqrt(3 + 4*cos(c + d*x)), x, 5, (21*sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(20*d) - (sqrt(7)*Elliptic.F((1/2)*(c + d*x), 8/7))/(20*d) - (sqrt(3 + 4*cos(c + d*x))*sin(c + d*x))/(5*d) + ((3 + 4*cos(c + d*x))^(3/2)*sin(c + d*x))/(10*d)]
    @test_int [cos(c + d*x)^1*sqrt(3 + 4*cos(c + d*x)), x, 4, (sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(2*d) + (sqrt(7)*Elliptic.F((1/2)*(c + d*x), 8/7))/(6*d) + (2*sqrt(3 + 4*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^0*sqrt(3 + 4*cos(c + d*x)), x, 1, (2*sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/d]
    @test_int [sec(c + d*x)^1*sqrt(3 + 4*cos(c + d*x)), x, 3, (8*Elliptic.F((1/2)*(c + d*x), 8/7))/(sqrt(7)*d) + (6*Elliptic.Pi(2, (1/2)*(c + d*x), 8/7))/(sqrt(7)*d)]
    @test_int [sec(c + d*x)^2*sqrt(3 + 4*cos(c + d*x)), x, 6, -((sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/d) + (3*Elliptic.F((1/2)*(c + d*x), 8/7))/(sqrt(7)*d) + (4*Elliptic.Pi(2, (1/2)*(c + d*x), 8/7))/(sqrt(7)*d) + (sqrt(3 + 4*cos(c + d*x))*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*sqrt(3 + 4*cos(c + d*x)), x, 7, -((sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(3*d)) + (3*Elliptic.F((1/2)*(c + d*x), 8/7))/(sqrt(7)*d) + (5*Elliptic.Pi(2, (1/2)*(c + d*x), 8/7))/(3*sqrt(7)*d) + (sqrt(3 + 4*cos(c + d*x))*tan(c + d*x))/(3*d) + (sqrt(3 + 4*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)]


    @test_int [cos(c + d*x)^3*sqrt(3 - 4*cos(c + d*x)), x, 6, -((47*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(20*sqrt(7)*d)) - (59*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(60*sqrt(7)*d) + (59*sqrt(3 - 4*cos(c + d*x))*sin(c + d*x))/(105*d) - (3*(3 - 4*cos(c + d*x))^(3/2)*sin(c + d*x))/(70*d) - ((3 - 4*cos(c + d*x))^(3/2)*cos(c + d*x)*sin(c + d*x))/(14*d)]
    @test_int [cos(c + d*x)^2*sqrt(3 - 4*cos(c + d*x)), x, 5, (21*sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(20*d) - (sqrt(7)*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(20*d) + (sqrt(3 - 4*cos(c + d*x))*sin(c + d*x))/(5*d) - ((3 - 4*cos(c + d*x))^(3/2)*sin(c + d*x))/(10*d)]
    @test_int [cos(c + d*x)^1*sqrt(3 - 4*cos(c + d*x)), x, 4, -((sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(2*d)) - (sqrt(7)*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(6*d) + (2*sqrt(3 - 4*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^0*sqrt(3 - 4*cos(c + d*x)), x, 1, (2*sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/d]
    @test_int [sec(c + d*x)^1*sqrt(3 - 4*cos(c + d*x)), x, 3, -((8*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(sqrt(7)*d)) - (6*Elliptic.Pi(2, (1/2)*(c + pi + d*x), 8/7))/(sqrt(7)*d)]
    @test_int [sec(c + d*x)^2*sqrt(3 - 4*cos(c + d*x)), x, 6, -((sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/d) + (3*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(sqrt(7)*d) + (4*Elliptic.Pi(2, (1/2)*(c + pi + d*x), 8/7))/(sqrt(7)*d) + (sqrt(3 - 4*cos(c + d*x))*tan(c + d*x))/d]
    @test_int [sec(c + d*x)^3*sqrt(3 - 4*cos(c + d*x)), x, 7, (sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(3*d) - (3*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(sqrt(7)*d) - (5*Elliptic.Pi(2, (1/2)*(c + pi + d*x), 8/7))/(3*sqrt(7)*d) - (sqrt(3 - 4*cos(c + d*x))*tan(c + d*x))/(3*d) + (sqrt(3 - 4*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*d)]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^3/sqrt(a + b*cos(c + d*x)), x, 7, (2*(8*a^2 + 9*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(15*b^3*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*a*(8*a^2 + 7*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(15*b^3*d*sqrt(a + b*cos(c + d*x))) - (8*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^2*d) + (2*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b*d)]
    @test_int [cos(c + d*x)^2/sqrt(a + b*cos(c + d*x)), x, 6, (-4*a*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(3*b^2*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (2*(2*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(3*b^2*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b*d)]
    @test_int [cos(c + d*x)^1/sqrt(a + b*cos(c + d*x)), x, 5, (2*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(b*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*a*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/sqrt(a + b*cos(c + d*x)), x, 2, (2*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/sqrt(a + b*cos(c + d*x)), x, 2, (2*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/sqrt(a + b*cos(c + d*x)), x, 9, -((sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(d*sqrt(a + b*cos(c + d*x))) - (b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(a*d)]
    @test_int [sec(c + d*x)^3/sqrt(a + b*cos(c + d*x)), x, 10, (3*b*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(4*a*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 + 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) - (3*b*sqrt(a + b*cos(c + d*x))*tan(c + d*x))/(4*a^2*d) + (sqrt(a + b*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(2*a*d)]


    @test_int [cos(c + d*x)^4/(a + b*cos(c + d*x))^(3/2), x, 8, (2*(16*a^4 - 8*a^2*b^2 - 3*b^4)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b^4*(a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (8*a*(4*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(5*b^4*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a*(8*a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^3*(a^2 - b^2)*d) + (2*(6*a^2 - b^2)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*b^2*(a^2 - b^2)*d)]
    @test_int [cos(c + d*x)^3/(a + b*cos(c + d*x))^(3/2), x, 7, -((2*a*(8*a^2 - 5*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (2*(8*a^2 + b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(4*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d)]
    @test_int [cos(c + d*x)^2/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(2*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(b^2*(a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (4*a*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(b^2*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*cos(c + d*x))^(3/2), x, 6, (-2*a*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(b*(a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (2*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(b*d*sqrt(a + b*cos(c + d*x))) + (2*a*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + b*cos(c + d*x))^(3/2), x, 4, (2*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/((a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*b*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*cos(c + d*x))^(3/2), x, 7, (-2*b*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(a*(a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (2*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*cos(c + d*x))^(3/2), x, 10, -(((a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*(a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(a*d*sqrt(a + b*cos(c + d*x))) - (3*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (b*(a^2 - 3*b^2)*sin(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + tan(c + d*x)/(a*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3/(a + b*cos(c + d*x))^(3/2), x, 11, (b*(7*a^2 - 15*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^3*(a^2 - b^2)*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (5*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) + ((4*a^2 + 15*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(4*a^3*d*sqrt(a + b*cos(c + d*x))) - (b^2*(7*a^2 - 15*b^2)*sin(c + d*x))/(4*a^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (5*b*tan(c + d*x))/(4*a^2*d*sqrt(a + b*cos(c + d*x))) + (sec(c + d*x)*tan(c + d*x))/(2*a*d*sqrt(a + b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^5/(a + b*cos(c + d*x))^(5/2), x, 9, (2*(128*a^6 - 212*a^4*b^2 + 55*a^2*b^4 + 9*b^6)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^5*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*a*(128*a^4 - 116*a^2*b^2 - 17*b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(15*b^5*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)^3*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*a^2*(2*a^2 - 3*b^2)*cos(c + d*x)^2*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (4*a*(32*a^4 - 49*a^2*b^2 + 7*b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^4*(a^2 - b^2)^2*d) + (2*(48*a^4 - 71*a^2*b^2 + 3*b^4)*cos(c + d*x)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*b^3*(a^2 - b^2)^2*d)]
    @test_int [cos(c + d*x)^4/(a + b*cos(c + d*x))^(5/2), x, 8, -((8*a*(4*a^4 - 7*a^2*b^2 + 2*b^4)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^4*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (2*(16*a^4 - 16*a^2*b^2 - b^4)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^4*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*a^3*(3*a^2 - 5*b^2)*sin(c + d*x))/(3*b^3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(2*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*b^3*(a^2 - b^2)*d)]
    @test_int [cos(c + d*x)^3/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(8*a^4 - 15*a^2*b^2 + 3*b^4)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*a*(8*a^2 - 9*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*cos(c + d*x)*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*a^2*(a^2 - 2*b^2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^2/(a + b*cos(c + d*x))^(5/2), x, 7, -((4*a*(a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + (2*(2*a^2 - 3*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(3*b^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*a*(a^2 - 3*b^2)*sin(c + d*x))/(3*b*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1/(a + b*cos(c + d*x))^(5/2), x, 7, (-2*(a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(3*b*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (2*a*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(3*b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*a*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(a^2 + 3*b^2)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0/(a + b*cos(c + d*x))^(5/2), x, 7, (8*a*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(3*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (2*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(3*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (2*b*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*a*b*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1/(a + b*cos(c + d*x))^(5/2), x, 10, (-2*b*(7*a^2 - 3*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(3*a^2*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b))) + (2*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(3*a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (c + d*x)/2, (2*b)/(a + b)))/(a^2*d*sqrt(a + b*cos(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*b^2*(7*a^2 - 3*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2/(a + b*cos(c + d*x))^(5/2), x, 11, -(((3*a^4 - 26*a^2*b^2 + 15*b^4)*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^3*(a^2 - b^2)^2*d*sqrt((a + b*cos(c + d*x))/(a + b)))) + ((3*a^2 - 5*b^2)*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((1/2)*(c + d*x), (2*b)/(a + b)))/(3*a^2*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) - (5*b*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.Pi(2, (1/2)*(c + d*x), (2*b)/(a + b)))/(a^3*d*sqrt(a + b*cos(c + d*x))) + (b*(3*a^2 - 5*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (b*(3*a^4 - 26*a^2*b^2 + 15*b^4)*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + tan(c + d*x)/(a*d*(a + b*cos(c + d*x))^(3/2))]


    @test_int [cos(c + d*x)^0/(a + b*cos(c + d*x))^(7/2), x, 8, (2*(23*a^2 + 9*b^2)*sqrt(a + b*cos(c + d*x))*Elliptic.E((c + d*x)/2, (2*b)/(a + b)))/(15*(a^2 - b^2)^3*d*sqrt((a + b*cos(c + d*x))/(a + b))) - (16*a*sqrt((a + b*cos(c + d*x))/(a + b))*Elliptic.F((c + d*x)/2, (2*b)/(a + b)))/(15*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) - (2*b*sin(c + d*x))/(5*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(5/2)) - (16*a*b*sin(c + d*x))/(15*(a^2 - b^2)^2*d*(a + b*cos(c + d*x))^(3/2)) - (2*b*(23*a^2 + 9*b^2)*sin(c + d*x))/(15*(a^2 - b^2)^3*d*sqrt(a + b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^3/sqrt(3 + 4*cos(c + d*x)), x, 5, (9*sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(20*d) - (23*Elliptic.F((1/2)*(c + d*x), 8/7))/(20*sqrt(7)*d) - (sqrt(3 + 4*cos(c + d*x))*sin(c + d*x))/(10*d) + (cos(c + d*x)*sqrt(3 + 4*cos(c + d*x))*sin(c + d*x))/(10*d)]
    @test_int [cos(c + d*x)^2/sqrt(3 + 4*cos(c + d*x)), x, 4, -((sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(4*d)) + (17*Elliptic.F((1/2)*(c + d*x), 8/7))/(12*sqrt(7)*d) + (sqrt(3 + 4*cos(c + d*x))*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^1/sqrt(3 + 4*cos(c + d*x)), x, 3, (sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(2*d) - (3*Elliptic.F((1/2)*(c + d*x), 8/7))/(2*sqrt(7)*d)]
    @test_int [cos(c + d*x)^0/sqrt(3 + 4*cos(c + d*x)), x, 1, (2*Elliptic.F((1/2)*(c + d*x), 8/7))/(sqrt(7)*d)]
    @test_int [sec(c + d*x)^1/sqrt(3 + 4*cos(c + d*x)), x, 1, (2*Elliptic.Pi(2, (1/2)*(c + d*x), 8/7))/(sqrt(7)*d)]
    @test_int [sec(c + d*x)^2/sqrt(3 + 4*cos(c + d*x)), x, 6, -((sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(3*d)) + Elliptic.F((1/2)*(c + d*x), 8/7)/(sqrt(7)*d) - (4*Elliptic.Pi(2, (1/2)*(c + d*x), 8/7))/(3*sqrt(7)*d) + (sqrt(3 + 4*cos(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^3/sqrt(3 + 4*cos(c + d*x)), x, 7, (sqrt(7)*Elliptic.E((1/2)*(c + d*x), 8/7))/(3*d) - Elliptic.F((1/2)*(c + d*x), 8/7)/(3*sqrt(7)*d) + (sqrt(7)*Elliptic.Pi(2, (1/2)*(c + d*x), 8/7))/(3*d) - (sqrt(3 + 4*cos(c + d*x))*tan(c + d*x))/(3*d) + (sqrt(3 + 4*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(6*d)]


    @test_int [cos(c + d*x)^3/sqrt(3 - 4*cos(c + d*x)), x, 5, -((9*sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(20*d)) + (23*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(20*sqrt(7)*d) - (sqrt(3 - 4*cos(c + d*x))*sin(c + d*x))/(10*d) - (sqrt(3 - 4*cos(c + d*x))*cos(c + d*x)*sin(c + d*x))/(10*d)]
    @test_int [cos(c + d*x)^2/sqrt(3 - 4*cos(c + d*x)), x, 4, -((sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(4*d)) + (17*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(12*sqrt(7)*d) - (sqrt(3 - 4*cos(c + d*x))*sin(c + d*x))/(6*d)]
    @test_int [cos(c + d*x)^1/sqrt(3 - 4*cos(c + d*x)), x, 3, -((sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(2*d)) + (3*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(2*sqrt(7)*d)]
    @test_int [cos(c + d*x)^0/sqrt(3 - 4*cos(c + d*x)), x, 1, (2*Elliptic.F((1/2)*(c + pi + d*x), 8/7))/(sqrt(7)*d)]
    @test_int [sec(c + d*x)^1/sqrt(3 - 4*cos(c + d*x)), x, 1, -((2*Elliptic.Pi(2, (1/2)*(c + pi + d*x), 8/7))/(sqrt(7)*d))]
    @test_int [sec(c + d*x)^2/sqrt(3 - 4*cos(c + d*x)), x, 6, -((sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(3*d)) + Elliptic.F((1/2)*(c + pi + d*x), 8/7)/(sqrt(7)*d) - (4*Elliptic.Pi(2, (1/2)*(c + pi + d*x), 8/7))/(3*sqrt(7)*d) + (sqrt(3 - 4*cos(c + d*x))*tan(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^3/sqrt(3 - 4*cos(c + d*x)), x, 7, -((sqrt(7)*Elliptic.E((1/2)*(c + pi + d*x), 8/7))/(3*d)) + Elliptic.F((1/2)*(c + pi + d*x), 8/7)/(3*sqrt(7)*d) - (sqrt(7)*Elliptic.Pi(2, (1/2)*(c + pi + d*x), 8/7))/(3*d) + (sqrt(3 - 4*cos(c + d*x))*tan(c + d*x))/(3*d) + (sqrt(3 - 4*cos(c + d*x))*sec(c + d*x)*tan(c + d*x))/(6*d)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/2)*(a+b*cos(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^(5/2)*(A + B*cos(c + d*x)), x, 6, (6*A*Elliptic.E((c + d*x)/2, 2))/(5*d) + (10*B*Elliptic.F((c + d*x)/2, 2))/(21*d) + (10*B*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*A*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*B*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(3/2)*(A + B*cos(c + d*x)), x, 5, (6*B*Elliptic.E((c + d*x)/2, 2))/(5*d) + (2*A*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*A*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*B*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [sqrt(cos(c + d*x))*(A + B*cos(c + d*x)), x, 4, (2*A*Elliptic.E((c + d*x)/2, 2))/d + (2*B*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*B*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [(A + B*cos(c + d*x))/sqrt(cos(c + d*x)), x, 3, (2*B*Elliptic.E((c + d*x)/2, 2))/d + (2*A*Elliptic.F((c + d*x)/2, 2))/d]
    @test_int [(A + B*cos(c + d*x))/cos(c + d*x)^(3/2), x, 4, (-2*A*Elliptic.E((c + d*x)/2, 2))/d + (2*B*Elliptic.F((c + d*x)/2, 2))/d + (2*A*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(A + B*cos(c + d*x))/cos(c + d*x)^(5/2), x, 5, (-2*B*Elliptic.E((c + d*x)/2, 2))/d + (2*A*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*A*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*B*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [(A + B*cos(c + d*x))/cos(c + d*x)^(7/2), x, 6, (-6*A*Elliptic.E((c + d*x)/2, 2))/(5*d) + (2*B*Elliptic.F((c + d*x)/2, 2))/(3*d) + (2*A*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*B*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (6*A*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2, x, 7, (2*(9*a^2 + 7*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (20*a*b*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (20*a*b*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*(9*a^2 + 7*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (4*a*b*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*b^2*cos(c + d*x)^(7/2)*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2, x, 6, (12*a*b*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*(7*a^2 + 5*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*(7*a^2 + 5*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (4*a*b*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*b^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^2, x, 5, (2*(5*a^2 + 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (4*a*b*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (4*a*b*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*b^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^2, x, 4, (4*a*b*Elliptic.E((1/2)*(c + d*x), 2))/d + (2*(3*a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^2, x, 4, -((2*(a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), 2))/d) + (4*a*b*Elliptic.F((1/2)*(c + d*x), 2))/d + (2*a^2*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^2, x, 5, -((4*a*b*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*(a^2 + 3*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (4*a*b*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^2, x, 6, -((2*(3*a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (4*a*b*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (2*a^2*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*a*b*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2)) + (2*(3*a^2 + 5*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3, x, 7, (2*b*(27*a^2 + 7*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(15*d) + (2*a*(7*a^2 + 15*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*a*(7*a^2 + 15*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (2*b*(27*a^2 + 7*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(45*d) + (40*a*b^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(63*d) + (2*b^2*cos(c + d*x)^(5/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(9*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^3, x, 6, (2*a*(5*a^2 + 9*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*b*(21*a^2 + 5*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (2*b*(21*a^2 + 5*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(21*d) + (32*a*b^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(35*d) + (2*b^2*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^3, x, 5, (6*b*(5*a^2 + b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d) + (2*a*(a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/d + (8*a*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(5*d) + (2*b^2*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d)]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^3, x, 5, -((2*a*(a^2 - 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*b*(9*a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) - (2*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^3, x, 5, -((2*b*(3*a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), 2))/d) + (2*a*(a^2 + 9*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*d) + (16*a^2*b*sin(c + d*x))/(3*d*sqrt(cos(c + d*x))) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^3, x, 6, -((6*a*(a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (2*b*(a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/d + (8*a^2*b*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2)) + (6*a*(a^2 + 5*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2))]
    @test_int [cos(c + d*x)^(-9/2)*(a + b*cos(c + d*x))^3, x, 7, -((2*b*(9*a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(5*d)) + (2*a*(5*a^2 + 21*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(21*d) + (32*a^2*b*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*a*(5*a^2 + 21*b^2)*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2)) + (2*b*(9*a^2 + 5*b^2)*sin(c + d*x))/(5*d*sqrt(cos(c + d*x))) + (2*a^2*(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(5/2)/(a + b*cos(c + d*x)), x, 6, -((2*a*Elliptic.E((1/2)*(c + d*x), 2))/(b^2*d)) + (2*(3*a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*b^3*d) - (2*a^3*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^3*(a + b)*d) + (2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*d)]
    @test_int [cos(c + d*x)^(3/2)/(a + b*cos(c + d*x)), x, 5, (2*Elliptic.E((1/2)*(c + d*x), 2))/(b*d) - (2*a*Elliptic.F((1/2)*(c + d*x), 2))/(b^2*d) + (2*a^2*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b^2*(a + b)*d)]
    @test_int [cos(c + d*x)^(1/2)/(a + b*cos(c + d*x)), x, 3, (2*Elliptic.F((1/2)*(c + d*x), 2))/(b*d) - (2*a*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(b*(a + b)*d)]
    @test_int [cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x)), x, 1, (2*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a + b)*d)]
    @test_int [cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x)), x, 5, -((2*Elliptic.E((1/2)*(c + d*x), 2))/(a*d)) - (2*b*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a + b)*d) + (2*sin(c + d*x))/(a*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x)), x, 7, (2*b*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*d) + (2*Elliptic.F((1/2)*(c + d*x), 2))/(3*a*d) + (2*b^2*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a + b)*d) + (2*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2)) - (2*b*sin(c + d*x))/(a^2*d*sqrt(cos(c + d*x)))]


    @test_int [cos(c + d*x)^(7/2)/(a + b*cos(c + d*x))^2, x, 7, -((a*(5*a^2 - 4*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d)) + ((15*a^4 - 16*a^2*b^2 - 2*b^4)*Elliptic.F((1/2)*(c + d*x), 2))/(3*b^4*(a^2 - b^2)*d) - (a^3*(5*a^2 - 7*b^2)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^4*(a + b)^2*d) + ((5*a^2 - 2*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)*d) - (a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^2, x, 6, ((3*a^2 - 2*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^3*(a + b)^2*d) - (a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^2, x, 6, -((a*Elliptic.E((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b^2*(a + b)^2*d) + (a*sqrt(cos(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^2, x, 6, Elliptic.E((1/2)*(c + d*x), 2)/((a^2 - b^2)*d) + (a*Elliptic.F((1/2)*(c + d*x), 2))/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/((a - b)*b*(a + b)^2*d) - (b*sqrt(cos(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^2, x, 6, -((b*Elliptic.E((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d)) - Elliptic.F((1/2)*(c + d*x), 2)/((a^2 - b^2)*d) + ((3*a^2 - b^2)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a*(a - b)*(a + b)^2*d) + (b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^2, x, 7, -(((2*a^2 - 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(a^2*(a^2 - b^2)*d)) + (b*Elliptic.F((1/2)*(c + d*x), 2))/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*sin(c + d*x))/(a^2*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^2, x, 8, (b*(4*a^2 - 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(a^3*(a - b)*(a + b)^2*d) + ((2*a^2 - 5*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2)) - (b*(4*a^2 - 5*b^2)*sin(c + d*x))/(a^3*(a^2 - b^2)*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)/(a + b*cos(c + d*x))^3, x, 8, -((a*(35*a^4 - 65*a^2*b^2 + 24*b^4)*Elliptic.E((1/2)*(c + d*x), 2))/(4*b^4*(a^2 - b^2)^2*d)) + ((105*a^6 - 223*a^4*b^2 + 128*a^2*b^4 + 8*b^6)*Elliptic.F((1/2)*(c + d*x), 2))/(12*b^5*(a^2 - b^2)^2*d) - (a^3*(35*a^4 - 86*a^2*b^2 + 63*b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^5*(a + b)^3*d) + ((35*a^4 - 61*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*sin(c + d*x))/(12*b^3*(a^2 - b^2)^2*d) - (a^2*cos(c + d*x)^(5/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(7*a^2 - 13*b^2)*cos(c + d*x)^(3/2)*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(7/2)/(a + b*cos(c + d*x))^3, x, 7, ((15*a^4 - 29*a^2*b^2 + 8*b^4)*Elliptic.E((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(5*a^4 - 11*a^2*b^2 + 8*b^4)*Elliptic.F((1/2)*(c + d*x), 2))/(4*b^4*(a^2 - b^2)^2*d) + (a^2*(15*a^4 - 38*a^2*b^2 + 35*b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^4*(a + b)^3*d) - (a^2*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (a^2*(5*a^2 - 11*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^3, x, 7, -((3*a*(a^2 - 3*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d)) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*Elliptic.F((1/2)*(c + d*x), 2))/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^3*(a + b)^3*d) - (a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(2*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (3*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^3, x, 7, -(((a^2 + 5*b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*b*(a^2 - b^2)^2*d)) + (a*(a^2 - 7*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*(a - b)^2*b^2*(a + b)^3*d) + (a*sqrt(cos(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + ((a^2 + 5*b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^3, x, 7, ((5*a^2 + b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a*(a - b)^2*b*(a + b)^3*d) - (b*sqrt(cos(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) - (b*(5*a^2 + b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^3, x, 7, -((3*b*(3*a^2 - b^2)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2 - b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^2) + (3*b^2*(3*a^2 - b^2)*sqrt(cos(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^3, x, 8, -(((8*a^4 - 29*a^2*b^2 + 15*b^4)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a^3*(a^2 - b^2)^2*d)) + (b*(11*a^2 - 5*b^2)*Elliptic.F((1/2)*(c + d*x), 2))/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^2) + (b^2*(11*a^2 - 5*b^2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^3, x, 9, (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*Elliptic.E((1/2)*(c + d*x), 2))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*Elliptic.F((1/2)*(c + d*x), 2))/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2))/(4*a^4*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))) + (b^2*sin(c + d*x))/(2*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2) + (b^2*(13*a^2 - 7*b^2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/2)*(a+b*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(1/2), x, 7, -(((a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b*d)) + (sqrt(a + b)*(a + 2*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b*d) + (sqrt(a + b)*(a^2 - 4*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b^2*d) + (a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*b*d*sqrt(cos(c + d*x))) + (sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(1/2), x, 7, -(((a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d)) + (sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d - (a*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*d) + (sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(1/2), x, 1, -((2*sqrt((a*(1 - cos(c + d*x)))/(a + b*cos(c + d*x)))*sqrt((a*(1 + cos(c + d*x)))/(a + b*cos(c + d*x)))*(a + b*cos(c + d*x))*csc(c + d*x)*Elliptic.Pi(b/(a + b), asin((sqrt(a + b)*sqrt(cos(c + d*x)))/sqrt(a + b*cos(c + d*x))), -((a - b)/(a + b))))/(sqrt(a + b)*d))]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(1/2), x, 3, (2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d) - (2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d)]
    @test_int [cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^(1/2), x, 4, (2*(a - b)*b*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*d) + (2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*d) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^(1/2), x, 5, (2*(a - b)*sqrt(a + b)*(9*a^2 - 2*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a^3*d) - (2*(a - b)*sqrt(a + b)*(9*a + 2*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a^2*d) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (2*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*a*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-9/2)*(a + b*cos(c + d*x))^(1/2), x, 6, (2*(a - b)*b*sqrt(a + b)*(19*a^2 + 8*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(105*a^4*d) + (2*(a - b)*sqrt(a + b)*(25*a^2 + 6*a*b + 8*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(105*a^3*d) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (2*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*a*d*cos(c + d*x)^(5/2)) + (2*(25*a^2 - 4*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a^2*d*cos(c + d*x)^(3/2))]


    @test_int [cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2), x, 8, -(((a - b)*sqrt(a + b)*(3*a^2 + 16*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*a*b*d)) + (sqrt(a + b)*(a + 2*b)*(3*a + 8*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*b*d) + (a*sqrt(a + b)*(a^2 - 12*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(8*b^2*d) + ((3*a^2 + 16*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*b*d*sqrt(cos(c + d*x))) + (a*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d) + (sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(3/2), x, 8, -((5*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d)) + (sqrt(a + b)*(5*a + 2*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d) - (sqrt(a + b)*(3*a^2 + 4*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b*d) + (3*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))) + ((a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(2*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(3/2), x, 6, -(((a - b)*b*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d)) + (sqrt(a + b)*(2*a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d - (3*a*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d + (b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(3/2), x, 5, (2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d - (2*(a - 2*b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d - (2*b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d]
    @test_int [cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^(3/2), x, 4, (8*(a - b)*b*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*d) + (2*(a - 3*b)*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*d) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^(3/2), x, 5, (2*(a - b)*sqrt(a + b)*(3*a^2 + b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(5*a^2*d) - (2*(a - b)*(3*a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(5*a*d) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (4*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-9/2)*(a + b*cos(c + d*x))^(3/2), x, 6, (4*(a - b)*b*sqrt(a + b)*(41*a^2 - 3*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(105*a^3*d) + (2*(a - b)*sqrt(a + b)*(25*a^2 - 57*a*b - 6*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(105*a^2*d) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (16*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(35*d*cos(c + d*x)^(5/2)) + (2*(25*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(105*a*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-11/2)*(a + b*cos(c + d*x))^(3/2), x, 7, (2*(a - b)*sqrt(a + b)*(147*a^4 + 33*a^2*b^2 + 8*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(315*a^4*d) - (2*(a - b)*sqrt(a + b)*(147*a^3 - 39*a^2*b - 6*a*b^2 - 8*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(315*a^3*d) + (2*a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (20*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*(49*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(5/2)) + (8*b*(22*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a^2*d*cos(c + d*x)^(3/2))]


    @test_int [cos(c + d*x)^(1/2)*(a + b*cos(c + d*x))^(5/2), x, 8, -(((a - b)*sqrt(a + b)*(33*a^2 + 16*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*a*d)) + (sqrt(a + b)*(33*a^2 + 26*a*b + 16*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*d) - (5*a*sqrt(a + b)*(a^2 + 4*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(8*b*d) + ((33*a^2 + 16*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d*sqrt(cos(c + d*x))) + (13*a*b*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d) + (b^2*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [cos(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^(5/2), x, 7, -((9*(a - b)*b*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d)) + (sqrt(a + b)*(8*a^2 + 9*a*b + 2*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d) - (sqrt(a + b)*(15*a^2 + 4*b^2)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d) + (9*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(cos(c + d*x))) + (b^2*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^(5/2), x, 7, ((a - b)*sqrt(a + b)*(2*a^2 - b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d) - (sqrt(a + b)*(2*a^2 - 6*a*b - b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d - (5*a*b*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) - ((2*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^(5/2), x, 6, (14*(a - b)*b*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*d) + (2*sqrt(a + b)*(a^2 - 7*a*b + 9*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*d) - (2*b^2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/d + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-7/2)*(a + b*cos(c + d*x))^(5/2), x, 5, (2*(a - b)*sqrt(a + b)*(9*a^2 + 23*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a*d) - (2*(a - b)*sqrt(a + b)*(9*a^2 - 8*a*b + 15*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*d*cos(c + d*x)^(5/2)) + (22*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(15*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-9/2)*(a + b*cos(c + d*x))^(5/2), x, 6, (2*(a - b)*b*sqrt(a + b)*(29*a^2 + 3*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(21*a^2*d) + (2*(a - b)*sqrt(a + b)*(5*a^2 - 24*a*b + 3*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(21*a*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(7/2)) + (6*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(7*d*cos(c + d*x)^(5/2)) + (2*(5*a^2 + 9*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(21*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-11/2)*(a + b*cos(c + d*x))^(5/2), x, 7, (2*(a - b)*sqrt(a + b)*(147*a^4 + 279*a^2*b^2 - 10*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(315*a^3*d) - (2*(a - b)*sqrt(a + b)*(147*a^3 - 114*a^2*b + 165*a*b^2 + 10*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(315*a^2*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(9*d*cos(c + d*x)^(9/2)) + (38*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(63*d*cos(c + d*x)^(7/2)) + (2*(49*a^2 + 75*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*d*cos(c + d*x)^(5/2)) + (2*b*(163*a^2 + 5*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(315*a*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-13/2)*(a + b*cos(c + d*x))^(5/2), x, 8, (2*(a - b)*b*sqrt(a + b)*(741*a^4 + 51*a^2*b^2 + 8*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(693*a^4*d) + (2*(a - b)*sqrt(a + b)*(135*a^4 - 606*a^3*b + 57*a^2*b^2 + 6*a*b^3 + 8*b^4)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(693*a^3*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(11*d*cos(c + d*x)^(11/2)) + (46*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(99*d*cos(c + d*x)^(9/2)) + (2*(81*a^2 + 113*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*d*cos(c + d*x)^(7/2)) + (2*b*(229*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*a*d*cos(c + d*x)^(5/2)) + (2*(135*a^4 + 205*a^2*b^2 - 4*b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(693*a^2*d*cos(c + d*x)^(3/2))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(1/2), x, 8, -(((a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*b*d)) + (sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*d) + (a*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^2*d) + (sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(b*d*sqrt(cos(c + d*x))), -(((a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*b*d)) + (sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*d) + (a*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^2*d) + (a*sin(c + d*x))/(b*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x))) + (sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(1/2), x, 1, -((2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*d))]
    @test_int [cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(1/2), x, 1, (2*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d)]
    @test_int [cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(1/2), x, 3, (2*(a - b)*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^2*d) - (2*sqrt(a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d)]
    @test_int [cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(1/2), x, 4, -((4*(a - b)*b*sqrt(a + b)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*d)) + (2*sqrt(a + b)*(a + 2*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*d) + (2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a*d*cos(c + d*x)^(3/2))]


    @test_int [cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(3/2), x, 7, -(((3*a^2 - b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*b^2*sqrt(a + b)*d)) + ((3*a + b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^2*sqrt(a + b)*d) + (3*a*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^3*d) - (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + ((3*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(b^2*(a^2 - b^2)*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(3/2), x, 6, (2*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*sqrt(a + b)*d) - (2*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*sqrt(a + b)*d) - (2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^2*d) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(3/2), x, 4, -((2*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*sqrt(a + b)*d)) + (2*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*sqrt(a + b)*d) + (2*a*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(3/2), x, 4, (2*b*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^2*sqrt(a + b)*d) + (2*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*sqrt(a + b)*d) - (2*b*sin(c + d*x))/((a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(3/2), x, 4, (2*(a^2 - 2*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^3*sqrt(a + b)*d) - (2*(a + 2*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^2*sqrt(a + b)*d) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(3/2), x, 5, -((2*b*(5*a^2 - 8*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^4*sqrt(a + b)*d)) + (2*(a + 2*b)*(a + 4*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*sqrt(a + b)*d) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-7/2)/(a + b*cos(c + d*x))^(3/2), x, 6, (2*(3*a^4 + 8*a^2*b^2 - 16*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(5*a^5*sqrt(a + b)*d) - (2*(3*a + 4*b)*(a^2 + 4*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(5*a^4*sqrt(a + b)*d) + (2*b^2*sin(c + d*x))/(a*(a^2 - b^2)*d*cos(c + d*x)^(5/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 6*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a^2*(a^2 - b^2)*d*cos(c + d*x)^(5/2)) - (2*b*(3*a^2 - 8*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(5*a^3*(a^2 - b^2)*d*cos(c + d*x)^(3/2))]


    @test_int [cos(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(5/2), x, 7, (2*(3*a^2 - 7*b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*(a - b)*b^2*(a + b)^(3/2)*d) - (2*(3*a^2 + a*b - 6*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*(a - b)*b^2*(a + b)^(3/2)*d) - (2*sqrt(a + b)*cot(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^3*d) - (2*a^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (2*a^2*(3*a^2 - 7*b^2)*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(5/2), x, 5, (8*b*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*(a - b)*(a + b)^(3/2)*d) + (2*(a - 3*b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*(a - b)*(a + b)^(3/2)*d) + (2*a*sqrt(cos(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (8*a*b*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)/(a + b*cos(c + d*x))^(5/2), x, 5, -((2*(3*a^2 + b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*(a - b)*(a + b)^(3/2)*d)) + (2*(3*a - b)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*b*sqrt(cos(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (2*(3*a^2 + b^2)*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^(5/2), x, 5, (4*b*(3*a^2 - b^2)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*(a - b)*(a + b)^(3/2)*d) + (2*(3*a^2 - 3*a*b - 2*b^2)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*(a - b)*(a + b)^(3/2)*d) + (2*b^2*sqrt(cos(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) - (4*b*(3*a^2 - b^2)*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^(5/2), x, 5, (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^4*(a - b)*(a + b)^(3/2)*d) - (2*(3*a^3 + 9*a^2*b - 6*a*b^2 - 8*b^3)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*(a - b)*(a + b)^(3/2)*d) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*sqrt(cos(c + d*x))*(a + b*cos(c + d*x))^(3/2)) + (8*b^2*(2*a^2 - b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(cos(c + d*x))*sqrt(a + b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^(5/2), x, 6, -((8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*cot(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^5*(a - b)*(a + b)^(3/2)*d)) + (2*(a^4 + 9*a^3*b + 16*a^2*b^2 - 12*a*b^3 - 16*b^4)*cot(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^4*(a - b)*(a + b)^(3/2)*d) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*cos(c + d*x)^(3/2)*(a + b*cos(c + d*x))^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2)*sqrt(a + b*cos(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d*cos(c + d*x)^(3/2))]


    @test_int [1/(sqrt(cos(c + d*x))*sqrt(2 + 3*cos(c + d*x))), x, 1, (2*Elliptic.F(asin(sin(c + d*x)/(1 + cos(c + d*x))), 1/5))/(sqrt(5)*d)]
    @test_int [1/(sqrt(cos(c + d*x))*sqrt(-2 + 3*cos(c + d*x))), x, 1, (2*Elliptic.F(asin(sin(c + d*x)/(1 + cos(c + d*x))), 5))/d]

    @test_int [1/(sqrt(cos(c + d*x))*sqrt(2 - 3*cos(c + d*x))), x, 2, -((2*sqrt(-cos(c + d*x))*Elliptic.F(asin(sin(c + d*x)/(1 - cos(c + d*x))), 1/5))/(sqrt(5)*d*sqrt(cos(c + d*x))))]
    @test_int [1/(sqrt(cos(c + d*x))*sqrt(-2 - 3*cos(c + d*x))), x, 2, -((2*sqrt(-cos(c + d*x))*Elliptic.F(asin(sin(c + d*x)/(1 - cos(c + d*x))), 5))/(d*sqrt(cos(c + d*x))))]

    @test_int [1/(sqrt(cos(c + d*x))*sqrt(3 + 2*cos(c + d*x))), x, 1, (2*cot(c + d*x)*Elliptic.F(asin(sqrt(3 + 2*cos(c + d*x))/(sqrt(5)*sqrt(cos(c + d*x)))), -5)*sqrt(-tan(c + d*x)^2))/d]
    @test_int [1/(sqrt(cos(c + d*x))*sqrt(3 - 2*cos(c + d*x))), x, 1, (2*cot(c + d*x)*Elliptic.F(asin(sqrt(3 - 2*cos(c + d*x))/sqrt(cos(c + d*x))), -(1/5))*sqrt(-tan(c + d*x)^2))/(sqrt(5)*d)]

    @test_int [1/(sqrt(cos(c + d*x))*sqrt(-3 + 2*cos(c + d*x))), x, 2, -((2*sqrt(-cos(c + d*x))*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(-3 + 2*cos(c + d*x))/sqrt(-cos(c + d*x))), -(1/5))*sqrt(-tan(c + d*x)^2))/(sqrt(5)*d))]
    @test_int [1/(sqrt(cos(c + d*x))*sqrt(-3 - 2*cos(c + d*x))), x, 2, -((2*sqrt(-cos(c + d*x))*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(-3 - 2*cos(c + d*x))/(sqrt(5)*sqrt(-cos(c + d*x)))), -5)*sqrt(-tan(c + d*x)^2))/d)]


    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(2 + 3*cos(c + d*x))), x, 2, (2*sqrt(cos(c + d*x))*Elliptic.F(asin(sin(c + d*x)/(1 + cos(c + d*x))), 1/5))/(sqrt(5)*d*sqrt(-cos(c + d*x)))]
    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(-2 + 3*cos(c + d*x))), x, 2, (2*sqrt(cos(c + d*x))*Elliptic.F(asin(sin(c + d*x)/(1 + cos(c + d*x))), 5))/(d*sqrt(-cos(c + d*x)))]

    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(2 - 3*cos(c + d*x))), x, 1, -((2*sqrt(-cos(c + d*x))*Elliptic.F(asin(sin(c + d*x)/(1 - cos(c + d*x))), 1/5))/(sqrt(5)*d*sqrt(-cos(c + d*x))))]
    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(-2 - 3*cos(c + d*x))), x, 1, -((2*sqrt(-cos(c + d*x))*Elliptic.F(asin(sin(c + d*x)/(1 - cos(c + d*x))), 5))/(d*sqrt(-cos(c + d*x))))]

    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(3 + 2*cos(c + d*x))), x, 2, (2*cos(c + d*x)^(3/2)*csc(c + d*x)*Elliptic.F(asin(sqrt(3 + 2*cos(c + d*x))/(sqrt(5)*sqrt(cos(c + d*x)))), -5)*sqrt(-tan(c + d*x)^2))/(d*sqrt(-cos(c + d*x)))]
    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(3 - 2*cos(c + d*x))), x, 2, (2*cos(c + d*x)^(3/2)*csc(c + d*x)*Elliptic.F(asin(sqrt(3 - 2*cos(c + d*x))/sqrt(cos(c + d*x))), -(1/5))*sqrt(-tan(c + d*x)^2))/(sqrt(5)*d*sqrt(-cos(c + d*x)))]

    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(-3 + 2*cos(c + d*x))), x, 1, -((2*cot(c + d*x)*Elliptic.F(asin(sqrt(-3 + 2*cos(c + d*x))/sqrt(-cos(c + d*x))), -(1/5))*sqrt(-tan(c + d*x)^2))/(sqrt(5)*d))]
    @test_int [1/(sqrt(-cos(c + d*x))*sqrt(-3 - 2*cos(c + d*x))), x, 1, -((2*cot(c + d*x)*Elliptic.F(asin(sqrt(-3 - 2*cos(c + d*x))/(sqrt(5)*sqrt(-cos(c + d*x)))), -5)*sqrt(-tan(c + d*x)^2))/d)]


    @test_int [sqrt(cos(c + d*x))/sqrt(2 + 3*cos(c + d*x)), x, 1, -((4*cot(c + d*x)*Elliptic.Pi(5/3, asin(sqrt(2 + 3*cos(c + d*x))/(sqrt(5)*sqrt(cos(c + d*x)))), 5)*sqrt(-1 - sec(c + d*x))*sqrt(1 - sec(c + d*x)))/(3*d))]
    @test_int [sqrt(cos(c + d*x))/sqrt(-2 + 3*cos(c + d*x)), x, 1, -((4*cot(c + d*x)*Elliptic.Pi(1/3, asin(sqrt(-2 + 3*cos(c + d*x))/sqrt(cos(c + d*x))), 1/5)*sqrt(-1 + sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(3*sqrt(5)*d))]

    @test_int [sqrt(cos(c + d*x))/sqrt(2 - 3*cos(c + d*x)), x, 2, -((4*cos(c + d*x)^(3/2)*csc(c + d*x)*Elliptic.Pi(1/3, asin(sqrt(2 - 3*cos(c + d*x))/sqrt(-cos(c + d*x))), 1/5)*sqrt(-1 + sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(3*sqrt(5)*d*sqrt(-cos(c + d*x))))]
    @test_int [sqrt(cos(c + d*x))/sqrt(-2 - 3*cos(c + d*x)), x, 2, -((4*cos(c + d*x)^(3/2)*csc(c + d*x)*Elliptic.Pi(5/3, asin(sqrt(-2 - 3*cos(c + d*x))/(sqrt(5)*sqrt(-cos(c + d*x)))), 5)*sqrt(-1 - sec(c + d*x))*sqrt(1 - sec(c + d*x)))/(3*d*sqrt(-cos(c + d*x))))]

    @test_int [sqrt(cos(c + d*x))/sqrt(3 + 2*cos(c + d*x)), x, 1, -((3*cot(c + d*x)*Elliptic.Pi(5/2, asin(sqrt(3 + 2*cos(c + d*x))/(sqrt(5)*sqrt(cos(c + d*x)))), -5)*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/d)]
    @test_int [sqrt(cos(c + d*x))/sqrt(3 - 2*cos(c + d*x)), x, 1, (3*cot(c + d*x)*Elliptic.Pi(-(1/2), asin(sqrt(3 - 2*cos(c + d*x))/sqrt(cos(c + d*x))), -(1/5))*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(sqrt(5)*d)]

    @test_int [sqrt(cos(c + d*x))/sqrt(-3 + 2*cos(c + d*x)), x, 2, (3*cos(c + d*x)^(3/2)*csc(c + d*x)*Elliptic.Pi(-(1/2), asin(sqrt(-3 + 2*cos(c + d*x))/sqrt(-cos(c + d*x))), -(1/5))*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(sqrt(5)*d*sqrt(-cos(c + d*x)))]
    @test_int [sqrt(cos(c + d*x))/sqrt(-3 - 2*cos(c + d*x)), x, 2, -((3*cos(c + d*x)^(3/2)*csc(c + d*x)*Elliptic.Pi(5/2, asin(sqrt(-3 - 2*cos(c + d*x))/(sqrt(5)*sqrt(-cos(c + d*x)))), -5)*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(d*sqrt(-cos(c + d*x))))]


    @test_int [sqrt(-cos(c + d*x))/sqrt(2 + 3*cos(c + d*x)), x, 2, -((4*sqrt(-cos(c + d*x))*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi(5/3, asin(sqrt(2 + 3*cos(c + d*x))/(sqrt(5)*sqrt(cos(c + d*x)))), 5)*sqrt(-1 - sec(c + d*x))*sqrt(1 - sec(c + d*x)))/(3*d))]
    @test_int [sqrt(-cos(c + d*x))/sqrt(-2 + 3*cos(c + d*x)), x, 2, -((4*sqrt(-cos(c + d*x))*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi(1/3, asin(sqrt(-2 + 3*cos(c + d*x))/sqrt(cos(c + d*x))), 1/5)*sqrt(-1 + sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(3*sqrt(5)*d))]

    @test_int [sqrt(-cos(c + d*x))/sqrt(2 - 3*cos(c + d*x)), x, 1, -((4*cot(c + d*x)*Elliptic.Pi(1/3, asin(sqrt(2 - 3*cos(c + d*x))/sqrt(-cos(c + d*x))), 1/5)*sqrt(-1 + sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(3*sqrt(5)*d))]
    @test_int [sqrt(-cos(c + d*x))/sqrt(-2 - 3*cos(c + d*x)), x, 1, -((4*cot(c + d*x)*Elliptic.Pi(5/3, asin(sqrt(-2 - 3*cos(c + d*x))/(sqrt(5)*sqrt(-cos(c + d*x)))), 5)*sqrt(-1 - sec(c + d*x))*sqrt(1 - sec(c + d*x)))/(3*d))]

    #= For*some*reason, Mathematica*cannot*verify*the*following*two*antiderivatives*correct. =#
    @test_int [sqrt(-cos(c + d*x))/sqrt(3 + 2*cos(c + d*x)), x, 2, -((3*sqrt(-cos(c + d*x))*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi(5/2, asin(sqrt(3 + 2*cos(c + d*x))/(sqrt(5)*sqrt(cos(c + d*x)))), -5)*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/d)]
    @test_int [sqrt(-cos(c + d*x))/sqrt(3 - 2*cos(c + d*x)), x, 2, (3*sqrt(-cos(c + d*x))*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi(-(1/2), asin(sqrt(3 - 2*cos(c + d*x))/sqrt(cos(c + d*x))), -(1/5))*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(sqrt(5)*d)]

    @test_int [sqrt(-cos(c + d*x))/sqrt(-3 + 2*cos(c + d*x)), x, 1, (3*cot(c + d*x)*Elliptic.Pi(-(1/2), asin(sqrt(-3 + 2*cos(c + d*x))/sqrt(-cos(c + d*x))), -(1/5))*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/(sqrt(5)*d)]
    @test_int [sqrt(-cos(c + d*x))/sqrt(-3 - 2*cos(c + d*x)), x, 1, -((3*cot(c + d*x)*Elliptic.Pi(5/2, asin(sqrt(-3 - 2*cos(c + d*x))/(sqrt(5)*sqrt(-cos(c + d*x)))), -5)*sqrt(1 - sec(c + d*x))*sqrt(1 + sec(c + d*x)))/d)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/3)*(a+b*cos(e+f*x))^n=#


    #= ::Subsubsection:: =#
    #=n>0=#


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(2/3)/(a + b*cos(c + d*x)), x, 5, -((b*AppellF1(1/2, -(1/3), 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*cos(c + d*x)^(2/3)*sin(c + d*x))/((a^2 - b^2)*d*(cos(c + d*x)^2)^(1/3))) + (a*AppellF1(1/2, 1/6, 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*(cos(c + d*x)^2)^(1/6)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(1/3))]
    @test_int [cos(c + d*x)^(1/3)/(a + b*cos(c + d*x)), x, 5, -((b*AppellF1(1/2, -(1/6), 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*cos(c + d*x)^(1/3)*sin(c + d*x))/((a^2 - b^2)*d*(cos(c + d*x)^2)^(1/6))) + (a*AppellF1(1/2, 1/3, 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*(cos(c + d*x)^2)^(1/3)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(2/3))]
    @test_int [cos(c + d*x)^(-1/3)/(a + b*cos(c + d*x)), x, 5, -((b*AppellF1(1/2, 1/6, 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*(cos(c + d*x)^2)^(1/6)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(1/3))) + (a*AppellF1(1/2, 2/3, 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*(cos(c + d*x)^2)^(2/3)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(4/3))]
    @test_int [cos(c + d*x)^(-2/3)/(a + b*cos(c + d*x)), x, 5, -((b*AppellF1(1/2, 1/3, 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*(cos(c + d*x)^2)^(1/3)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(2/3))) + (a*AppellF1(1/2, 5/6, 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*(cos(c + d*x)^2)^(5/6)*sin(c + d*x))/((a^2 - b^2)*d*cos(c + d*x)^(5/3))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^(m/3)*(a+b*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection:: =#
    #=n>0=#


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(7/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(cos(c + d*x)^(7/3)/sqrt(a + b*cos(c + d*x)), x)]
    @test_int [cos(c + d*x)^(5/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(cos(c + d*x)^(5/3)/sqrt(a + b*cos(c + d*x)), x)]
    @test_int [cos(c + d*x)^(4/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(cos(c + d*x)^(4/3)/sqrt(a + b*cos(c + d*x)), x)]
    @test_int [cos(c + d*x)^(2/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(cos(c + d*x)^(2/3)/sqrt(a + b*cos(c + d*x)), x)]
    @test_int [cos(c + d*x)^(1/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(cos(c + d*x)^(1/3)/sqrt(a + b*cos(c + d*x)), x)]
    @test_int [cos(c + d*x)^(-1/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(1/(cos(c + d*x)^(1/3)*sqrt(a + b*cos(c + d*x))), x)]
    @test_int [cos(c + d*x)^(-2/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(1/(cos(c + d*x)^(2/3)*sqrt(a + b*cos(c + d*x))), x)]
    @test_int [cos(c + d*x)^(-4/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(1/(cos(c + d*x)^(4/3)*sqrt(a + b*cos(c + d*x))), x)]
    @test_int [cos(c + d*x)^(-5/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(1/(cos(c + d*x)^(5/3)*sqrt(a + b*cos(c + d*x))), x)]
    @test_int [cos(c + d*x)^(-7/3)/sqrt(a + b*cos(c + d*x)), x, 0, Unintegrable(1/(cos(c + d*x)^(7/3)*sqrt(a + b*cos(c + d*x))), x)]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^(m/2)*(a+b*cos(e+f*x))^n=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [(A + B*cos(c + d*x))*sec(c + d*x)^(7/2), x, 9, (-6*A*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (6*A*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*B*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*A*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(A + B*cos(c + d*x))*sec(c + d*x)^(5/2), x, 8, (-2*B*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*A*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*A*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(A + B*cos(c + d*x))*sec(c + d*x)^(3/2), x, 7, (-2*A*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*A*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(A + B*cos(c + d*x))*sqrt(sec(c + d*x)), x, 6, (2*B*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*A*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d]
    @test_int [(A + B*cos(c + d*x))/sqrt(sec(c + d*x)), x, 7, (2*A*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/d + (2*B*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(A + B*cos(c + d*x))/sec(c + d*x)^(3/2), x, 8, (6*B*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(3*d) + (2*B*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*A*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [(A + B*cos(c + d*x))/sec(c + d*x)^(5/2), x, 9, (6*A*sqrt(cos(c + d*x))*Elliptic.E((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(5*d) + (10*B*sqrt(cos(c + d*x))*Elliptic.F((c + d*x)/2, 2)*sqrt(sec(c + d*x)))/(21*d) + (2*B*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*A*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (10*B*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(9/2)*(a + b*cos(c + d*x))^2, x, 10, -((12*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*(5*a^2 + 7*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (12*a*b*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*(5*a^2 + 7*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (4*a*b*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d) + (2*a^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^(7/2)*(a + b*cos(c + d*x))^2, x, 9, -((2*(3*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (4*a*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*(3*a^2 + 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (4*a*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d) + (2*a^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^(5/2)*(a + b*cos(c + d*x))^2, x, 8, -((4*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*(a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (4*a*b*sqrt(sec(c + d*x))*sin(c + d*x))/d + (2*a^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^2, x, 7, -((2*(a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (4*a*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^2, x, 7, (4*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b^2*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^2, x, 8, (2*(5*a^2 + 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (4*a*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*b^2*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (4*a*b*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^2, x, 9, (12*a*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*(7*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b^2*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (4*a*b*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2)) + (2*(7*a^2 + 5*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)*(a + b*cos(c + d*x))^2, x, 10, (2*(9*a^2 + 7*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (20*a*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b^2*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2)) + (4*a*b*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2)) + (2*(9*a^2 + 7*b^2)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (20*a*b*sin(c + d*x))/(21*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(9/2)*(a + b*cos(c + d*x))^3, x, 10, -((2*b*(9*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*a*(5*a^2 + 21*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (2*b*(9*a^2 + 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (2*a*(5*a^2 + 21*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (32*a^2*b*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a^2*sec(c + d*x)^(5/2)*(b + a*sec(c + d*x))*sin(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^(7/2)*(a + b*cos(c + d*x))^3, x, 9, -((6*a*(a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d)) + (2*b*(a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (6*a*(a^2 + 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(5*d) + (8*a^2*b*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a^2*sec(c + d*x)^(3/2)*(b + a*sec(c + d*x))*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^(5/2)*(a + b*cos(c + d*x))^3, x, 8, -((2*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*a*(a^2 + 9*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (16*a^2*b*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*a^2*sqrt(sec(c + d*x))*(b + a*sec(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^(3/2)*(a + b*cos(c + d*x))^3, x, 8, -((2*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d) + (2*b*(9*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*d) + (2*a*(3*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*d) + (2*b^2*(b + a*sec(c + d*x))*sin(c + d*x))/(3*d*sqrt(sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)*(a + b*cos(c + d*x))^3, x, 8, (6*b*(5*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*a*(a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/d + (8*a*b^2*sin(c + d*x))/(5*d*sqrt(sec(c + d*x))) + (2*b^2*(b + a*sec(c + d*x))*sin(c + d*x))/(5*d*sec(c + d*x)^(3/2))]
    @test_int [sec(c + d*x)^(-1/2)*(a + b*cos(c + d*x))^3, x, 9, (2*a*(5*a^2 + 9*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(5*d) + (2*b*(21*a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (32*a*b^2*sin(c + d*x))/(35*d*sec(c + d*x)^(3/2)) + (2*b*(21*a^2 + 5*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*b^2*(b + a*sec(c + d*x))*sin(c + d*x))/(7*d*sec(c + d*x)^(5/2))]
    @test_int [sec(c + d*x)^(-3/2)*(a + b*cos(c + d*x))^3, x, 10, (2*b*(27*a^2 + 7*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(15*d) + (2*a*(7*a^2 + 15*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(21*d) + (40*a*b^2*sin(c + d*x))/(63*d*sec(c + d*x)^(5/2)) + (2*b*(27*a^2 + 7*b^2)*sin(c + d*x))/(45*d*sec(c + d*x)^(3/2)) + (2*a*(7*a^2 + 15*b^2)*sin(c + d*x))/(21*d*sqrt(sec(c + d*x))) + (2*b^2*(b + a*sec(c + d*x))*sin(c + d*x))/(9*d*sec(c + d*x)^(7/2))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [sec(c + d*x)^(5/2)/(a + b*cos(c + d*x)), x, 11, (2*b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*d) + (2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a*d) + (2*b^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a + b)*d) - (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*d) + (2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d)]
    @test_int [sec(c + d*x)^(3/2)/(a + b*cos(c + d*x)), x, 7, -((2*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*d)) - (2*b*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a + b)*d) + (2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*d)]
    @test_int [sec(c + d*x)^(1/2)/(a + b*cos(c + d*x)), x, 3, (2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a + b)*d)]
    @test_int [sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x)), x, 5, (2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*d) - (2*a*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*(a + b)*d)]
    @test_int [sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x)), x, 9, (2*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*d) - (2*a*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*d) + (2*a^2*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a + b)*d)]
    @test_int [sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x)), x, 10, -((2*a*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*d)) + (2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*b^3*d) - (2*a^3*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^3*(a + b)*d) + (2*sin(c + d*x))/(3*b*d*sqrt(sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^2, x, 12, (b*(4*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(3*a^2*(a^2 - b^2)*d) + (b^2*(7*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^3*(a - b)*(a + b)^2*d) - (b*(4*a^2 - 5*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^3*(a^2 - b^2)*d) + ((2*a^2 - 5*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^2, x, 11, -(((2*a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a^2 - b^2)*d)) + (b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d) - (b*(5*a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a^2*(a - b)*(a + b)^2*d) + ((2*a^2 - 3*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(a^2*(a^2 - b^2)*d) + (b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^2, x, 10, -((b*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a^2 - b^2)*d)) - (sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a^2 - b^2)*d) + ((3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(a*(a - b)*(a + b)^2*d) + (b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^2, x, 10, (sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a^2 - b^2)*d) + (a*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d) - ((a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b*(a + b)^2*d) - (b*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^2, x, 10, -((a*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b*(a^2 - b^2)*d)) + ((a^2 - 2*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) - (a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b^2*(a + b)^2*d) + (a*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^2, x, 10, ((3*a^2 - 2*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^2*(a^2 - b^2)*d) - (a*(3*a^2 - 4*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(b^3*(a^2 - b^2)*d) + (a^2*(3*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/((a - b)*b^3*(a + b)^2*d) - (a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*(b + a*sec(c + d*x)))]


    @test_int [sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^3, x, 13, (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(12*a^3*(a^2 - b^2)^2*d) + (b^2*(63*a^4 - 86*a^2*b^2 + 35*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^4*(a - b)^2*(a + b)^3*d) - (b*(24*a^4 - 65*a^2*b^2 + 35*b^4)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^4*(a^2 - b^2)^2*d) + ((8*a^4 - 61*a^2*b^2 + 35*b^4)*sec(c + d*x)^(3/2)*sin(c + d*x))/(12*a^3*(a^2 - b^2)^2*d) + (b^2*sec(c + d*x)^(7/2)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(b + a*sec(c + d*x))^2) + (b^2*(13*a^2 - 7*b^2)*sec(c + d*x)^(5/2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^3, x, 12, -(((8*a^4 - 29*a^2*b^2 + 15*b^4)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a^2 - b^2)^2*d)) + (b*(11*a^2 - 5*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d) - (b*(35*a^4 - 38*a^2*b^2 + 15*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^3*(a - b)^2*(a + b)^3*d) + ((8*a^4 - 29*a^2*b^2 + 15*b^4)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^3*(a^2 - b^2)^2*d) + (b^2*sec(c + d*x)^(5/2)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(b + a*sec(c + d*x))^2) + (b^2*(11*a^2 - 5*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(1/2)/(a + b*cos(c + d*x))^3, x, 11, -((3*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a^2 - b^2)^2*d)) - ((7*a^2 - b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a^2 - b^2)^2*d) + (3*(5*a^4 - 2*a^2*b^2 + b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a^2*(a - b)^2*(a + b)^3*d) + (b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(b + a*sec(c + d*x))^2) + (3*b^2*(3*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a^2*(a^2 - b^2)^2*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-1/2)/(a + b*cos(c + d*x))^3, x, 11, ((5*a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a^2 - b^2)^2*d) + (3*(a^2 + b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b*(a^2 - b^2)^2*d) - ((3*a^4 + 10*a^2*b^2 - b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*a*(a - b)^2*b*(a + b)^3*d) + (b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(2*a*(a^2 - b^2)*d*(b + a*sec(c + d*x))^2) - (b*(7*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*a*(a^2 - b^2)^2*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-3/2)/(a + b*cos(c + d*x))^3, x, 11, -(((a^2 + 5*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b*(a^2 - b^2)^2*d)) + (a*(a^2 - 7*b^2)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d) - ((a^4 - 10*a^2*b^2 - 3*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^2*(a + b)^3*d) - (b*sqrt(sec(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(b + a*sec(c + d*x))^2) + (3*(a^2 + b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*(a^2 - b^2)^2*d*(b + a*sec(c + d*x)))]
    @test_int [sec(c + d*x)^(-5/2)/(a + b*cos(c + d*x))^3, x, 11, -((3*a*(a^2 - 3*b^2)*sqrt(cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^2*(a^2 - b^2)^2*d)) + ((3*a^4 - 5*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*b^3*(a^2 - b^2)^2*d) - (3*a*(a^4 - 2*a^2*b^2 + 5*b^4)*sqrt(cos(c + d*x))*Elliptic.Pi((2*b)/(a + b), (1/2)*(c + d*x), 2)*sqrt(sec(c + d*x)))/(4*(a - b)^2*b^3*(a + b)^3*d) + (a*sqrt(sec(c + d*x))*sin(c + d*x))/(2*(a^2 - b^2)*d*(b + a*sec(c + d*x))^2) + (a*(a^2 - 7*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(4*b*(a^2 - b^2)^2*d*(b + a*sec(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*sec(e+f*x)^(m/2)*(a+b*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2), x, 6, (2*(a - b)*sqrt(a + b)*(9*a^2 - 2*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a^3*d*sqrt(sec(c + d*x))) - (2*(a - b)*sqrt(a + b)*(9*a + 2*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a^2*d*sqrt(sec(c + d*x))) + (2*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*a*d) + (2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2), x, 5, (2*(a - b)*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*d*sqrt(sec(c + d*x))) + (2*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*d*sqrt(sec(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2), x, 4, (2*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d*sqrt(sec(c + d*x))) - (2*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d*sqrt(sec(c + d*x)))]
    @test_int [sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x)), x, 2, -((1/(sqrt(a + b)*d))*(2*sqrt(cos(c + d*x))*sqrt((a*(1 - cos(c + d*x)))/(a + b*cos(c + d*x)))*sqrt((a*(1 + cos(c + d*x)))/(a + b*cos(c + d*x)))*(a + b*cos(c + d*x))*csc(c + d*x)*Elliptic.Pi(b/(a + b), asin((sqrt(a + b)*sqrt(cos(c + d*x)))/sqrt(a + b*cos(c + d*x))), -((a - b)/(a + b)))*sqrt(sec(c + d*x))))]
    @test_int [sqrt(a + b*cos(c + d*x))/sqrt(sec(c + d*x)), x, 8, -(((a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d*sqrt(sec(c + d*x)))) + (sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) - (a*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*d*sqrt(sec(c + d*x))) + (sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [sqrt(a + b*cos(c + d*x))/sec(c + d*x)^(3/2), x, 8, -((a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(a + 2*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(a^2 - 4*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b^2*d*sqrt(sec(c + d*x))) + (sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (a*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(4*b*d)]


    @test_int [(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(9/2), x, 7, (4*(a - b)*b*sqrt(a + b)*(41*a^2 - 3*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(105*a^3*d*sqrt(sec(c + d*x))) + (2*(a - b)*sqrt(a + b)*(25*a^2 - 57*a*b - 6*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(105*a^2*d*sqrt(sec(c + d*x))) + (2*(25*a^2 + 3*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(105*a*d) + (16*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(35*d) + (2*a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(7/2), x, 6, (2*(a - b)*sqrt(a + b)*(3*a^2 + b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(5*a^2*d*sqrt(sec(c + d*x))) - (2*(a - b)*(3*a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(5*a*d*sqrt(sec(c + d*x))) + (4*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(5*d) + (2*a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2), x, 5, (8*(a - b)*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*d*sqrt(sec(c + d*x))) + (2*(a - 3*b)*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*d*sqrt(sec(c + d*x))) + (2*a*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2), x, 6, (2*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) - (2*(a - 2*b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) - (2*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x)))]
    @test_int [(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x)), x, 7, -(((a - b)*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d*sqrt(sec(c + d*x)))) + (sqrt(a + b)*(2*a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) - (3*a*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) + (b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + b*cos(c + d*x))^(3/2)/sqrt(sec(c + d*x)), x, 9, (-5*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(5*a + 2*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d*sqrt(sec(c + d*x))) - (sqrt(a + b)*(3*a^2 + 4*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b*d*sqrt(sec(c + d*x))) + (3*a*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d) + ((a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [(a + b*cos(c + d*x))^(3/2)/sec(c + d*x)^(3/2), x, 9, -((a - b)*sqrt(a + b)*(3*a^2 + 16*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*a*b*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(a + 2*b)*(3*a + 8*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*b*d*sqrt(sec(c + d*x))) + (a*sqrt(a + b)*(a^2 - 12*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(8*b^2*d*sqrt(sec(c + d*x))) + (a*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sqrt(sec(c + d*x))) + ((a + b*cos(c + d*x))^(3/2)*sin(c + d*x))/(3*d*sqrt(sec(c + d*x))) + ((3*a^2 + 16*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(24*b*d)]


    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(11/2), x, 8, (2*(a - b)*sqrt(a + b)*(147*a^4 + 279*a^2*b^2 - 10*b^4)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(315*a^3*d*sqrt(sec(c + d*x))) - (2*(a - b)*sqrt(a + b)*(147*a^3 - 114*a^2*b + 165*a*b^2 + 10*b^3)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(315*a^2*d*sqrt(sec(c + d*x))) + (2*b*(163*a^2 + 5*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(315*a*d) + (2*(49*a^2 + 75*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(315*d) + (38*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(63*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(9/2)*sin(c + d*x))/(9*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(9/2), x, 7, (2*(a - b)*b*sqrt(a + b)*(29*a^2 + 3*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(21*a^2*d*sqrt(sec(c + d*x))) + (2*(a - b)*sqrt(a + b)*(5*a^2 - 24*a*b + 3*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(21*a*d*sqrt(sec(c + d*x))) + (2*(5*a^2 + 9*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(21*d) + (6*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(7*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(7/2)*sin(c + d*x))/(7*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(7/2), x, 6, (2*(a - b)*sqrt(a + b)*(9*a^2 + 23*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a*d*sqrt(sec(c + d*x))) - (2*(a - b)*sqrt(a + b)*(9*a^2 - 8*a*b + 15*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(15*a*d*sqrt(sec(c + d*x))) + (22*a*b*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(15*d) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)*sin(c + d*x))/(5*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2), x, 7, (14*(a - b)*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*d*sqrt(sec(c + d*x))) + (2*sqrt(a + b)*(a^2 - 7*a*b + 9*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*d*sqrt(sec(c + d*x))) - (2*b^2*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2), x, 8, ((a - b)*sqrt(a + b)*(2*a^2 - b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d*sqrt(sec(c + d*x))) - (sqrt(a + b)*(2*a^2 - 6*a*b - b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) - (5*a*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(d*sqrt(sec(c + d*x))) + (2*a^2*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d - ((2*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/d]
    @test_int [(a + b*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x)), x, 8, (-9*(a - b)*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(8*a^2 + 9*a*b + 2*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d*sqrt(sec(c + d*x))) - (sqrt(a + b)*(15*a^2 + 4*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*d*sqrt(sec(c + d*x))) + (b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*d*sqrt(sec(c + d*x))) + (9*a*b*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(4*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)/sqrt(sec(c + d*x)), x, 9, -((a - b)*sqrt(a + b)*(33*a^2 + 16*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*a*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(33*a^2 + 26*a*b + 16*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(24*d*sqrt(sec(c + d*x))) - (5*a*sqrt(a + b)*(a^2 + 4*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(8*b*d*sqrt(sec(c + d*x))) + (b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(3*d*sec(c + d*x)^(3/2)) + (13*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(12*d*sqrt(sec(c + d*x))) + ((33*a^2 + 16*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(24*d)]
    @test_int [(a + b*cos(c + d*x))^(5/2)/sec(c + d*x)^(3/2), x, 10, -((a - b)*sqrt(a + b)*(15*a^2 + 284*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(192*b*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(15*a^3 + 118*a^2*b + 284*a*b^2 + 72*b^3)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(192*b*d*sqrt(sec(c + d*x))) + (sqrt(a + b)*(5*a^4 - 120*a^2*b^2 - 48*b^4)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(64*b^2*d*sqrt(sec(c + d*x))) + (b^2*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(4*d*sec(c + d*x)^(5/2)) + (17*a*b*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(24*d*sec(c + d*x)^(3/2)) + ((59*a^2 + 36*b^2)*sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(96*d*sqrt(sec(c + d*x))) + (a*(15*a^2 + 284*b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(192*b*d)]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [sec(c + d*x)^(5/2)/sqrt(a + b*cos(c + d*x)), x, 5, (-4*(a - b)*b*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*d*sqrt(sec(c + d*x))) + (2*sqrt(a + b)*(a + 2*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*d*sqrt(sec(c + d*x))) + (2*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*d)]
    @test_int [sec(c + d*x)^(3/2)/sqrt(a + b*cos(c + d*x)), x, 4, (2*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^2*d*sqrt(sec(c + d*x))) - (2*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d*sqrt(sec(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/sqrt(a + b*cos(c + d*x)), x, 2, (2*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*d*sqrt(sec(c + d*x)))]
    @test_int [1/(sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))), x, 2, (-2*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*d*sqrt(sec(c + d*x)))]
    @test_int [1/(sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)), x, 9, -(((a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*b*d*sqrt(sec(c + d*x)))) + (sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*d*sqrt(sec(c + d*x))) + (a*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^2*d*sqrt(sec(c + d*x))) + sin(c + d*x)/(d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) + (a*sqrt(sec(c + d*x))*sin(c + d*x))/(b*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [1/(sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(5/2)), x, 8, (3*(a - b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b^2*d*sqrt(sec(c + d*x))) - ((3*a - 2*b)*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b^2*d*sqrt(sec(c + d*x))) - (sqrt(a + b)*(3*a^2 + 4*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(4*b^3*d*sqrt(sec(c + d*x))) + (sqrt(a + b*cos(c + d*x))*sin(c + d*x))/(2*b*d*sqrt(sec(c + d*x))) - (3*a*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(4*b^2*d)]


    @test_int [sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(3/2), x, 6, -((2*b*(5*a^2 - 8*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^4*sqrt(a + b)*d*sqrt(sec(c + d*x)))) + (2*(a + 2*b)*(a + 4*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*sqrt(a + b)*d*sqrt(sec(c + d*x))) + (2*b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))) + (2*(a^2 - 4*b^2)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)*d)]
    @test_int [sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(3/2), x, 5, (2*(a^2 - 2*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^3*sqrt(a + b)*d*sqrt(sec(c + d*x))) - (2*(a + 2*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^2*sqrt(a + b)*d*sqrt(sec(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(a*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/(a + b*cos(c + d*x))^(3/2), x, 5, (2*b*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a^2*sqrt(a + b)*d*sqrt(sec(c + d*x))) + (2*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*sqrt(a + b)*d*sqrt(sec(c + d*x))) - (2*b*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [1/((a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))), x, 5, -((2*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*sqrt(a + b)*d*sqrt(sec(c + d*x)))) + (2*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*sqrt(a + b)*d*sqrt(sec(c + d*x))) + (2*a*sqrt(sec(c + d*x))*sin(c + d*x))/((a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [1/((a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(3/2)), x, 7, (2*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*sqrt(a + b)*d*sqrt(sec(c + d*x))) - (2*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b*sqrt(a + b)*d*sqrt(sec(c + d*x))) - (2*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^2*d*sqrt(sec(c + d*x))) - (2*a^2*sqrt(sec(c + d*x))*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [1/((a + b*cos(c + d*x))^(3/2)*sec(c + d*x)^(5/2)), x, 8, -(((3*a^2 - b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(a*b^2*sqrt(a + b)*d*sqrt(sec(c + d*x)))) + ((3*a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^2*sqrt(a + b)*d*sqrt(sec(c + d*x))) + (3*a*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^3*d*sqrt(sec(c + d*x))) - (2*a^2*sin(c + d*x))/(b*(a^2 - b^2)*d*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))) + ((3*a^2 - b^2)*sqrt(a + b*cos(c + d*x))*sqrt(sec(c + d*x))*sin(c + d*x))/(b^2*(a^2 - b^2)*d)]


    @test_int [sec(c + d*x)^(5/2)/(a + b*cos(c + d*x))^(5/2), x, 7, -((8*b*(2*a^4 - 7*a^2*b^2 + 4*b^4)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^5*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x)))) + (2*(a^4 + 9*a^3*b + 16*a^2*b^2 - 12*a*b^3 - 16*b^4)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^4*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) + (2*b^2*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (4*b^2*(5*a^2 - 3*b^2)*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x))) + (2*(a^4 - 13*a^2*b^2 + 8*b^4)*sqrt(a + b*cos(c + d*x))*sec(c + d*x)^(3/2)*sin(c + d*x))/(3*a^3*(a^2 - b^2)^2*d)]
    @test_int [sec(c + d*x)^(3/2)/(a + b*cos(c + d*x))^(5/2), x, 6, (2*(3*a^4 - 15*a^2*b^2 + 8*b^4)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^4*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) - (2*(3*a^3 + 9*a^2*b - 6*a*b^2 - 8*b^3)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) + (2*b^2*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)) + (8*b^2*(2*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [sqrt(sec(c + d*x))/(a + b*cos(c + d*x))^(5/2), x, 6, (4*b*(3*a^2 - b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^3*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) + (2*(3*a^2 - 3*a*b - 2*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) + (2*b^2*sin(c + d*x))/(3*a*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (4*b*(3*a^2 - b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*a*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [1/((a + b*cos(c + d*x))^(5/2)*sqrt(sec(c + d*x))), x, 6, -((2*(3*a^2 + b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a^2*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x)))) + (2*(3*a - b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) - (2*b*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) + (2*(3*a^2 + b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [1/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(3/2)), x, 6, (8*b*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) + (2*(a - 3*b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*a*(a - b)*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) + (2*a*sin(c + d*x))/(3*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (8*a*b*sqrt(sec(c + d*x))*sin(c + d*x))/(3*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]
    @test_int [1/((a + b*cos(c + d*x))^(5/2)*sec(c + d*x)^(5/2)), x, 8, (2*(3*a^2 - 7*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.E(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*(a - b)*b^2*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) - (2*(3*a^2 + a*b - 6*b^2)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.F(asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(3*(a - b)*b^2*(a + b)^(3/2)*d*sqrt(sec(c + d*x))) - (2*sqrt(a + b)*sqrt(cos(c + d*x))*csc(c + d*x)*Elliptic.Pi((a + b)/b, asin(sqrt(a + b*cos(c + d*x))/(sqrt(a + b)*sqrt(cos(c + d*x)))), -((a + b)/(a - b)))*sqrt((a*(1 - sec(c + d*x)))/(a + b))*sqrt((a*(1 + sec(c + d*x)))/(a - b)))/(b^3*d*sqrt(sec(c + d*x))) - (2*a^2*sin(c + d*x))/(3*b*(a^2 - b^2)*d*(a + b*cos(c + d*x))^(3/2)*sqrt(sec(c + d*x))) - (2*a^2*(3*a^2 - 7*b^2)*sqrt(sec(c + d*x))*sin(c + d*x))/(3*b^2*(a^2 - b^2)^2*d*sqrt(a + b*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*cos(e+f*x)^m*(a+b*cos(e+f*x))^n*with*m*and/or*n*symbolic=#


    @test_int [cos(c + d*x)^m*(a + b*cos(c + d*x))^4, x, 6, If(13>=8, (b^2*(b^2*(3 + m) + a^2*(22 + 5*m))*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m)*(4 + m)) + (2*a*b^3*(5 + m)*cos(c + d*x)^(2 + m)*sin(c + d*x))/(d*(3 + m)*(4 + m)) + (b^2*cos(c + d*x)^(1 + m)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*(4 + m)) - ((b^4*(3 + 4*m + m^2) + 6*a^2*b^2*(4 + 5*m + m^2) + a^4*(8 + 6*m + m^2))*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*(2 + m)*(4 + m)*sqrt(sin(c + d*x)^2)) - (4*a*b*(b^2*(2 + m) + a^2*(3 + m))*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*(3 + m)*sqrt(sin(c + d*x)^2)), (b^2*(b^2*(3 + m) + a^2*(22 + 5*m))*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m)*(4 + m)) + (2*a*b^3*(5 + m)*cos(c + d*x)^(2 + m)*sin(c + d*x))/(d*(3 + m)*(4 + m)) + (b^2*cos(c + d*x)^(1 + m)*(a + b*cos(c + d*x))^2*sin(c + d*x))/(d*(4 + m)) - ((b^4*(3 + 4*m + m^2) + 6*a^2*b^2*(4 + 5*m + m^2) + a^4*(8 + 6*m + m^2))*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(4 + m)*(2 + 3*m + m^2)*sqrt(sin(c + d*x)^2)) - (4*a*b*(b^2*(2 + m) + a^2*(3 + m))*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*(3 + m)*sqrt(sin(c + d*x)^2)))]
    @test_int [cos(c + d*x)^m*(a + b*cos(c + d*x))^3, x, 5, (a*b^2*(7 + 2*m)*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m)*(3 + m)) + (b^2*cos(c + d*x)^(1 + m)*(a + b*cos(c + d*x))*sin(c + d*x))/(d*(3 + m)) - (a*(3*b^2*(1 + m) + a^2*(2 + m))*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*(2 + m)*sqrt(sin(c + d*x)^2)) - (b*(b^2*(2 + m) + 3*a^2*(3 + m))*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*(3 + m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m*(a + b*cos(c + d*x))^2, x, 4, (b^2*cos(c + d*x)^(1 + m)*sin(c + d*x))/(d*(2 + m)) - ((b^2*(1 + m) + a^2*(2 + m))*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*(2 + m)*sqrt(sin(c + d*x)^2)) - (2*a*b*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m*(a + b*cos(c + d*x))^1, x, 3, -((a*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m)*sqrt(sin(c + d*x)^2))) - (b*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m/(a + b*cos(c + d*x))^1, x, 5, (a*AppellF1(1/2, (1 - m)/2, 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*cos(c + d*x)^(-1 + m)*(cos(c + d*x)^2)^((1 - m)/2)*sin(c + d*x))/((a^2 - b^2)*d) - (b*AppellF1(1/2, -(m/2), 1, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*cos(c + d*x)^m*sin(c + d*x))/((cos(c + d*x)^2)^(m/2)*((a^2 - b^2)*d))]
    @test_int [cos(c + d*x)^m/(a + b*cos(c + d*x))^2, x, 8, (b^2*AppellF1(1/2, (1/2)*(-1 - m), 2, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*cos(c + d*x)^(1 + m)*(cos(c + d*x)^2)^((1/2)*(-1 - m))*sin(c + d*x))/((a^2 - b^2)^2*d) + (a^2*AppellF1(1/2, (1 - m)/2, 2, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*cos(c + d*x)^(-1 + m)*(cos(c + d*x)^2)^((1 - m)/2)*sin(c + d*x))/((a^2 - b^2)^2*d) - (2*a*b*AppellF1(1/2, -(m/2), 2, 3/2, sin(c + d*x)^2, -((b^2*sin(c + d*x)^2)/(a^2 - b^2)))*cos(c + d*x)^m*sin(c + d*x))/((cos(c + d*x)^2)^(m/2)*((a^2 - b^2)^2*d))]


    @test_int [sec(c + d*x)^m*(a + b*cos(c + d*x))^3, x, 8, If(13>=8, -((a^2*b*(1 - 2*m)*sec(c + d*x)^(-2 + m)*sin(c + d*x))/(d*(1 - m)*(2 - m))) - (a^2*sec(c + d*x)^(-2 + m)*(b + a*sec(c + d*x))*sin(c + d*x))/(d*(1 - m)) - (b*(b^2*(2 - m) + 3*a^2*(3 - m))*HypergeometricFunctions._₂F₁(1/2, (4 - m)/2, (6 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-4 + m)*sin(c + d*x))/(d*(2 - m)*(4 - m)*sqrt(sin(c + d*x)^2)) - (a*(3*b^2*(1 - m) + a^2*(2 - m))*HypergeometricFunctions._₂F₁(1/2, (3 - m)/2, (5 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-3 + m)*sin(c + d*x))/(d*(1 - m)*(3 - m)*sqrt(sin(c + d*x)^2)), -((a^2*b*(1 - 2*m)*sec(c + d*x)^(-2 + m)*sin(c + d*x))/(d*(2 - 3*m + m^2))) - (a^2*sec(c + d*x)^(-2 + m)*(b + a*sec(c + d*x))*sin(c + d*x))/(d*(1 - m)) - (b*(b^2*(2 - m) + 3*a^2*(3 - m))*HypergeometricFunctions._₂F₁(1/2, (4 - m)/2, (6 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-4 + m)*sin(c + d*x))/(d*(8 - 6*m + m^2)*sqrt(sin(c + d*x)^2)) - (a*(3*b^2*(1 - m) + a^2*(2 - m))*HypergeometricFunctions._₂F₁(1/2, (3 - m)/2, (5 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-3 + m)*sin(c + d*x))/(d*(3 - 4*m + m^2)*sqrt(sin(c + d*x)^2)))]
    @test_int [sec(c + d*x)^m*(a + b*cos(c + d*x))^2, x, 7, If(13>=8, -((a^2*sec(c + d*x)^(-1 + m)*sin(c + d*x))/(d*(1 - m))) - ((b^2*(1 - m) + a^2*(2 - m))*HypergeometricFunctions._₂F₁(1/2, (3 - m)/2, (5 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-3 + m)*sin(c + d*x))/(d*(1 - m)*(3 - m)*sqrt(sin(c + d*x)^2)) - (2*a*b*HypergeometricFunctions._₂F₁(1/2, (2 - m)/2, (4 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-2 + m)*sin(c + d*x))/(d*(2 - m)*sqrt(sin(c + d*x)^2)), -((a^2*sec(c + d*x)^(-1 + m)*sin(c + d*x))/(d*(1 - m))) - ((b^2*(1 - m) + a^2*(2 - m))*HypergeometricFunctions._₂F₁(1/2, (3 - m)/2, (5 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-3 + m)*sin(c + d*x))/(d*(3 - 4*m + m^2)*sqrt(sin(c + d*x)^2)) - (2*a*b*HypergeometricFunctions._₂F₁(1/2, (2 - m)/2, (4 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-2 + m)*sin(c + d*x))/(d*(2 - m)*sqrt(sin(c + d*x)^2)))]
    @test_int [sec(c + d*x)^m*(a + b*cos(c + d*x))^1, x, 6, -((b*HypergeometricFunctions._₂F₁(1/2, (2 - m)/2, (4 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-2 + m)*sin(c + d*x))/(d*(2 - m)*sqrt(sin(c + d*x)^2))) - (a*HypergeometricFunctions._₂F₁(1/2, (1 - m)/2, (3 - m)/2, cos(c + d*x)^2)*sec(c + d*x)^(-1 + m)*sin(c + d*x))/(d*(1 - m)*sqrt(sin(c + d*x)^2))]


    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*cos(e+f*x))^m*(c+d*cos(e+f*x))^n=#


    #= ::Section:: =#
    #=Integrands*of*the*form*(a+a*cos(e+f*x))^m*(c-c*cos(e+f*x))^n=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+a*cos(e+f*x))^m*(c+d*cos(e+f*x))^n=#


    @test_int [sqrt(1 - cos(x))/sqrt(a - cos(x)), x, 2, -2*atan(sin(x)/(sqrt(1 - cos(x))*sqrt(a - cos(x))))]
    @test_int [sqrt((1 - cos(x))/(a - cos(x))), x, 3, -((2*atan(sin(x)/(sqrt(1 - cos(x))*sqrt(a - cos(x))))*sqrt((1 - cos(x))/(a - cos(x)))*sqrt(a - cos(x)))/sqrt(1 - cos(x)))]


    @test_int [(-B*(1/(1 + 1)) + B*cos(c + d*x))*(a + a*cos(c + d*x)), x, 1, (a*B*sin(c + d*x))/(2*d) + (a*B*cos(c + d*x)*sin(c + d*x))/(2*d)]
    @test_int [(-B*(4/(4 + 1)) + B*cos(c + d*x))*(a + a*cos(c + d*x))^4, x, 1, (B*(a + a*cos(c + d*x))^4*sin(c + d*x))/(5*d)]
    @test_int [(-B*(n/(n + 1)) + B*cos(c + d*x))*(a + a*cos(c + d*x))^n, x, 1, (B*(a + a*cos(c + d*x))^n*sin(c + d*x))/(d*(1 + n))]


    @test_int [(-B*(-3/(-3 + 1)) + B*cos(c + d*x))/(a + a*cos(c + d*x))^3, x, 1, -((B*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^3))]


    @test_int [(-B*(3/2/(3/2 + 1)) + B*cos(c + d*x))*(a + a*cos(c + d*x))^(3/2), x, 1, (2*B*(a + a*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]


    @test_int [(-B*(-1/2/(-1/2 + 1)) + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/2), x, 2, (2*B*sin(c + d*x))/(d*sqrt(a + a*cos(c + d*x)))]
    @test_int [(-B*(-5/2/(-5/2 + 1)) + B*cos(c + d*x))/(a + a*cos(c + d*x))^(5/2), x, 1, -((2*B*sin(c + d*x))/(3*d*(a + a*cos(c + d*x))^(5/2)))]


    @test_int [(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(2/3), x, 3, (3*B*(a + a*cos(c + d*x))^(2/3)*sin(c + d*x))/(5*d) + (2*2^(1/6)*(5*A + 2*B)*(a + a*cos(c + d*x))^(2/3)*HypergeometricFunctions._₂F₁(-(1/6), 1/2, 3/2, (1/2)*(1 - cos(c + d*x)))*sin(c + d*x))/(5*d*(1 + cos(c + d*x))^(7/6))]
    @test_int [(A + B*cos(c + d*x))*(a + a*cos(c + d*x))^(1/3), x, 3, (3*B*(a + a*cos(c + d*x))^(1/3)*sin(c + d*x))/(4*d) + ((4*A + B)*(a + a*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 3/2, (1/2)*(1 - cos(c + d*x)))*sin(c + d*x))/(2*2^(1/6)*d*(1 + cos(c + d*x))^(5/6))]


    @test_int [(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(1/3), x, 3, (3*B*sin(c + d*x))/(2*d*(a + a*cos(c + d*x))^(1/3)) + ((2*A - B)*HypergeometricFunctions._₂F₁(1/2, 5/6, 3/2, (1/2)*(1 - cos(c + d*x)))*sin(c + d*x))/(2^(5/6)*d*(1 + cos(c + d*x))^(1/6)*(a + a*cos(c + d*x))^(1/3))]
    @test_int [(A + B*cos(c + d*x))/(a + a*cos(c + d*x))^(2/3), x, 3, (3*(A - B)*sin(c + d*x))/(d*(a + a*cos(c + d*x))^(2/3)) - (2^(5/6)*(A - 2*B)*(a + a*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 3/2, (1/2)*(1 - cos(c + d*x)))*sin(c + d*x))/(a*d*(1 + cos(c + d*x))^(5/6))]


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*cos(e+f*x))^m*(c+d*cos(e+f*x))^n=#


    @test_int [(b*B/a + B*cos(c + d*x))/(a + b*cos(c + d*x)), x, 3, (B*x)/b - (2*sqrt(a - b)*sqrt(a + b)*B*atan((sqrt(a - b)*tan((1/2)*(c + d*x)))/sqrt(a + b)))/(a*b*d)]
    @test_int [(a + b*cos(c + d*x))/(b + a*cos(c + d*x))^2, x, 2, sin(c + d*x)/(d*(b + a*cos(c + d*x)))]
    @test_int [(3 + cos(c + d*x))/(2 - cos(c + d*x)), x, 2, -x + (5*x)/sqrt(3) + (10*atan(sin(c + d*x)/(2 + sqrt(3) - cos(c + d*x))))/(sqrt(3)*d)]


    @test_int [(a*B + b*B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/2), x, 3, (2*B*sqrt(a + b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), (2*b)/(a + b)))/(d*sqrt((a + b*cos(c + d*x))/(a + b)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(a+b*cos(e+f*x))^(n/3)*(c+d*cos(e+f*x))^n=#
    #==#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(2/3), x, 7, (sqrt(2)*(a + b)*B*AppellF1(1/2, 1/2, -(5/3), 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*(a + b*cos(c + d*x))^(2/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*((a + b*cos(c + d*x))/(a + b))^(2/3)) + (sqrt(2)*(A*b - a*B)*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*(a + b*cos(c + d*x))^(2/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*((a + b*cos(c + d*x))/(a + b))^(2/3))]
    @test_int [(A + B*cos(c + d*x))*(a + b*cos(c + d*x))^(1/3), x, 7, (sqrt(2)*(a + b)*B*AppellF1(1/2, 1/2, -(4/3), 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*(a + b*cos(c + d*x))^(1/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*((a + b*cos(c + d*x))/(a + b))^(1/3)) + (sqrt(2)*(A*b - a*B)*AppellF1(1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*(a + b*cos(c + d*x))^(1/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*((a + b*cos(c + d*x))/(a + b))^(1/3))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(1/3), x, 7, (sqrt(2)*B*AppellF1(1/2, 1/2, -(2/3), 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*(a + b*cos(c + d*x))^(2/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*((a + b*cos(c + d*x))/(a + b))^(2/3)) + (sqrt(2)*(A*b - a*B)*AppellF1(1/2, 1/2, 1/3, 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*((a + b*cos(c + d*x))/(a + b))^(1/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*(a + b*cos(c + d*x))^(1/3))]
    @test_int [(A + B*cos(c + d*x))/(a + b*cos(c + d*x))^(2/3), x, 7, (sqrt(2)*B*AppellF1(1/2, 1/2, -(1/3), 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*(a + b*cos(c + d*x))^(1/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*((a + b*cos(c + d*x))/(a + b))^(1/3)) + (sqrt(2)*(A*b - a*B)*AppellF1(1/2, 1/2, 2/3, 3/2, (1/2)*(1 - cos(c + d*x)), (b*(1 - cos(c + d*x)))/(a + b))*((a + b*cos(c + d*x))/(a + b))^(2/3)*sin(c + d*x))/(b*d*sqrt(1 + cos(c + d*x))*(a + b*cos(c + d*x))^(2/3))]


    #= ::Title::Closed:: =#
    #=Integrands*of*the*form*(a+b*cos(e+f*x))^m*(c*cos(e+f*x))^n*(d*cos(e+f*x))^p=#


    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(a+b*cos(e+f*x))*(c*cos(e+f*x))^m*(d*cos(e+f*x))^n=#


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the  form*(a+b*cos(e+f*x))*(c*cos(e+f*x))^m*(d*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^2*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 9, (6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x))) + (10*b*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*sqrt(b*cos(c + d*x))) + (10*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*A*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d) + (2*B*(b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b^2*d)]
    @test_int [cos(c + d*x)^1*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 8, (6*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x))) + (2*A*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*d) + (2*B*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b*d)]
    @test_int [cos(c + d*x)^0*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 6, (2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x))) + (2*b*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^1*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 6, (2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x))) + (2*A*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 7, -((2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x)))) + (2*b*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x))) + (2*A*b*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 8, -((2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x)))) + (2*A*b*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^2*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (2*b*B*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 9, -((6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x)))) + (2*b*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^3*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/2)) + (2*b^2*B*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (6*A*b*sin(c + d*x))/(5*d*sqrt(b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^1*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 9, (6*A*b*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x))) + (10*b^2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*sqrt(b*cos(c + d*x))) + (10*b*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*A*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d) + (2*B*(b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b*d)]
    @test_int [cos(c + d*x)^0*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 7, (6*b*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x))) + (2*A*b^2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*d) + (2*B*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^1*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 7, (2*A*b*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x))) + (2*b^2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*b*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^2*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 6, (2*b*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x))) + (2*A*b^2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 7, -((2*A*b*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x)))) + (2*b^2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x))) + (2*A*b^2*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 8, -((2*b*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x)))) + (2*A*b^2*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^3*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (2*b^2*B*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^5*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 9, -((6*A*b*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x)))) + (2*b^2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^4*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/2)) + (2*b^3*B*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (6*A*b^2*sin(c + d*x))/(5*d*sqrt(b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^0*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 8, (6*A*b^2*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x))) + (10*b^3*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*sqrt(b*cos(c + d*x))) + (10*b^2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(21*d) + (2*A*b*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d) + (2*B*(b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*d)]
    @test_int [sec(c + d*x)^1*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 8, (6*b^2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x))) + (2*A*b^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*d) + (2*b*B*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*d)]
    @test_int [sec(c + d*x)^2*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 7, (2*A*b^2*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x))) + (2*b^3*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*b^2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*d)]
    @test_int [sec(c + d*x)^3*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 6, (2*b^2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x))) + (2*A*b^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^4*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 7, -((2*A*b^2*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x)))) + (2*b^3*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x))) + (2*A*b^3*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^5*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 8, -((2*b^2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(d*sqrt(cos(c + d*x)))) + (2*A*b^3*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^4*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (2*b^3*B*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^6*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 9, -((6*A*b^2*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*d*sqrt(cos(c + d*x)))) + (2*b^3*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^5*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/2)) + (2*b^4*B*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (6*A*b^3*sin(c + d*x))/(5*d*sqrt(b*cos(c + d*x)))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^3*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 9, (6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b*d*sqrt(cos(c + d*x))) + (10*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*d*sqrt(b*cos(c + d*x))) + (10*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(21*b*d) + (2*A*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b^2*d) + (2*B*(b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b^3*d)]
    @test_int [cos(c + d*x)^2*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 8, (6*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*b*d) + (2*B*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b^2*d)]
    @test_int [cos(c + d*x)^1*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 7, (2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b*d*sqrt(cos(c + d*x))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*b*d)]
    @test_int [cos(c + d*x)^0*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 5, (2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 7, -((2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b*d*sqrt(cos(c + d*x)))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(d*sqrt(b*cos(c + d*x))) + (2*A*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 8, -((2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b*d*sqrt(cos(c + d*x)))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (2*B*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^3*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 9, -((6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b*d*sqrt(cos(c + d*x)))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*d*sqrt(b*cos(c + d*x))) + (2*A*b^2*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/2)) + (2*b*B*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (6*A*sin(c + d*x))/(5*d*sqrt(b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^4*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 9, (6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b^2*d*sqrt(cos(c + d*x))) + (10*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*b*d*sqrt(b*cos(c + d*x))) + (10*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(21*b^2*d) + (2*A*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b^3*d) + (2*B*(b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b^4*d)]
    @test_int [cos(c + d*x)^3*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 8, (6*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b^2*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b*d*sqrt(b*cos(c + d*x))) + (2*A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*b^2*d) + (2*B*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b^3*d)]
    @test_int [cos(c + d*x)^2*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 7, (2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^2*d*sqrt(cos(c + d*x))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b*d*sqrt(b*cos(c + d*x))) + (2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*b^2*d)]
    @test_int [cos(c + d*x)^1*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 6, (2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^2*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(b*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 6, -((2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^2*d*sqrt(cos(c + d*x)))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(b*d*sqrt(b*cos(c + d*x))) + (2*A*sin(c + d*x))/(b*d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 8, -((2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^2*d*sqrt(cos(c + d*x)))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b*d*sqrt(b*cos(c + d*x))) + (2*A*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (2*B*sin(c + d*x))/(b*d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^2*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 9, -((6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b^2*d*sqrt(cos(c + d*x)))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b*d*sqrt(b*cos(c + d*x))) + (2*A*b*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/2)) + (2*B*sin(c + d*x))/(3*d*(b*cos(c + d*x))^(3/2)) + (6*A*sin(c + d*x))/(5*b*d*sqrt(b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^5*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 9, (6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b^3*d*sqrt(cos(c + d*x))) + (10*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(21*b^2*d*sqrt(b*cos(c + d*x))) + (10*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(21*b^3*d) + (2*A*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b^4*d) + (2*B*(b*cos(c + d*x))^(5/2)*sin(c + d*x))/(7*b^5*d)]
    @test_int [cos(c + d*x)^4*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 8, (6*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b^3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b^2*d*sqrt(b*cos(c + d*x))) + (2*A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*b^3*d) + (2*B*(b*cos(c + d*x))^(3/2)*sin(c + d*x))/(5*b^4*d)]
    @test_int [cos(c + d*x)^3*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 7, (2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^3*d*sqrt(cos(c + d*x))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b^2*d*sqrt(b*cos(c + d*x))) + (2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(3*b^3*d)]
    @test_int [cos(c + d*x)^2*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 6, (2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^3*d*sqrt(cos(c + d*x))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(b^2*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^1*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 7, -((2*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^3*d*sqrt(cos(c + d*x)))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(b^2*d*sqrt(b*cos(c + d*x))) + (2*A*sin(c + d*x))/(b^2*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^0*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 7, -((2*B*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(b^3*d*sqrt(cos(c + d*x)))) + (2*A*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b^2*d*sqrt(b*cos(c + d*x))) + (2*A*sin(c + d*x))/(3*b*d*(b*cos(c + d*x))^(3/2)) + (2*B*sin(c + d*x))/(b^2*d*sqrt(b*cos(c + d*x)))]
    @test_int [sec(c + d*x)^1*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 9, -((6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b^3*d*sqrt(cos(c + d*x)))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b^2*d*sqrt(b*cos(c + d*x))) + (2*A*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/2)) + (2*B*sin(c + d*x))/(3*b*d*(b*cos(c + d*x))^(3/2)) + (6*A*sin(c + d*x))/(5*b^2*d*sqrt(b*cos(c + d*x)))]


    @test_int [(A + B*cos(c + d*x))/(b*cos(c + d*x))^(7/2), x, 8, -((6*A*sqrt(b*cos(c + d*x))*Elliptic.E((1/2)*(c + d*x), 2))/(5*b^4*d*sqrt(cos(c + d*x)))) + (2*B*sqrt(cos(c + d*x))*Elliptic.F((1/2)*(c + d*x), 2))/(3*b^3*d*sqrt(b*cos(c + d*x))) + (2*A*sin(c + d*x))/(5*b*d*(b*cos(c + d*x))^(5/2)) + (2*B*sin(c + d*x))/(3*b^2*d*(b*cos(c + d*x))^(3/2)) + (6*A*sin(c + d*x))/(5*b^3*d*sqrt(b*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the  form*(a+b*cos(e+f*x))*(c*cos(e+f*x))^(m/2)*(d*cos(e+f*x))^(n/2)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 7, (3*B*x*sqrt(b*cos(c + d*x)))/(8*sqrt(cos(c + d*x))) + (A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (3*B*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(8*d) + (B*cos(c + d*x)^(5/2)*sqrt(b*cos(c + d*x))*sin(c + d*x))/(4*d) - (A*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 6, (A*x*sqrt(b*cos(c + d*x)))/(2*sqrt(cos(c + d*x))) + (B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (A*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d) - (B*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 2, (B*x*sqrt(b*cos(c + d*x)))/(2*sqrt(cos(c + d*x))) + (A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 3, (A*x*sqrt(b*cos(c + d*x)))/sqrt(cos(c + d*x)) + (B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 3, (B*x*sqrt(b*cos(c + d*x)))/sqrt(cos(c + d*x)) + (A*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 5, (B*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(d*sqrt(cos(c + d*x))) + (A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 6, (A*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(2*d*sqrt(cos(c + d*x))) + (A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)) + (B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(1/2), x, 6, (B*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(2*d*sqrt(cos(c + d*x))) + (B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)) + (A*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2)) + (A*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*cos(c + d*x)^(7/2))]


    @test_int [cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 7, (3*b*B*x*sqrt(b*cos(c + d*x)))/(8*sqrt(cos(c + d*x))) + (A*b*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (3*b*B*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(8*d) + (b*B*cos(c + d*x)^(5/2)*sqrt(b*cos(c + d*x))*sin(c + d*x))/(4*d) - (A*b*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 6, (A*b*x*sqrt(b*cos(c + d*x)))/(2*sqrt(cos(c + d*x))) + (b*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (A*b*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d) - (b*B*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 2, (b*B*x*sqrt(b*cos(c + d*x)))/(2*sqrt(cos(c + d*x))) + (A*b*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (b*B*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 3, (A*b*x*sqrt(b*cos(c + d*x)))/sqrt(cos(c + d*x)) + (b*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 3, (b*B*x*sqrt(b*cos(c + d*x)))/sqrt(cos(c + d*x)) + (A*b*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 5, (b*B*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(d*sqrt(cos(c + d*x))) + (A*b*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 6, (A*b*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(2*d*sqrt(cos(c + d*x))) + (A*b*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)) + (b*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(3/2), x, 6, (b*B*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(2*d*sqrt(cos(c + d*x))) + (b*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)) + (A*b*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2)) + (A*b*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*cos(c + d*x)^(7/2))]


    @test_int [cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 7, (3*b^2*B*x*sqrt(b*cos(c + d*x)))/(8*sqrt(cos(c + d*x))) + (A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (3*b^2*B*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(8*d) + (b^2*B*cos(c + d*x)^(5/2)*sqrt(b*cos(c + d*x))*sin(c + d*x))/(4*d) - (A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 6, (A*b^2*x*sqrt(b*cos(c + d*x)))/(2*sqrt(cos(c + d*x))) + (b^2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (A*b^2*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d) - (b^2*B*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 2, (b^2*B*x*sqrt(b*cos(c + d*x)))/(2*sqrt(cos(c + d*x))) + (A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x))) + (b^2*B*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d)]
    @test_int [cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 3, (A*b^2*x*sqrt(b*cos(c + d*x)))/sqrt(cos(c + d*x)) + (b^2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 3, (b^2*B*x*sqrt(b*cos(c + d*x)))/sqrt(cos(c + d*x)) + (A*b^2*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(d*sqrt(cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-9/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 5, (b^2*B*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(d*sqrt(cos(c + d*x))) + (A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-11/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 6, (A*b^2*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(2*d*sqrt(cos(c + d*x))) + (A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)) + (b^2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2))]
    @test_int [cos(c + d*x)^(-13/2)*(A + B*cos(c + d*x))*(b*cos(c + d*x))^(5/2), x, 6, (b^2*B*atanh(sin(c + d*x))*sqrt(b*cos(c + d*x)))/(2*d*sqrt(cos(c + d*x))) + (b^2*B*sqrt(b*cos(c + d*x))*sin(c + d*x))/(2*d*cos(c + d*x)^(5/2)) + (A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x))/(d*cos(c + d*x)^(3/2)) + (A*b^2*sqrt(b*cos(c + d*x))*sin(c + d*x)^3)/(3*d*cos(c + d*x)^(7/2))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 6, (A*x*sqrt(cos(c + d*x)))/(2*sqrt(b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(b*cos(c + d*x))) + (A*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(b*cos(c + d*x))) - (B*sqrt(cos(c + d*x))*sin(c + d*x)^3)/(3*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 2, (B*x*sqrt(cos(c + d*x)))/(2*sqrt(b*cos(c + d*x))) + (A*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(b*cos(c + d*x))) + (B*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 3, (A*x*sqrt(cos(c + d*x)))/sqrt(b*cos(c + d*x)) + (B*sqrt(cos(c + d*x))*sin(c + d*x))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 3, (B*x*sqrt(cos(c + d*x)))/sqrt(b*cos(c + d*x)) + (A*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 5, (B*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(d*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 6, (A*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(2*d*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*sqrt(b*cos(c + d*x))) + (B*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-7/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(1/2), x, 6, (B*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(2*d*sqrt(b*cos(c + d*x))) + (B*sin(c + d*x))/(2*d*cos(c + d*x)^(3/2)*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x)^3)/(3*d*cos(c + d*x)^(5/2)*sqrt(b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(7/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 6, (A*x*sqrt(cos(c + d*x)))/(2*b*sqrt(b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sin(c + d*x))/(b*d*sqrt(b*cos(c + d*x))) + (A*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*d*sqrt(b*cos(c + d*x))) - (B*sqrt(cos(c + d*x))*sin(c + d*x)^3)/(3*b*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 2, (B*x*sqrt(cos(c + d*x)))/(2*b*sqrt(b*cos(c + d*x))) + (A*sqrt(cos(c + d*x))*sin(c + d*x))/(b*d*sqrt(b*cos(c + d*x))) + (B*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 3, (A*x*sqrt(cos(c + d*x)))/(b*sqrt(b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sin(c + d*x))/(b*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 3, (B*x*sqrt(cos(c + d*x)))/(b*sqrt(b*cos(c + d*x))) + (A*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(b*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 5, (B*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(b*d*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(b*d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 6, (A*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(2*b*d*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(2*b*d*cos(c + d*x)^(3/2)*sqrt(b*cos(c + d*x))) + (B*sin(c + d*x))/(b*d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-5/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(3/2), x, 6, (B*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(2*b*d*sqrt(b*cos(c + d*x))) + (B*sin(c + d*x))/(2*b*d*cos(c + d*x)^(3/2)*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(b*d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x)^3)/(3*b*d*cos(c + d*x)^(5/2)*sqrt(b*cos(c + d*x)))]


    @test_int [cos(c + d*x)^(9/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 6, (A*x*sqrt(cos(c + d*x)))/(2*b^2*sqrt(b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sin(c + d*x))/(b^2*d*sqrt(b*cos(c + d*x))) + (A*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b^2*d*sqrt(b*cos(c + d*x))) - (B*sqrt(cos(c + d*x))*sin(c + d*x)^3)/(3*b^2*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(7/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 2, (B*x*sqrt(cos(c + d*x)))/(2*b^2*sqrt(b*cos(c + d*x))) + (A*sqrt(cos(c + d*x))*sin(c + d*x))/(b^2*d*sqrt(b*cos(c + d*x))) + (B*cos(c + d*x)^(3/2)*sin(c + d*x))/(2*b^2*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(5/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 3, (A*x*sqrt(cos(c + d*x)))/(b^2*sqrt(b*cos(c + d*x))) + (B*sqrt(cos(c + d*x))*sin(c + d*x))/(b^2*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(3/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 3, (B*x*sqrt(cos(c + d*x)))/(b^2*sqrt(b*cos(c + d*x))) + (A*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(b^2*d*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(1/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 5, (B*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(b^2*d*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(b^2*d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-1/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 6, (A*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(2*b^2*d*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(2*b^2*d*cos(c + d*x)^(3/2)*sqrt(b*cos(c + d*x))) + (B*sin(c + d*x))/(b^2*d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x)))]
    @test_int [cos(c + d*x)^(-3/2)*(A + B*cos(c + d*x))/(b*cos(c + d*x))^(5/2), x, 6, (B*atanh(sin(c + d*x))*sqrt(cos(c + d*x)))/(2*b^2*d*sqrt(b*cos(c + d*x))) + (B*sin(c + d*x))/(2*b^2*d*cos(c + d*x)^(3/2)*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x))/(b^2*d*sqrt(cos(c + d*x))*sqrt(b*cos(c + d*x))) + (A*sin(c + d*x)^3)/(3*b^2*d*cos(c + d*x)^(5/2)*sqrt(b*cos(c + d*x)))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the  form*(a+b*cos(e+f*x))*(c*cos(e+f*x))^m*(d*cos(e+f*x))^(n/3)=#


    #= ::Subsubsection::Closed:: =#
    #=n>0=#


    @test_int [cos(c + d*x)^2*(b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x)), x, 4, -((3*A*(b*cos(c + d*x))^(10/3)*HypergeometricFunctions._₂F₁(1/2, 5/3, 8/3, cos(c + d*x)^2)*sin(c + d*x))/(10*b^3*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(13/3)*HypergeometricFunctions._₂F₁(1/2, 13/6, 19/6, cos(c + d*x)^2)*sin(c + d*x))/(13*b^4*d*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)*(b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x)), x, 4, -((3*A*(b*cos(c + d*x))^(7/3)*HypergeometricFunctions._₂F₁(1/2, 7/6, 13/6, cos(c + d*x)^2)*sin(c + d*x))/(7*b^2*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(10/3)*HypergeometricFunctions._₂F₁(1/2, 5/3, 8/3, cos(c + d*x)^2)*sin(c + d*x))/(10*b^3*d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x)), x, 3, -((3*A*(b*cos(c + d*x))^(4/3)*HypergeometricFunctions._₂F₁(1/2, 2/3, 5/3, cos(c + d*x)^2)*sin(c + d*x))/(4*b*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(7/3)*HypergeometricFunctions._₂F₁(1/2, 7/6, 13/6, cos(c + d*x)^2)*sin(c + d*x))/(7*b^2*d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x))*sec(c + d*x), x, 4, (-3*A*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 7/6, cos(c + d*x)^2)*sin(c + d*x))/(d*sqrt(sin(c + d*x)^2)) - (3*B*(b*cos(c + d*x))^(4/3)*HypergeometricFunctions._₂F₁(1/2, 2/3, 5/3, cos(c + d*x)^2)*sin(c + d*x))/(4*b*d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x))*sec(c + d*x)^2, x, 4, (3*A*b*HypergeometricFunctions._₂F₁(-(1/3), 1/2, 2/3, cos(c + d*x)^2)*sin(c + d*x))/(2*d*(b*cos(c + d*x))^(2/3)*sqrt(sin(c + d*x)^2)) - (3*B*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 7/6, cos(c + d*x)^2)*sin(c + d*x))/(d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x))*sec(c + d*x)^3, x, 4, (3*A*b^2*HypergeometricFunctions._₂F₁(-(5/6), 1/2, 1/6, cos(c + d*x)^2)*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/3)*sqrt(sin(c + d*x)^2)) + (3*b*B*HypergeometricFunctions._₂F₁(-(1/3), 1/2, 2/3, cos(c + d*x)^2)*sin(c + d*x))/(2*d*(b*cos(c + d*x))^(2/3)*sqrt(sin(c + d*x)^2))]


    @test_int [cos(c + d*x)^2*(b*cos(c + d*x))^(4/3)*(A + B*cos(c + d*x)), x, 4, -((3*A*(b*cos(c + d*x))^(13/3)*HypergeometricFunctions._₂F₁(1/2, 13/6, 19/6, cos(c + d*x)^2)*sin(c + d*x))/(13*b^3*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(16/3)*HypergeometricFunctions._₂F₁(1/2, 8/3, 11/3, cos(c + d*x)^2)*sin(c + d*x))/(16*b^4*d*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)*(b*cos(c + d*x))^(4/3)*(A + B*cos(c + d*x)), x, 4, -((3*A*(b*cos(c + d*x))^(10/3)*HypergeometricFunctions._₂F₁(1/2, 5/3, 8/3, cos(c + d*x)^2)*sin(c + d*x))/(10*b^2*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(13/3)*HypergeometricFunctions._₂F₁(1/2, 13/6, 19/6, cos(c + d*x)^2)*sin(c + d*x))/(13*b^3*d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(4/3)*(A + B*cos(c + d*x)), x, 3, -((3*A*(b*cos(c + d*x))^(7/3)*HypergeometricFunctions._₂F₁(1/2, 7/6, 13/6, cos(c + d*x)^2)*sin(c + d*x))/(7*b*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(10/3)*HypergeometricFunctions._₂F₁(1/2, 5/3, 8/3, cos(c + d*x)^2)*sin(c + d*x))/(10*b^2*d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(4/3)*(A + B*cos(c + d*x))*sec(c + d*x), x, 4, -((3*A*(b*cos(c + d*x))^(4/3)*HypergeometricFunctions._₂F₁(1/2, 2/3, 5/3, cos(c + d*x)^2)*sin(c + d*x))/(4*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(7/3)*HypergeometricFunctions._₂F₁(1/2, 7/6, 13/6, cos(c + d*x)^2)*sin(c + d*x))/(7*b*d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(4/3)*(A + B*cos(c + d*x))*sec(c + d*x)^2, x, 4, (-3*A*b*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 7/6, cos(c + d*x)^2)*sin(c + d*x))/(d*sqrt(sin(c + d*x)^2)) - (3*B*(b*cos(c + d*x))^(4/3)*HypergeometricFunctions._₂F₁(1/2, 2/3, 5/3, cos(c + d*x)^2)*sin(c + d*x))/(4*d*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^(4/3)*(A + B*cos(c + d*x))*sec(c + d*x)^3, x, 4, (3*A*b^2*HypergeometricFunctions._₂F₁(-(1/3), 1/2, 2/3, cos(c + d*x)^2)*sin(c + d*x))/(2*d*(b*cos(c + d*x))^(2/3)*sqrt(sin(c + d*x)^2)) - (3*b*B*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 7/6, cos(c + d*x)^2)*sin(c + d*x))/(d*sqrt(sin(c + d*x)^2))]


    #= ::Subsubsection::Closed:: =#
    #=n<0=#


    @test_int [(cos(c + d*x)^2*(A + B*cos(c + d*x)))/(b*cos(c + d*x))^(2/3), x, 4, -((3*A*(b*cos(c + d*x))^(7/3)*HypergeometricFunctions._₂F₁(1/2, 7/6, 13/6, cos(c + d*x)^2)*sin(c + d*x))/(7*b^3*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(10/3)*HypergeometricFunctions._₂F₁(1/2, 5/3, 8/3, cos(c + d*x)^2)*sin(c + d*x))/(10*b^4*d*sqrt(sin(c + d*x)^2))]
    @test_int [(cos(c + d*x)*(A + B*cos(c + d*x)))/(b*cos(c + d*x))^(2/3), x, 4, -((3*A*(b*cos(c + d*x))^(4/3)*HypergeometricFunctions._₂F₁(1/2, 2/3, 5/3, cos(c + d*x)^2)*sin(c + d*x))/(4*b^2*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(7/3)*HypergeometricFunctions._₂F₁(1/2, 7/6, 13/6, cos(c + d*x)^2)*sin(c + d*x))/(7*b^3*d*sqrt(sin(c + d*x)^2))]
    @test_int [(A + B*cos(c + d*x))/(b*cos(c + d*x))^(2/3), x, 3, (-3*A*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 7/6, cos(c + d*x)^2)*sin(c + d*x))/(b*d*sqrt(sin(c + d*x)^2)) - (3*B*(b*cos(c + d*x))^(4/3)*HypergeometricFunctions._₂F₁(1/2, 2/3, 5/3, cos(c + d*x)^2)*sin(c + d*x))/(4*b^2*d*sqrt(sin(c + d*x)^2))]
    @test_int [((A + B*cos(c + d*x))*sec(c + d*x))/(b*cos(c + d*x))^(2/3), x, 4, (3*A*HypergeometricFunctions._₂F₁(-(1/3), 1/2, 2/3, cos(c + d*x)^2)*sin(c + d*x))/(2*d*(b*cos(c + d*x))^(2/3)*sqrt(sin(c + d*x)^2)) - (3*B*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/6, 1/2, 7/6, cos(c + d*x)^2)*sin(c + d*x))/(b*d*sqrt(sin(c + d*x)^2))]
    @test_int [((A + B*cos(c + d*x))*sec(c + d*x)^2)/(b*cos(c + d*x))^(2/3), x, 4, (3*A*b*HypergeometricFunctions._₂F₁(-(5/6), 1/2, 1/6, cos(c + d*x)^2)*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/3)*sqrt(sin(c + d*x)^2)) + (3*B*HypergeometricFunctions._₂F₁(-(1/3), 1/2, 2/3, cos(c + d*x)^2)*sin(c + d*x))/(2*d*(b*cos(c + d*x))^(2/3)*sqrt(sin(c + d*x)^2))]
    @test_int [((A + B*cos(c + d*x))*sec(c + d*x)^3)/(b*cos(c + d*x))^(2/3), x, 4, (3*A*b^2*HypergeometricFunctions._₂F₁(-(4/3), 1/2, -(1/3), cos(c + d*x)^2)*sin(c + d*x))/(8*d*(b*cos(c + d*x))^(8/3)*sqrt(sin(c + d*x)^2)) + (3*b*B*HypergeometricFunctions._₂F₁(-(5/6), 1/2, 1/6, cos(c + d*x)^2)*sin(c + d*x))/(5*d*(b*cos(c + d*x))^(5/3)*sqrt(sin(c + d*x)^2))]


    @test_int [(cos(c + d*x)^2*(A + B*cos(c + d*x)))/(b*cos(c + d*x))^(4/3), x, 4, -((3*A*(b*cos(c + d*x))^(5/3)*HypergeometricFunctions._₂F₁(1/2, 5/6, 11/6, cos(c + d*x)^2)*sin(c + d*x))/(5*b^3*d*sqrt(sin(c + d*x)^2))) - (3*B*(b*cos(c + d*x))^(8/3)*HypergeometricFunctions._₂F₁(1/2, 4/3, 7/3, cos(c + d*x)^2)*sin(c + d*x))/(8*b^4*d*sqrt(sin(c + d*x)^2))]
    @test_int [(cos(c + d*x)*(A + B*cos(c + d*x)))/(b*cos(c + d*x))^(4/3), x, 4, (-3*A*(b*cos(c + d*x))^(2/3)*HypergeometricFunctions._₂F₁(1/3, 1/2, 4/3, cos(c + d*x)^2)*sin(c + d*x))/(2*b^2*d*sqrt(sin(c + d*x)^2)) - (3*B*(b*cos(c + d*x))^(5/3)*HypergeometricFunctions._₂F₁(1/2, 5/6, 11/6, cos(c + d*x)^2)*sin(c + d*x))/(5*b^3*d*sqrt(sin(c + d*x)^2))]
    @test_int [(A + B*cos(c + d*x))/(b*cos(c + d*x))^(4/3), x, 3, (3*A*HypergeometricFunctions._₂F₁(-(1/6), 1/2, 5/6, cos(c + d*x)^2)*sin(c + d*x))/(b*d*(b*cos(c + d*x))^(1/3)*sqrt(sin(c + d*x)^2)) - (3*B*(b*cos(c + d*x))^(2/3)*HypergeometricFunctions._₂F₁(1/3, 1/2, 4/3, cos(c + d*x)^2)*sin(c + d*x))/(2*b^2*d*sqrt(sin(c + d*x)^2))]
    @test_int [((A + B*cos(c + d*x))*sec(c + d*x))/(b*cos(c + d*x))^(4/3), x, 4, (3*A*HypergeometricFunctions._₂F₁(-(2/3), 1/2, 1/3, cos(c + d*x)^2)*sin(c + d*x))/(4*d*(b*cos(c + d*x))^(4/3)*sqrt(sin(c + d*x)^2)) + (3*B*HypergeometricFunctions._₂F₁(-(1/6), 1/2, 5/6, cos(c + d*x)^2)*sin(c + d*x))/(b*d*(b*cos(c + d*x))^(1/3)*sqrt(sin(c + d*x)^2))]
    @test_int [((A + B*cos(c + d*x))*sec(c + d*x)^2)/(b*cos(c + d*x))^(4/3), x, 4, (3*A*b*HypergeometricFunctions._₂F₁(-(7/6), 1/2, -(1/6), cos(c + d*x)^2)*sin(c + d*x))/(7*d*(b*cos(c + d*x))^(7/3)*sqrt(sin(c + d*x)^2)) + (3*B*HypergeometricFunctions._₂F₁(-(2/3), 1/2, 1/3, cos(c + d*x)^2)*sin(c + d*x))/(4*d*(b*cos(c + d*x))^(4/3)*sqrt(sin(c + d*x)^2))]
    @test_int [((A + B*cos(c + d*x))*sec(c + d*x)^3)/(b*cos(c + d*x))^(4/3), x, 4, (3*A*b^2*HypergeometricFunctions._₂F₁(-(5/3), 1/2, -(2/3), cos(c + d*x)^2)*sin(c + d*x))/(10*d*(b*cos(c + d*x))^(10/3)*sqrt(sin(c + d*x)^2)) + (3*b*B*HypergeometricFunctions._₂F₁(-(7/6), 1/2, -(1/6), cos(c + d*x)^2)*sin(c + d*x))/(7*d*(b*cos(c + d*x))^(7/3)*sqrt(sin(c + d*x)^2))]


    #= ::Subsection::Closed:: =#
    #=Integrands*of*the  form*(a+b*cos(e+f*x))*(c*cos(e+f*x))^m*(d*cos(e+f*x))^n)*with*m*and/or*n*symbolic=#


    @test_int [cos(c + d*x)^m*(b*cos(c + d*x))^n*(A + B*cos(c + d*x)), x, 4, -((A*cos(c + d*x)^(1 + m)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (1 + m + n)/2, (3 + m + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + m + n)*sqrt(sin(c + d*x)^2))) - (B*cos(c + d*x)^(2 + m)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (2 + m + n)/2, (4 + m + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + m + n)*sqrt(sin(c + d*x)^2))]


    @test_int [cos(c + d*x)^2*(b*cos(c + d*x))^n*(A + B*cos(c + d*x)), x, 4, -((A*(b*cos(c + d*x))^(3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 + n)/2, (5 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(b^3*d*(3 + n)*sqrt(sin(c + d*x)^2))) - (B*(b*cos(c + d*x))^(4 + n)*HypergeometricFunctions._₂F₁(1/2, (4 + n)/2, (6 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(b^4*d*(4 + n)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^1*(b*cos(c + d*x))^n*(A + B*cos(c + d*x)), x, 4, -((A*(b*cos(c + d*x))^(2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 + n)/2, (4 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(b^2*d*(2 + n)*sqrt(sin(c + d*x)^2))) - (B*(b*cos(c + d*x))^(3 + n)*HypergeometricFunctions._₂F₁(1/2, (3 + n)/2, (5 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(b^3*d*(3 + n)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^0*(b*cos(c + d*x))^n*(A + B*cos(c + d*x)), x, 3, -((A*(b*cos(c + d*x))^(1 + n)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(b*d*(1 + n)*sqrt(sin(c + d*x)^2))) - (B*(b*cos(c + d*x))^(2 + n)*HypergeometricFunctions._₂F₁(1/2, (2 + n)/2, (4 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(b^2*d*(2 + n)*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^n*(A + B*cos(c + d*x))*sec(c + d*x)^1, x, 4, -((A*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, n/2, (2 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*n*sqrt(sin(c + d*x)^2))) - (B*(b*cos(c + d*x))^(1 + n)*HypergeometricFunctions._₂F₁(1/2, (1 + n)/2, (3 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(b*d*(1 + n)*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^n*(A + B*cos(c + d*x))*sec(c + d*x)^2, x, 4, (A*b*(b*cos(c + d*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1/2, (-1 + n)/2, (1 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 - n)*sqrt(sin(c + d*x)^2)) - (B*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, n/2, (2 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*n*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^n*(A + B*cos(c + d*x))*sec(c + d*x)^3, x, 4, (A*b^2*(b*cos(c + d*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (-2 + n)/2, n/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 - n)*sqrt(sin(c + d*x)^2)) + (b*B*(b*cos(c + d*x))^(-1 + n)*HypergeometricFunctions._₂F₁(1/2, (-1 + n)/2, (1 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 - n)*sqrt(sin(c + d*x)^2))]
    @test_int [(b*cos(c + d*x))^n*(A + B*cos(c + d*x))*sec(c + d*x)^4, x, 4, (A*b^3*(b*cos(c + d*x))^(-3 + n)*HypergeometricFunctions._₂F₁(1/2, (-3 + n)/2, (-1 + n)/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(3 - n)*sqrt(sin(c + d*x)^2)) + (b^2*B*(b*cos(c + d*x))^(-2 + n)*HypergeometricFunctions._₂F₁(1/2, (-2 + n)/2, n/2, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 - n)*sqrt(sin(c + d*x)^2))]


    @test_int [cos(c + d*x)^(5/2)*(b*cos(c + d*x))^n*(A + B*cos(c + d*x)), x, 4, (-2*A*cos(c + d*x)^(7/2)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (7 + 2*n)/4, (11 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(7 + 2*n)*sqrt(sin(c + d*x)^2)) - (2*B*cos(c + d*x)^(9/2)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (9 + 2*n)/4, (13 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(9 + 2*n)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^(3/2)*(b*cos(c + d*x))^n*(A + B*cos(c + d*x)), x, 4, (-2*A*cos(c + d*x)^(5/2)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (5 + 2*n)/4, (9 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(5 + 2*n)*sqrt(sin(c + d*x)^2)) - (2*B*cos(c + d*x)^(7/2)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (7 + 2*n)/4, (11 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(7 + 2*n)*sqrt(sin(c + d*x)^2))]
    @test_int [sqrt(cos(c + d*x))*(b*cos(c + d*x))^n*(A + B*cos(c + d*x)), x, 4, (-2*A*cos(c + d*x)^(3/2)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (3 + 2*n)/4, (7 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(3 + 2*n)*sqrt(sin(c + d*x)^2)) - (2*B*cos(c + d*x)^(5/2)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (5 + 2*n)/4, (9 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(5 + 2*n)*sqrt(sin(c + d*x)^2))]
    @test_int [((b*cos(c + d*x))^n*(A + B*cos(c + d*x)))/sqrt(cos(c + d*x)), x, 4, (-2*A*sqrt(cos(c + d*x))*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (1 + 2*n)/4, (5 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + 2*n)*sqrt(sin(c + d*x)^2)) - (2*B*cos(c + d*x)^(3/2)*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (3 + 2*n)/4, (7 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(3 + 2*n)*sqrt(sin(c + d*x)^2))]
    @test_int [((b*cos(c + d*x))^n*(A + B*cos(c + d*x)))/cos(c + d*x)^(3/2), x, 4, (2*A*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (-1 + 2*n)/4, (3 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 - 2*n)*sqrt(cos(c + d*x))*sqrt(sin(c + d*x)^2)) - (2*B*sqrt(cos(c + d*x))*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (1 + 2*n)/4, (5 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + 2*n)*sqrt(sin(c + d*x)^2))]
    @test_int [((b*cos(c + d*x))^n*(A + B*cos(c + d*x)))/cos(c + d*x)^(5/2), x, 4, (2*A*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (-3 + 2*n)/4, (1 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(3 - 2*n)*cos(c + d*x)^(3/2)*sqrt(sin(c + d*x)^2)) + (2*B*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (-1 + 2*n)/4, (3 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 - 2*n)*sqrt(cos(c + d*x))*sqrt(sin(c + d*x)^2))]
    @test_int [((b*cos(c + d*x))^n*(A + B*cos(c + d*x)))/cos(c + d*x)^(7/2), x, 4, (2*A*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(5 - 2*n)*cos(c + d*x)^(5/2)*sqrt(sin(c + d*x)^2)) + (2*B*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (-3 + 2*n)/4, (1 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(3 - 2*n)*cos(c + d*x)^(3/2)*sqrt(sin(c + d*x)^2))]
    @test_int [((b*cos(c + d*x))^n*(A + B*cos(c + d*x)))/cos(c + d*x)^(9/2), x, 4, (2*A*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (-7 + 2*n)/4, (-3 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(7 - 2*n)*cos(c + d*x)^(7/2)*sqrt(sin(c + d*x)^2)) + (2*B*(b*cos(c + d*x))^n*HypergeometricFunctions._₂F₁(1/2, (-5 + 2*n)/4, (-1 + 2*n)/4, cos(c + d*x)^2)*sin(c + d*x))/(d*(5 - 2*n)*cos(c + d*x)^(5/2)*sqrt(sin(c + d*x)^2))]


    @test_int [cos(c + d*x)^m*(b*cos(c + d*x))^(4/3)*(A + B*cos(c + d*x)), x, 4, (-3*A*b*cos(c + d*x)^(2 + m)*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/2, (7 + 3*m)/6, (13 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(7 + 3*m)*sqrt(sin(c + d*x)^2)) - (3*b*B*cos(c + d*x)^(3 + m)*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/2, (10 + 3*m)/6, (16 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(10 + 3*m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m*(b*cos(c + d*x))^(2/3)*(A + B*cos(c + d*x)), x, 4, (-3*A*cos(c + d*x)^(1 + m)*(b*cos(c + d*x))^(2/3)*HypergeometricFunctions._₂F₁(1/2, (5 + 3*m)/6, (11 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(5 + 3*m)*sqrt(sin(c + d*x)^2)) - (3*B*cos(c + d*x)^(2 + m)*(b*cos(c + d*x))^(2/3)*HypergeometricFunctions._₂F₁(1/2, (8 + 3*m)/6, (14 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(8 + 3*m)*sqrt(sin(c + d*x)^2))]
    @test_int [cos(c + d*x)^m*(b*cos(c + d*x))^(1/3)*(A + B*cos(c + d*x)), x, 4, (-3*A*cos(c + d*x)^(1 + m)*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/2, (4 + 3*m)/6, (10 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(4 + 3*m)*sqrt(sin(c + d*x)^2)) - (3*B*cos(c + d*x)^(2 + m)*(b*cos(c + d*x))^(1/3)*HypergeometricFunctions._₂F₁(1/2, (7 + 3*m)/6, (13 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(7 + 3*m)*sqrt(sin(c + d*x)^2))]
    @test_int [(cos(c + d*x)^m*(A + B*cos(c + d*x)))/(b*cos(c + d*x))^(1/3), x, 4, (-3*A*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + 3*m)/6, (8 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(2 + 3*m)*(b*cos(c + d*x))^(1/3)*sqrt(sin(c + d*x)^2)) - (3*B*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (5 + 3*m)/6, (11 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(5 + 3*m)*(b*cos(c + d*x))^(1/3)*sqrt(sin(c + d*x)^2))]
    @test_int [(cos(c + d*x)^m*(A + B*cos(c + d*x)))/(b*cos(c + d*x))^(2/3), x, 4, (-3*A*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1 + 3*m)/6, (7 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(1 + 3*m)*(b*cos(c + d*x))^(2/3)*sqrt(sin(c + d*x)^2)) - (3*B*cos(c + d*x)^(2 + m)*HypergeometricFunctions._₂F₁(1/2, (4 + 3*m)/6, (10 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(d*(4 + 3*m)*(b*cos(c + d*x))^(2/3)*sqrt(sin(c + d*x)^2))]
    @test_int [(cos(c + d*x)^m*(A + B*cos(c + d*x)))/(b*cos(c + d*x))^(4/3), x, 4, (3*A*cos(c + d*x)^m*HypergeometricFunctions._₂F₁(1/2, (-1 + 3*m)/6, (5 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(b*d*(1 - 3*m)*(b*cos(c + d*x))^(1/3)*sqrt(sin(c + d*x)^2)) - (3*B*cos(c + d*x)^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (2 + 3*m)/6, (8 + 3*m)/6, cos(c + d*x)^2)*sin(c + d*x))/(b*d*(2 + 3*m)*(b*cos(c + d*x))^(1/3)*sqrt(sin(c + d*x)^2))]
    end
