text:
@testset "4.5.2.3 (g sec)^p (a+b sec)^m (c+d sec)^n" begin
    (a, b, c, d, e, f, g, m, n, p, x, ) = @variables a b c d e f g m n p x

    #= ::Package:: =#

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

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

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^4, x, 12, (7*a*c^4*atanh(sin(e + f*x)))/(8*f) - (a*c^4*sec(e + f*x)*tan(e + f*x))/(8*f) - (3*a*c^4*sec(e + f*x)^3*tan(e + f*x))/(4*f) + (4*a*c^4*tan(e + f*x)^3)/(3*f) + (a*c^4*tan(e + f*x)^5)/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^3, x, 9, (5*a*c^3*atanh(sin(e + f*x)))/(8*f) - (3*a*c^3*sec(e + f*x)*tan(e + f*x))/(8*f) - (a*c^3*sec(e + f*x)^3*tan(e + f*x))/(4*f) + (2*a*c^3*tan(e + f*x)^3)/(3*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^2, x, 6, (a*c^2*atanh(sin(e + f*x)))/(2*f) - (a*c^2*sec(e + f*x)*tan(e + f*x))/(2*f) + (a*c^2*tan(e + f*x)^3)/(3*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^1, x, 3, (a*c*atanh(sin(e + f*x)))/(2*f) - (a*c*sec(e + f*x)*tan(e + f*x))/(2*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^1, x, 2, -((a*atanh(sin(e + f*x)))/(c*f)) - (2*a*tan(e + f*x))/(f*(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^2, x, 1, -(((a + a*sec(e + f*x))*tan(e + f*x))/(3*f*(c - c*sec(e + f*x))^2))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^3, x, 2, -(((a + a*sec(e + f*x))*tan(e + f*x))/(5*f*(c - c*sec(e + f*x))^3)) - ((a + a*sec(e + f*x))*tan(e + f*x))/(15*c*f*(c - c*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^4, x, 3, -(((a + a*sec(e + f*x))*tan(e + f*x))/(7*f*(c - c*sec(e + f*x))^4)) - (2*(a + a*sec(e + f*x))*tan(e + f*x))/(35*c*f*(c - c*sec(e + f*x))^3) - (2*(a + a*sec(e + f*x))*tan(e + f*x))/(105*f*(c^2 - c^2*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^5, x, 4, -(((a + a*sec(e + f*x))*tan(e + f*x))/(9*f*(c - c*sec(e + f*x))^5)) - ((a + a*sec(e + f*x))*tan(e + f*x))/(21*c*f*(c - c*sec(e + f*x))^4) - (2*(a + a*sec(e + f*x))*tan(e + f*x))/(105*c^2*f*(c - c*sec(e + f*x))^3) - (2*(a + a*sec(e + f*x))*tan(e + f*x))/(315*c*f*(c^2 - c^2*sec(e + f*x))^2)]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^5, x, 14, (9*a^2*c^5*atanh(sin(e + f*x)))/(16*f) - (3*a^2*c^5*sec(e + f*x)*tan(e + f*x))/(16*f) - (3*a^2*c^5*sec(e + f*x)^3*tan(e + f*x))/(8*f) + (a^2*c^5*sec(e + f*x)*tan(e + f*x)^3)/(4*f) + (a^2*c^5*sec(e + f*x)^3*tan(e + f*x)^3)/(2*f) - (4*a^2*c^5*tan(e + f*x)^5)/(5*f) - (a^2*c^5*tan(e + f*x)^7)/(7*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^4, x, 11, (7*a^2*c^4*atanh(sin(e + f*x)))/(16*f) - (5*a^2*c^4*sec(e + f*x)*tan(e + f*x))/(16*f) - (a^2*c^4*sec(e + f*x)^3*tan(e + f*x))/(8*f) + (a^2*c^4*sec(e + f*x)*tan(e + f*x)^3)/(4*f) + (a^2*c^4*sec(e + f*x)^3*tan(e + f*x)^3)/(6*f) - (2*a^2*c^4*tan(e + f*x)^5)/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^3, x, 7, (3*a^2*c^3*atanh(sin(e + f*x)))/(8*f) - (3*a^2*c^3*sec(e + f*x)*tan(e + f*x))/(8*f) + (a^2*c^3*sec(e + f*x)*tan(e + f*x)^3)/(4*f) - (a^2*c^3*tan(e + f*x)^5)/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^2, x, 4, (3*a^2*c^2*atanh(sin(e + f*x)))/(8*f) - (3*a^2*c^2*sec(e + f*x)*tan(e + f*x))/(8*f) + (a^2*c^2*sec(e + f*x)*tan(e + f*x)^3)/(4*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^1, x, 6, (a^2*c*atanh(sin(e + f*x)))/(2*f) - (a^2*c*sec(e + f*x)*tan(e + f*x))/(2*f) - (a^2*c*tan(e + f*x)^3)/(3*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^1, x, 5, -((3*a^2*atanh(sin(e + f*x)))/(c*f)) - (3*a^2*tan(e + f*x))/(c*f) - (2*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(f*(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^2, x, 3, (a^2*atanh(sin(e + f*x)))/(c^2*f) - (2*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(3*f*(c - c*sec(e + f*x))^2) + (2*a^2*tan(e + f*x))/(f*(c^2 - c^2*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^3, x, 1, -(((a + a*sec(e + f*x))^2*tan(e + f*x))/(5*f*(c - c*sec(e + f*x))^3))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^4, x, 2, -(((a + a*sec(e + f*x))^2*tan(e + f*x))/(7*f*(c - c*sec(e + f*x))^4)) - ((a + a*sec(e + f*x))^2*tan(e + f*x))/(35*c*f*(c - c*sec(e + f*x))^3)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^5, x, 3, -(((a + a*sec(e + f*x))^2*tan(e + f*x))/(9*f*(c - c*sec(e + f*x))^5)) - (2*(a + a*sec(e + f*x))^2*tan(e + f*x))/(63*c*f*(c - c*sec(e + f*x))^4) - (2*(a + a*sec(e + f*x))^2*tan(e + f*x))/(315*c^2*f*(c - c*sec(e + f*x))^3)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^6, x, 4, -(((a + a*sec(e + f*x))^2*tan(e + f*x))/(11*f*(c - c*sec(e + f*x))^6)) - ((a + a*sec(e + f*x))^2*tan(e + f*x))/(33*c*f*(c - c*sec(e + f*x))^5) - (2*(a + a*sec(e + f*x))^2*tan(e + f*x))/(231*c^2*f*(c - c*sec(e + f*x))^4) - (2*(a + a*sec(e + f*x))^2*tan(e + f*x))/(1155*f*(c^2 - c^2*sec(e + f*x))^3)]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^6, x, 16, (55*a^3*c^6*atanh(sin(e + f*x)))/(128*f) - (25*a^3*c^6*sec(e + f*x)*tan(e + f*x))/(128*f) - (15*a^3*c^6*sec(e + f*x)^3*tan(e + f*x))/(64*f) + (5*a^3*c^6*sec(e + f*x)*tan(e + f*x)^3)/(24*f) + (5*a^3*c^6*sec(e + f*x)^3*tan(e + f*x)^3)/(16*f) - (a^3*c^6*sec(e + f*x)*tan(e + f*x)^5)/(6*f) - (3*a^3*c^6*sec(e + f*x)^3*tan(e + f*x)^5)/(8*f) + (4*a^3*c^6*tan(e + f*x)^7)/(7*f) + (a^3*c^6*tan(e + f*x)^9)/(9*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^5, x, 13, (45*a^3*c^5*atanh(sin(e + f*x)))/(128*f) - (35*a^3*c^5*sec(e + f*x)*tan(e + f*x))/(128*f) - (5*a^3*c^5*sec(e + f*x)^3*tan(e + f*x))/(64*f) + (5*a^3*c^5*sec(e + f*x)*tan(e + f*x)^3)/(24*f) + (5*a^3*c^5*sec(e + f*x)^3*tan(e + f*x)^3)/(48*f) - (a^3*c^5*sec(e + f*x)*tan(e + f*x)^5)/(6*f) - (a^3*c^5*sec(e + f*x)^3*tan(e + f*x)^5)/(8*f) + (2*a^3*c^5*tan(e + f*x)^7)/(7*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^4, x, 8, (5*a^3*c^4*atanh(sin(e + f*x)))/(16*f) - (5*a^3*c^4*sec(e + f*x)*tan(e + f*x))/(16*f) + (5*a^3*c^4*sec(e + f*x)*tan(e + f*x)^3)/(24*f) - (a^3*c^4*sec(e + f*x)*tan(e + f*x)^5)/(6*f) + (a^3*c^4*tan(e + f*x)^7)/(7*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^3, x, 5, (5*a^3*c^3*atanh(sin(e + f*x)))/(16*f) - (5*a^3*c^3*sec(e + f*x)*tan(e + f*x))/(16*f) + (5*a^3*c^3*sec(e + f*x)*tan(e + f*x)^3)/(24*f) - (a^3*c^3*sec(e + f*x)*tan(e + f*x)^5)/(6*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^2, x, 7, (3*a^3*c^2*atanh(sin(e + f*x)))/(8*f) - (3*a^3*c^2*sec(e + f*x)*tan(e + f*x))/(8*f) + (a^3*c^2*sec(e + f*x)*tan(e + f*x)^3)/(4*f) + (a^3*c^2*tan(e + f*x)^5)/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^1, x, 9, (5*a^3*c*atanh(sin(e + f*x)))/(8*f) - (3*a^3*c*sec(e + f*x)*tan(e + f*x))/(8*f) - (a^3*c*sec(e + f*x)^3*tan(e + f*x))/(4*f) - (2*a^3*c*tan(e + f*x)^3)/(3*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^1, x, 6, -((15*a^3*atanh(sin(e + f*x)))/(2*c*f)) - (10*a^3*tan(e + f*x))/(c*f) - (5*a^3*sec(e + f*x)*tan(e + f*x))/(2*c*f) - (2*a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(f*(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^2, x, 6, (5*a^3*atanh(sin(e + f*x)))/(c^2*f) + (5*a^3*tan(e + f*x))/(c^2*f) - (2*a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(3*f*(c - c*sec(e + f*x))^2) + (10*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(3*f*(c^2 - c^2*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^3, x, 4, -((a^3*atanh(sin(e + f*x)))/(c^3*f)) - (2*a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(5*f*(c - c*sec(e + f*x))^3) + (2*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(3*c*f*(c - c*sec(e + f*x))^2) - (2*a^3*tan(e + f*x))/(f*(c^3 - c^3*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^4, x, 1, -(((a + a*sec(e + f*x))^3*tan(e + f*x))/(7*f*(c - c*sec(e + f*x))^4))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^5, x, 2, -(((a + a*sec(e + f*x))^3*tan(e + f*x))/(9*f*(c - c*sec(e + f*x))^5)) - ((a + a*sec(e + f*x))^3*tan(e + f*x))/(63*c*f*(c - c*sec(e + f*x))^4)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^6, x, 3, -(((a + a*sec(e + f*x))^3*tan(e + f*x))/(11*f*(c - c*sec(e + f*x))^6)) - (2*(a + a*sec(e + f*x))^3*tan(e + f*x))/(99*c*f*(c - c*sec(e + f*x))^5) - (2*(a + a*sec(e + f*x))^3*tan(e + f*x))/(693*c^2*f*(c - c*sec(e + f*x))^4)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^7, x, 4, -(((a + a*sec(e + f*x))^3*tan(e + f*x))/(13*f*(c - c*sec(e + f*x))^7)) - (3*(a + a*sec(e + f*x))^3*tan(e + f*x))/(143*c*f*(c - c*sec(e + f*x))^6) - (2*(a + a*sec(e + f*x))^3*tan(e + f*x))/(429*c^2*f*(c - c*sec(e + f*x))^5) - (2*(a + a*sec(e + f*x))^3*tan(e + f*x))/(3003*c^3*f*(c - c*sec(e + f*x))^4)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^4/(a + a*sec(e + f*x)), x, 10, -((35*c^4*atanh(sin(e + f*x)))/(2*a*f)) + (28*c^4*tan(e + f*x))/(a*f) - (21*c^4*sec(e + f*x)*tan(e + f*x))/(2*a*f) + (2*c*(c - c*sec(e + f*x))^3*tan(e + f*x))/(f*(a + a*sec(e + f*x))) + (7*c^4*tan(e + f*x)^3)/(3*a*f)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^3/(a + a*sec(e + f*x)), x, 6, -((15*c^3*atanh(sin(e + f*x)))/(2*a*f)) + (10*c^3*tan(e + f*x))/(a*f) - (5*c^3*sec(e + f*x)*tan(e + f*x))/(2*a*f) + (2*c*(c - c*sec(e + f*x))^2*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^2/(a + a*sec(e + f*x)), x, 5, -((3*c^2*atanh(sin(e + f*x)))/(a*f)) + (3*c^2*tan(e + f*x))/(a*f) + (2*(c^2 - c^2*sec(e + f*x))*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^1/(a + a*sec(e + f*x)), x, 2, -((c*atanh(sin(e + f*x)))/(a*f)) + (2*c*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c - c*sec(e + f*x))), x, 3, csc(e + f*x)/(a*c*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c - c*sec(e + f*x))^2), x, 6, -(cot(e + f*x)^3/(3*a*c^2*f)) + csc(e + f*x)/(a*c^2*f) - csc(e + f*x)^3/(3*a*c^2*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c - c*sec(e + f*x))^3), x, 10, (2*cot(e + f*x)^5)/(5*a*c^3*f) + csc(e + f*x)/(a*c^3*f) - csc(e + f*x)^3/(a*c^3*f) + (2*csc(e + f*x)^5)/(5*a*c^3*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c - c*sec(e + f*x))^4), x, 13, -(cot(e + f*x)^5/(5*a*c^4*f)) - (4*cot(e + f*x)^7)/(7*a*c^4*f) + csc(e + f*x)/(a*c^4*f) - (2*csc(e + f*x)^3)/(a*c^4*f) + (9*csc(e + f*x)^5)/(5*a*c^4*f) - (4*csc(e + f*x)^7)/(7*a*c^4*f)]

    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^5/(a + a*sec(e + f*x))^2, x, 11, (105*c^5*atanh(sin(e + f*x)))/(2*a^2*f) - (84*c^5*tan(e + f*x))/(a^2*f) + (63*c^5*sec(e + f*x)*tan(e + f*x))/(2*a^2*f) - (6*c^2*(c - c*sec(e + f*x))^3*tan(e + f*x))/(f*(a^2 + a^2*sec(e + f*x))) + (2*c*(c - c*sec(e + f*x))^4*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - (7*c^5*tan(e + f*x)^3)/(a^2*f)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^4/(a + a*sec(e + f*x))^2, x, 7, (35*c^4*atanh(sin(e + f*x)))/(2*a^2*f) - (70*c^4*tan(e + f*x))/(3*a^2*f) + (35*c^4*sec(e + f*x)*tan(e + f*x))/(6*a^2*f) + (2*c*(c - c*sec(e + f*x))^3*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - (14*(c^2 - c^2*sec(e + f*x))^2*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^3/(a + a*sec(e + f*x))^2, x, 6, (5*c^3*atanh(sin(e + f*x)))/(a^2*f) - (5*c^3*tan(e + f*x))/(a^2*f) + (2*c*(c - c*sec(e + f*x))^2*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - (10*(c^3 - c^3*sec(e + f*x))*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^2/(a + a*sec(e + f*x))^2, x, 3, (c^2*atanh(sin(e + f*x)))/(a^2*f) - (2*c^2*tan(e + f*x))/(f*(a^2 + a^2*sec(e + f*x))) + (2*(c^2 - c^2*sec(e + f*x))*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^1/(a + a*sec(e + f*x))^2, x, 1, ((c - c*sec(e + f*x))*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^1), x, 6, cot(e + f*x)^3/(3*a^2*c*f) + csc(e + f*x)/(a^2*c*f) - csc(e + f*x)^3/(3*a^2*c*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^2), x, 3, csc(e + f*x)/(a^2*c^2*f) - csc(e + f*x)^3/(3*a^2*c^2*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^3), x, 7, cot(e + f*x)^5/(5*a^2*c^3*f) + csc(e + f*x)/(a^2*c^3*f) - (2*csc(e + f*x)^3)/(3*a^2*c^3*f) + csc(e + f*x)^5/(5*a^2*c^3*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^4), x, 10, -((2*cot(e + f*x)^7)/(7*a^2*c^4*f)) + csc(e + f*x)/(a^2*c^4*f) - (4*csc(e + f*x)^3)/(3*a^2*c^4*f) + csc(e + f*x)^5/(a^2*c^4*f) - (2*csc(e + f*x)^7)/(7*a^2*c^4*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^5), x, 13, cot(e + f*x)^7/(7*a^2*c^5*f) + (4*cot(e + f*x)^9)/(9*a^2*c^5*f) + csc(e + f*x)/(a^2*c^5*f) - (7*csc(e + f*x)^3)/(3*a^2*c^5*f) + (3*csc(e + f*x)^5)/(a^2*c^5*f) - (13*csc(e + f*x)^7)/(7*a^2*c^5*f) + (4*csc(e + f*x)^9)/(9*a^2*c^5*f)]

    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^6/(a + a*sec(e + f*x))^3, x, 12, -((231*c^6*atanh(sin(e + f*x)))/(2*a^3*f)) + (924*c^6*tan(e + f*x))/(5*a^3*f) - (693*c^6*sec(e + f*x)*tan(e + f*x))/(10*a^3*f) - (22*c^2*(c - c*sec(e + f*x))^4*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + (2*c*(c - c*sec(e + f*x))^5*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + (66*(c^2 - c^2*sec(e + f*x))^3*tan(e + f*x))/(5*f*(a^3 + a^3*sec(e + f*x))) + (77*c^6*tan(e + f*x)^3)/(5*a^3*f)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^5/(a + a*sec(e + f*x))^3, x, 8, -((63*c^5*atanh(sin(e + f*x)))/(2*a^3*f)) + (42*c^5*tan(e + f*x))/(a^3*f) - (21*c^5*sec(e + f*x)*tan(e + f*x))/(2*a^3*f) - (6*c^2*(c - c*sec(e + f*x))^3*tan(e + f*x))/(5*a*f*(a + a*sec(e + f*x))^2) + (2*c*(c - c*sec(e + f*x))^4*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + (42*c*(c^2 - c^2*sec(e + f*x))^2*tan(e + f*x))/(5*f*(a^3 + a^3*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^4/(a + a*sec(e + f*x))^3, x, 7, -((7*c^4*atanh(sin(e + f*x)))/(a^3*f)) + (7*c^4*tan(e + f*x))/(a^3*f) + (2*c*(c - c*sec(e + f*x))^3*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) - (14*(c^2 - c^2*sec(e + f*x))^2*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + (14*(c^4 - c^4*sec(e + f*x))*tan(e + f*x))/(3*f*(a^3 + a^3*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^3/(a + a*sec(e + f*x))^3, x, 4, -((c^3*atanh(sin(e + f*x)))/(a^3*f)) + (2*c^3*tan(e + f*x))/(f*(a^3 + a^3*sec(e + f*x))) + (2*c*(c - c*sec(e + f*x))^2*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) - (2*(c^3 - c^3*sec(e + f*x))*tan(e + f*x))/(3*a*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^2/(a + a*sec(e + f*x))^3, x, 1, ((c - c*sec(e + f*x))^2*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^1/(a + a*sec(e + f*x))^3, x, 2, ((c - c*sec(e + f*x))*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + ((c - c*sec(e + f*x))*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^1), x, 10, -((2*cot(e + f*x)^5)/(5*a^3*c*f)) + csc(e + f*x)/(a^3*c*f) - csc(e + f*x)^3/(a^3*c*f) + (2*csc(e + f*x)^5)/(5*a^3*c*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^2), x, 7, -(cot(e + f*x)^5/(5*a^3*c^2*f)) + csc(e + f*x)/(a^3*c^2*f) - (2*csc(e + f*x)^3)/(3*a^3*c^2*f) + csc(e + f*x)^5/(5*a^3*c^2*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^3), x, 4, csc(e + f*x)/(a^3*c^3*f) - (2*csc(e + f*x)^3)/(3*a^3*c^3*f) + csc(e + f*x)^5/(5*a^3*c^3*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^4), x, 7, -(cot(e + f*x)^7/(7*a^3*c^4*f)) + csc(e + f*x)/(a^3*c^4*f) - csc(e + f*x)^3/(a^3*c^4*f) + (3*csc(e + f*x)^5)/(5*a^3*c^4*f) - csc(e + f*x)^7/(7*a^3*c^4*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^5), x, 10, (2*cot(e + f*x)^9)/(9*a^3*c^5*f) + csc(e + f*x)/(a^3*c^5*f) - (5*csc(e + f*x)^3)/(3*a^3*c^5*f) + (9*csc(e + f*x)^5)/(5*a^3*c^5*f) - csc(e + f*x)^7/(a^3*c^5*f) + (2*csc(e + f*x)^9)/(9*a^3*c^5*f)]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^6), x, 13, -(cot(e + f*x)^9/(9*a^3*c^6*f)) - (4*cot(e + f*x)^11)/(11*a^3*c^6*f) + csc(e + f*x)/(a^3*c^6*f) - (8*csc(e + f*x)^3)/(3*a^3*c^6*f) + (22*csc(e + f*x)^5)/(5*a^3*c^6*f) - (4*csc(e + f*x)^7)/(a^3*c^6*f) + (17*csc(e + f*x)^9)/(9*a^3*c^6*f) - (4*csc(e + f*x)^11)/(11*a^3*c^6*f)]

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(7/2), x, 4, -((256*c^4*(a + a*sec(e + f*x))*tan(e + f*x))/(315*f*sqrt(c - c*sec(e + f*x)))) - (64*c^3*(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(105*f) - (8*c^2*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(21*f) - (2*c*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(9*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2), x, 3, -((64*c^3*(a + a*sec(e + f*x))*tan(e + f*x))/(105*f*sqrt(c - c*sec(e + f*x)))) - (16*c^2*(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(35*f) - (2*c*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(7*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2), x, 2, -((8*c^2*(a + a*sec(e + f*x))*tan(e + f*x))/(15*f*sqrt(c - c*sec(e + f*x)))) - (2*c*(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)), x, 1, -((2*c*(a + a*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(c - c*sec(e + f*x))))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x)))/sqrt(c - c*sec(e + f*x)), x, 3, (-2*sqrt(2)*a*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(sqrt(c)*f) + (2*a*tan(e + f*x))/(f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x)))/(c - c*sec(e + f*x))^(3/2), x, 3, (a*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(sqrt(2)*c^(3/2)*f) - (a*tan(e + f*x))/(f*(c - c*sec(e + f*x))^(3/2))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x)))/(c - c*sec(e + f*x))^(5/2), x, 4, (a*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(8*sqrt(2)*c^(5/2)*f) - (a*tan(e + f*x))/(2*f*(c - c*sec(e + f*x))^(5/2)) + (a*tan(e + f*x))/(8*c*f*(c - c*sec(e + f*x))^(3/2))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(7/2), x, 4, -((256*c^4*(a + a*sec(e + f*x))^2*tan(e + f*x))/(1155*f*sqrt(c - c*sec(e + f*x)))) - (64*c^3*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(231*f) - (8*c^2*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(33*f) - (2*c*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(11*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(5/2), x, 3, -((64*c^3*(a + a*sec(e + f*x))^2*tan(e + f*x))/(315*f*sqrt(c - c*sec(e + f*x)))) - (16*c^2*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(63*f) - (2*c*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(9*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(3/2), x, 2, -((8*c^2*(a + a*sec(e + f*x))^2*tan(e + f*x))/(35*f*sqrt(c - c*sec(e + f*x)))) - (2*c*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(7*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x)), x, 1, -((2*c*(a + a*sec(e + f*x))^2*tan(e + f*x))/(5*f*sqrt(c - c*sec(e + f*x))))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/sqrt(c - c*sec(e + f*x)), x, 4, -((4*sqrt(2)*a^2*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(sqrt(c)*f)) + (16*a^2*tan(e + f*x))/(3*f*sqrt(c - c*sec(e + f*x))) - (2*a^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(3*c*f), -((4*sqrt(2)*a^2*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(sqrt(c)*f)) + (4*a^2*tan(e + f*x))/(f*sqrt(c - c*sec(e + f*x))) + (2*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c - c*sec(e + f*x))^(3/2), x, 4, (3*sqrt(2)*a^2*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(c^(3/2)*f) - (2*a^2*tan(e + f*x))/(f*(c - c*sec(e + f*x))^(3/2)) - (2*a^2*tan(e + f*x))/(c*f*sqrt(c - c*sec(e + f*x))), (3*sqrt(2)*a^2*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(c^(3/2)*f) - ((a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(f*(c - c*sec(e + f*x))^(3/2)) - (3*a^2*tan(e + f*x))/(c*f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c - c*sec(e + f*x))^(5/2), x, 4, -((3*a^2*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(4*sqrt(2)*c^(5/2)*f)) - (a^2*tan(e + f*x))/(f*(c - c*sec(e + f*x))^(5/2)) + (5*a^2*tan(e + f*x))/(4*c*f*(c - c*sec(e + f*x))^(3/2)), -((3*a^2*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(4*sqrt(2)*c^(5/2)*f)) - ((a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(2*f*(c - c*sec(e + f*x))^(5/2)) + (3*a^2*tan(e + f*x))/(4*c*f*(c - c*sec(e + f*x))^(3/2))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c - c*sec(e + f*x))^(7/2), x, 5, -((a^2*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(16*sqrt(2)*c^(7/2)*f)) - ((a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(3*f*(c - c*sec(e + f*x))^(7/2)) + (a^2*tan(e + f*x))/(4*c*f*(c - c*sec(e + f*x))^(5/2)) - (a^2*tan(e + f*x))/(16*c^2*f*(c - c*sec(e + f*x))^(3/2))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(7/2), x, 4, -((256*c^4*(a + a*sec(e + f*x))^3*tan(e + f*x))/(3003*f*sqrt(c - c*sec(e + f*x)))) - (64*c^3*(a + a*sec(e + f*x))^3*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(429*f) - (24*c^2*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(143*f) - (2*c*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(13*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(5/2), x, 3, -((64*c^3*(a + a*sec(e + f*x))^3*tan(e + f*x))/(693*f*sqrt(c - c*sec(e + f*x)))) - (16*c^2*(a + a*sec(e + f*x))^3*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(99*f) - (2*c*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(11*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(3/2), x, 2, -((8*c^2*(a + a*sec(e + f*x))^3*tan(e + f*x))/(63*f*sqrt(c - c*sec(e + f*x)))) - (2*c*(a + a*sec(e + f*x))^3*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(9*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*sqrt(c - c*sec(e + f*x)), x, 1, -((2*c*(a + a*sec(e + f*x))^3*tan(e + f*x))/(7*f*sqrt(c - c*sec(e + f*x))))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/sqrt(c - c*sec(e + f*x)), x, 5, (-8*sqrt(2)*a^3*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(sqrt(c)*f) + (8*a^3*tan(e + f*x))/(f*sqrt(c - c*sec(e + f*x))) + (2*a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(5*f*sqrt(c - c*sec(e + f*x))) + (4*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/(c - c*sec(e + f*x))^(3/2), x, 5, (10*sqrt(2)*a^3*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(c^(3/2)*f) - (a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(f*(c - c*sec(e + f*x))^(3/2)) - (10*a^3*tan(e + f*x))/(c*f*sqrt(c - c*sec(e + f*x))) - (5*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(3*c*f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/(c - c*sec(e + f*x))^(5/2), x, 5, (-15*a^3*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(2*sqrt(2)*c^(5/2)*f) - (a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(2*f*(c - c*sec(e + f*x))^(5/2)) + (5*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(4*c*f*(c - c*sec(e + f*x))^(3/2)) + (15*a^3*tan(e + f*x))/(4*c^2*f*sqrt(c - c*sec(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(7/2)/(a + a*sec(e + f*x)), x, 4, (128*c^4*tan(e + f*x))/(5*a*f*sqrt(c - c*sec(e + f*x))) + (32*c^3*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(5*a*f) + (12*c^2*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(5*a*f) + (2*c*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(5/2)/(a + a*sec(e + f*x)), x, 3, (32*c^3*tan(e + f*x))/(3*a*f*sqrt(c - c*sec(e + f*x))) + (8*c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(3*a*f) + (2*c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(3/2)/(a + a*sec(e + f*x)), x, 2, (4*c^2*tan(e + f*x))/(a*f*sqrt(c - c*sec(e + f*x))) + (2*c*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)*sqrt(c - c*sec(e + f*x))/(a + a*sec(e + f*x)), x, 1, (2*c*tan(e + f*x))/(f*(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))), x, 3, -(atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x))))/(sqrt(2)*a*sqrt(c)*f)) + tan(e + f*x)/(f*(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)), x, 4, (-3*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(4*sqrt(2)*a*c^(3/2)*f) - (3*tan(e + f*x))/(4*a*f*(c - c*sec(e + f*x))^(3/2)) + tan(e + f*x)/(f*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)), x, 5, (-15*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(32*sqrt(2)*a*c^(5/2)*f) - (5*tan(e + f*x))/(8*a*f*(c - c*sec(e + f*x))^(5/2)) + tan(e + f*x)/(f*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)) - (15*tan(e + f*x))/(32*a*c*f*(c - c*sec(e + f*x))^(3/2))]

    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(7/2)/(a + a*sec(e + f*x))^2, x, 4, -((64*c^4*tan(e + f*x))/(3*a^2*f*sqrt(c - c*sec(e + f*x)))) - (16*c^3*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(3*a^2*f) - (4*c^2*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(f*(a^2 + a^2*sec(e + f*x))) + (2*c*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(5/2)/(a + a*sec(e + f*x))^2, x, 3, (-16*c^3*tan(e + f*x))/(3*a^2*f*sqrt(c - c*sec(e + f*x))) - (8*c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))) + (2*c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(3/2)/(a + a*sec(e + f*x))^2, x, 2, -((4*c^2*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))) + (2*c*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)*sqrt(c - c*sec(e + f*x))/(a + a*sec(e + f*x))^2, x, 1, (2*c*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x))), x, 4, -(atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x))))/(2*sqrt(2)*a^2*sqrt(c)*f)) + tan(e + f*x)/(3*f*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x))) + tan(e + f*x)/(2*f*(a^2 + a^2*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(3/2)), x, 5, (-5*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(8*sqrt(2)*a^2*c^(3/2)*f) - (5*tan(e + f*x))/(8*a^2*f*(c - c*sec(e + f*x))^(3/2)) + tan(e + f*x)/(3*f*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(3/2)) + (5*tan(e + f*x))/(6*f*(a^2 + a^2*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(5/2)), x, 6, (-35*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(64*sqrt(2)*a^2*c^(5/2)*f) - (35*tan(e + f*x))/(48*a^2*f*(c - c*sec(e + f*x))^(5/2)) + tan(e + f*x)/(3*f*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(5/2)) + (7*tan(e + f*x))/(6*f*(a^2 + a^2*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)) - (35*tan(e + f*x))/(64*a^2*c*f*(c - c*sec(e + f*x))^(3/2))]

    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(7/2)/(a + a*sec(e + f*x))^3, x, 4, (32*c^4*tan(e + f*x))/(5*a^3*f*sqrt(c - c*sec(e + f*x))) + (16*c^3*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(5*f*(a^3 + a^3*sec(e + f*x))) - (4*c^2*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(5*a*f*(a + a*sec(e + f*x))^2) + (2*c*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(5/2)/(a + a*sec(e + f*x))^3, x, 3, (16*c^3*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (8*c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + (2*c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3)]
    @test_int [sec(e + f*x)*(c - c*sec(e + f*x))^(3/2)/(a + a*sec(e + f*x))^3, x, 2, -((4*c^2*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x)))) + (2*c*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3)]
    @test_int [sec(e + f*x)*sqrt(c - c*sec(e + f*x))/(a + a*sec(e + f*x))^3, x, 1, (2*c*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*sqrt(c - c*sec(e + f*x))), x, 5, -(atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x))))/(4*sqrt(2)*a^3*sqrt(c)*f)) + tan(e + f*x)/(5*f*(a + a*sec(e + f*x))^3*sqrt(c - c*sec(e + f*x))) + tan(e + f*x)/(6*a*f*(a + a*sec(e + f*x))^2*sqrt(c - c*sec(e + f*x))) + tan(e + f*x)/(4*f*(a^3 + a^3*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(3/2)), x, 6, (-7*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(16*sqrt(2)*a^3*c^(3/2)*f) - (7*tan(e + f*x))/(16*a^3*f*(c - c*sec(e + f*x))^(3/2)) + tan(e + f*x)/(5*f*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(3/2)) + (7*tan(e + f*x))/(30*a*f*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(3/2)) + (7*tan(e + f*x))/(12*f*(a^3 + a^3*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(5/2)), x, 7, (-63*atan((sqrt(c)*tan(e + f*x))/(sqrt(2)*sqrt(c - c*sec(e + f*x)))))/(128*sqrt(2)*a^3*c^(5/2)*f) - (21*tan(e + f*x))/(32*a^3*f*(c - c*sec(e + f*x))^(5/2)) + tan(e + f*x)/(5*f*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(5/2)) + (3*tan(e + f*x))/(10*a*f*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(5/2)) + (21*tan(e + f*x))/(20*f*(a^3 + a^3*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)) - (63*tan(e + f*x))/(128*a^3*c*f*(c - c*sec(e + f*x))^(3/2))]

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(e + f*x)*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2), x, 1, (a*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2), x, 1, (a*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)), x, 1, -((c*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(f*sqrt(c - c*sec(e + f*x))))]
    @test_int [(sec(e + f*x)*sqrt(a + a*sec(e + f*x)))/sqrt(c - c*sec(e + f*x)), x, 1, (a*log(1 - sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c - c*sec(e + f*x))^(3/2), x, 1, -((sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(2*f*(c - c*sec(e + f*x))^(3/2)))]
    @test_int [(sec(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c - c*sec(e + f*x))^(5/2), x, 1, -((a*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(7/2), x, 2, (a^2*(c - c*sec(e + f*x))^(7/2)*tan(e + f*x))/(10*f*sqrt(a + a*sec(e + f*x))) + (a*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(7/2)*tan(e + f*x))/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(5/2), x, 2, (a^2*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(6*f*sqrt(a + a*sec(e + f*x))) + (a*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(4*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(3/2), x, 2, -((c^2*(a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(3*f*sqrt(c - c*sec(e + f*x)))) - (c*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(3*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x)), x, 1, -((c*(a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*sqrt(c - c*sec(e + f*x))))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(3/2))/sqrt(c - c*sec(e + f*x)), x, 2, (2*a^2*log(1 - sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (a*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(3/2))/(c - c*sec(e + f*x))^(3/2), x, 2, -((a*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(f*(c - c*sec(e + f*x))^(3/2))) - (a^2*log(1 - sec(e + f*x))*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(3/2))/(c - c*sec(e + f*x))^(5/2), x, 1, -((a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(4*f*(c - c*sec(e + f*x))^(5/2))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(3/2))/(c - c*sec(e + f*x))^(7/2), x, 2, -(((a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(6*f*(c - c*sec(e + f*x))^(7/2))) - ((a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(24*c*f*(c - c*sec(e + f*x))^(5/2))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(3/2))/(c - c*sec(e + f*x))^(9/2), x, 2, -((a*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(4*f*(c - c*sec(e + f*x))^(9/2))) + (a^2*tan(e + f*x))/(12*c*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(7/2))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(3/2))/(c - c*sec(e + f*x))^(11/2), x, 2, -((a*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(5*f*(c - c*sec(e + f*x))^(11/2))) + (a^2*tan(e + f*x))/(20*c*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(9/2))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(7/2), x, 3, (a^3*(c - c*sec(e + f*x))^(7/2)*tan(e + f*x))/(15*f*sqrt(a + a*sec(e + f*x))) + (2*a^2*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(7/2)*tan(e + f*x))/(15*f) + (a*(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(7/2)*tan(e + f*x))/(6*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(5/2), x, 3, -((2*c^3*(a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(15*f*sqrt(c - c*sec(e + f*x)))) - (c^2*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(5*f) - (c*(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(3/2), x, 2, -((c^2*(a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(6*f*sqrt(c - c*sec(e + f*x)))) - (c*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(4*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x)), x, 1, -((c*(a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(3*f*sqrt(c - c*sec(e + f*x))))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(5/2))/sqrt(c - c*sec(e + f*x)), x, 3, (4*a^3*log(1 - sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (2*a^2*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(f*sqrt(c - c*sec(e + f*x))) + (a*(a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(5/2))/(c - c*sec(e + f*x))^(3/2), x, 3, -((a*(a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(f*(c - c*sec(e + f*x))^(3/2))) - (4*a^3*log(1 - sec(e + f*x))*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (2*a^2*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(c*f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(5/2))/(c - c*sec(e + f*x))^(5/2), x, 3, -(a*(a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*(c - c*sec(e + f*x))^(5/2)) + (a^2*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(c*f*(c - c*sec(e + f*x))^(3/2)) + (a^3*log(1 - sec(e + f*x))*tan(e + f*x))/(c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(5/2))/(c - c*sec(e + f*x))^(7/2), x, 1, -(((a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(6*f*(c - c*sec(e + f*x))^(7/2)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(5/2))/(c - c*sec(e + f*x))^(9/2), x, 2, -(((a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(8*f*(c - c*sec(e + f*x))^(9/2))) - ((a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(48*c*f*(c - c*sec(e + f*x))^(7/2))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^(5/2))/(c - c*sec(e + f*x))^(11/2), x, 3, -(((a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(10*f*(c - c*sec(e + f*x))^(11/2))) - ((a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(40*c*f*(c - c*sec(e + f*x))^(9/2)) - ((a + a*sec(e + f*x))^(5/2)*tan(e + f*x))/(240*c^2*f*(c - c*sec(e + f*x))^(7/2))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(sec(e + f*x)*(c - c*sec(e + f*x))^(5/2))/sqrt(a + a*sec(e + f*x)), x, 3, (-4*c^3*log(1 + sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (2*c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) - (c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(c - c*sec(e + f*x))^(3/2))/sqrt(a + a*sec(e + f*x)), x, 2, (-2*c^2*log(1 + sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (c*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*sqrt(c - c*sec(e + f*x)))/sqrt(a + a*sec(e + f*x)), x, 1, -((c*log(1 + sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))))]
    @test_int [sec(e + f*x)/(sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))), x, 2, -((atanh(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))))]
    @test_int [sec(e + f*x)/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)), x, 3, -tan(e + f*x)/(2*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)) - (atanh(cos(e + f*x))*tan(e + f*x))/(2*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)), x, 4, -tan(e + f*x)/(4*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)) - tan(e + f*x)/(4*c*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)) - (atanh(cos(e + f*x))*tan(e + f*x))/(4*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    @test_int [(sec(e + f*x)*(c - c*sec(e + f*x))^(5/2))/(a + a*sec(e + f*x))^(3/2), x, 3, (4*c^3*log(1 + sec(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (2*c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))) + (c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [(sec(e + f*x)*(c - c*sec(e + f*x))^(3/2))/(a + a*sec(e + f*x))^(3/2), x, 2, (c^2*log(1 + sec(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (c*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [(sec(e + f*x)*sqrt(c - c*sec(e + f*x)))/(a + a*sec(e + f*x))^(3/2), x, 1, (sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(2*f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x))), x, 3, tan(e + f*x)/(2*f*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x))) - (atanh(cos(e + f*x))*tan(e + f*x))/(2*a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(3/2)), x, 3, csc(e + f*x)/(2*a*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (atanh(cos(e + f*x))*tan(e + f*x))/(2*a*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(5/2)), x, 4, (3*csc(e + f*x))/(8*a*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(4*f*(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(5/2)) - (3*atanh(cos(e + f*x))*tan(e + f*x))/(8*a*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    @test_int [(sec(e + f*x)*(c - c*sec(e + f*x))^(5/2))/(a + a*sec(e + f*x))^(5/2), x, 3, -((c^3*log(1 + sec(e + f*x))*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))) - (c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(a*f*(a + a*sec(e + f*x))^(3/2)) + (c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*(a + a*sec(e + f*x))^(5/2))]
    @test_int [(sec(e + f*x)*(c - c*sec(e + f*x))^(3/2))/(a + a*sec(e + f*x))^(5/2), x, 1, ((c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(4*f*(a + a*sec(e + f*x))^(5/2))]
    @test_int [(sec(e + f*x)*sqrt(c - c*sec(e + f*x)))/(a + a*sec(e + f*x))^(5/2), x, 1, (c*tan(e + f*x))/(2*f*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x))), x, 4, tan(e + f*x)/(4*f*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x))) + tan(e + f*x)/(4*a*f*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x))) - (atanh(cos(e + f*x))*tan(e + f*x))/(4*a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(3/2)), x, 4, (3*csc(e + f*x))/(8*a^2*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + tan(e + f*x)/(4*f*(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(3/2)) - (3*atanh(cos(e + f*x))*tan(e + f*x))/(8*a^2*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(5/2)), x, 4, (3*csc(e + f*x))/(8*a^2*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (cot(e + f*x)^2*csc(e + f*x))/(4*a^2*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (3*atanh(cos(e + f*x))*tan(e + f*x))/(8*a^2*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

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

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^n, x, 3, -((2^(1/2 + n)*c*HypergeometricFunctions._₂F₁(1/2 + m, 1/2 - n, 3/2 + m, (1/2)*(1 + sec(e + f*x)))*(1 - sec(e + f*x))^(1/2 - n)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(-1 + n)*tan(e + f*x))/(f*(1 + 2*m)))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^2, x, 3, (2^(1/2 + m)*a*HypergeometricFunctions._₂F₁(5/2, 1/2 - m, 7/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(1/2 - m)*(a + a*sec(e + f*x))^(-1 + m)*(c - c*sec(e + f*x))^2*tan(e + f*x))/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^1, x, 3, (2^(1/2 + m)*a*HypergeometricFunctions._₂F₁(3/2, 1/2 - m, 5/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(1/2 - m)*(a + a*sec(e + f*x))^(-1 + m)*(c - c*sec(e + f*x))*tan(e + f*x))/(3*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^1, x, 3, -((2^(1/2 + m)*a*HypergeometricFunctions._₂F₁(-(1/2), 1/2 - m, 1/2, (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(1/2 - m)*(a + a*sec(e + f*x))^(-1 + m)*tan(e + f*x))/(f*(c - c*sec(e + f*x))))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^2, x, 3, -((2^(1/2 + m)*a*HypergeometricFunctions._₂F₁(-(3/2), 1/2 - m, -(1/2), (1/2)*(1 - sec(e + f*x)))*(1 + sec(e + f*x))^(1/2 - m)*(a + a*sec(e + f*x))^(-1 + m)*tan(e + f*x))/(3*f*(c - c*sec(e + f*x))^2))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(5/2), x, 3, -((64*c^3*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(5 + 2*m)*(3 + 8*m + 4*m^2)*sqrt(c - c*sec(e + f*x)))) - (16*c^2*(a + a*sec(e + f*x))^m*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*(15 + 16*m + 4*m^2)) - (2*c*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(f*(5 + 2*m))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(3/2), x, 2, -((8*c^2*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(3 + 8*m + 4*m^2)*sqrt(c - c*sec(e + f*x)))) - (2*c*(a + a*sec(e + f*x))^m*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*(3 + 2*m))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(1/2), x, 1, -((2*c*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + 2*m)*sqrt(c - c*sec(e + f*x))))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(1/2), x, 2, -((HypergeometricFunctions._₂F₁(1, 1/2 + m, 3/2 + m, (1/2)*(1 + sec(e + f*x)))*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*(1 + 2*m)*sqrt(c - c*sec(e + f*x))))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(3/2), x, 2, -((HypergeometricFunctions._₂F₁(2, 1/2 + m, 3/2 + m, (1/2)*(1 + sec(e + f*x)))*(a + a*sec(e + f*x))^m*tan(e + f*x))/(2*c*f*(1 + 2*m)*sqrt(c - c*sec(e + f*x))))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(5/2), x, 2, -((HypergeometricFunctions._₂F₁(3, 1/2 + m, 3/2 + m, (1/2)*(1 + sec(e + f*x)))*(a + a*sec(e + f*x))^m*tan(e + f*x))/(4*c^2*f*(1 + 2*m)*sqrt(c - c*sec(e + f*x))))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(m + 3), x, 3, -(((a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(-3 - m)*tan(e + f*x))/(f*(1 + 2*m))) + (2*(a + a*sec(e + f*x))^(1 + m)*(c - c*sec(e + f*x))^(-3 - m)*tan(e + f*x))/(a*f*(3 + 8*m + 4*m^2)) - (2*(a + a*sec(e + f*x))^(2 + m)*(c - c*sec(e + f*x))^(-3 - m)*tan(e + f*x))/(a^2*f*(1 + 2*m)*(15 + 16*m + 4*m^2))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(m + 2), x, 2, -(((a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(-2 - m)*tan(e + f*x))/(f*(1 + 2*m))) + ((a + a*sec(e + f*x))^(1 + m)*(c - c*sec(e + f*x))^(-2 - m)*tan(e + f*x))/(a*f*(3 + 8*m + 4*m^2))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(m + 1), x, 1, -(((a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(-1 - m)*tan(e + f*x))/(f*(1 + 2*m)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(m + 0), x, 3, -((2^(1/2 - m)*c*HypergeometricFunctions._₂F₁(1/2 + m, 1/2 + m, 3/2 + m, (1/2)*(1 + sec(e + f*x)))*(1 - sec(e + f*x))^(1/2 + m)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(-1 - m)*tan(e + f*x))/(f*(1 + 2*m)))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(m - 1), x, 3, -((2^(3/2 - m)*c*HypergeometricFunctions._₂F₁(-(1/2) + m, 1/2 + m, 3/2 + m, (1/2)*(1 + sec(e + f*x)))*(1 - sec(e + f*x))^(-(1/2) + m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/((c - c*sec(e + f*x))^m*(f*(1 + 2*m))))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^m/(c - c*sec(e + f*x))^(m - 2), x, 3, -((2^(5/2 - m)*c^2*HypergeometricFunctions._₂F₁(-(3/2) + m, 1/2 + m, 3/2 + m, (1/2)*(1 + sec(e + f*x)))*(1 - sec(e + f*x))^(-(1/2) + m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/((c - c*sec(e + f*x))^m*(f*(1 + 2*m))))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+a*sec(e+f*x))^m*(c-c*sec(e+f*x))=#

    @test_int [sec(e + f*x)^2*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x)), x, 10, (a^3*c*atanh(sin(e + f*x)))/(4*f) + (a^3*c*sec(e + f*x)*tan(e + f*x))/(4*f) - (a^3*c*sec(e + f*x)^3*tan(e + f*x))/(2*f) - (2*a^3*c*tan(e + f*x)^3)/(3*f) - (a^3*c*tan(e + f*x)^5)/(5*f)]
    @test_int [sec(e + f*x)^2*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x)), x, 7, (a^2*c*atanh(sin(e + f*x)))/(8*f) + (a^2*c*sec(e + f*x)*tan(e + f*x))/(8*f) - (a^2*c*sec(e + f*x)^3*tan(e + f*x))/(4*f) - (a^2*c*tan(e + f*x)^3)/(3*f)]
    @test_int [sec(e + f*x)^2*(a + a*sec(e + f*x))^1*(c - c*sec(e + f*x)), x, 3, -((a*c*tan(e + f*x)^3)/(3*f))]
    @test_int [sec(e + f*x)^2/(a + a*sec(e + f*x))^1*(c - c*sec(e + f*x)), x, 5, (2*c*atanh(sin(e + f*x)))/(a*f) - (c*tan(e + f*x))/(a*f) - (2*c*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)^2/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x)), x, 4, -((c*atanh(sin(e + f*x)))/(a^2*f)) + (7*c*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x))) - (2*c*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]
    @test_int [sec(e + f*x)^2/(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x)), x, 3, -((2*c*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3)) + (11*c*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) - (4*c*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x)))]

    @test_int [(g*sec(e + f*x))^p*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x)), x, 5, -((a^2*c*(cos(e + f*x)^2)^((3 + p)/2)*HypergeometricFunctions._₂F₁(3/2, (3 + p)/2, 5/2, sin(e + f*x)^2)*(g*sec(e + f*x))^p*tan(e + f*x)^3)/(3*f)) - (a^2*c*(cos(e + f*x)^2)^((4 + p)/2)*HypergeometricFunctions._₂F₁(3/2, (4 + p)/2, 5/2, sin(e + f*x)^2)*(g*sec(e + f*x))^(1 + p)*tan(e + f*x)^3)/(3*f*g)]
    @test_int [(g*sec(e + f*x))^p*(a + a*sec(e + f*x))^1*(c - c*sec(e + f*x)), x, 2, -((a*c*(cos(e + f*x)^2)^((3 + p)/2)*HypergeometricFunctions._₂F₁(3/2, (3 + p)/2, 5/2, sin(e + f*x)^2)*(g*sec(e + f*x))^p*tan(e + f*x)^3)/(3*f))]
    @test_int [(g*sec(e + f*x))^p/(a + a*sec(e + f*x))^1*(c - c*sec(e + f*x)), x, 6, -((c*g*(1 - 2*p)*HypergeometricFunctions._₂F₁(1/2, (1 - p)/2, (3 - p)/2, cos(e + f*x)^2)*(g*sec(e + f*x))^(-1 + p)*sin(e + f*x))/(a*f*(1 - p)*sqrt(sin(e + f*x)^2))) + (2*c*HypergeometricFunctions._₂F₁(1/2, -(p/2), (2 - p)/2, cos(e + f*x)^2)*(g*sec(e + f*x))^p*sin(e + f*x))/(a*f*sqrt(sin(e + f*x)^2)) - (2*c*(g*sec(e + f*x))^p*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [(g*sec(e + f*x))^p/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x)), x, 7, -((c*g*(3 - 4*p)*HypergeometricFunctions._₂F₁(1/2, (1 - p)/2, (3 - p)/2, cos(e + f*x)^2)*(g*sec(e + f*x))^(-1 + p)*sin(e + f*x))/(3*a^2*f*sqrt(sin(e + f*x)^2))) + (c*(5 - 4*p)*HypergeometricFunctions._₂F₁(1/2, -(p/2), (2 - p)/2, cos(e + f*x)^2)*(g*sec(e + f*x))^p*sin(e + f*x))/(3*a^2*f*sqrt(sin(e + f*x)^2)) - (c*(5 - 4*p)*(g*sec(e + f*x))^p*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x))) - (2*c*(g*sec(e + f*x))^p*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+a*sec(e+f*x))^(m/2) / (c-c*sec(e+f*x))=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(g*sec(e + f*x))^(3/2)*sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x)), x, 5, -((2*sqrt(a)*g^(3/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(c*f)) + (2*g*cot(e + f*x)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))/(c*f), -((2*a*g*sqrt(g*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x)))) + (2*a*g^(3/2)*atan((sqrt(c)*sqrt(g*sec(e + f*x)))/(sqrt(g)*sqrt(c - c*sec(e + f*x))))*tan(e + f*x))/(sqrt(c)*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)^2/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))), x, 4, -(atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))))/(sqrt(2)*sqrt(a)*c*f)) + (cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(a*c*f), -(tan(e + f*x)/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x)))) - (atanh(sqrt(c - c*sec(e + f*x))/(sqrt(2)*sqrt(c)))*tan(e + f*x))/(sqrt(2)*sqrt(c)*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    @test_int [sec(e + f*x)^(5/2)/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))), x, 8, -((2*asinh((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*c*f)) + atanh((sqrt(a)*sqrt(sec(e + f*x))*sin(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))))/(sqrt(2)*sqrt(a)*c*f) + (csc(e + f*x)*sqrt(a + a*sec(e + f*x)))/(a*c*f*sqrt(sec(e + f*x))), -((sec(e + f*x)^(3/2)*sin(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x)))) + (2*atan((sqrt(c)*sqrt(sec(e + f*x)))/sqrt(c - c*sec(e + f*x)))*tan(e + f*x))/(sqrt(c)*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (atan((sqrt(2)*sqrt(c)*sqrt(sec(e + f*x)))/sqrt(c - c*sec(e + f*x)))*tan(e + f*x))/(sqrt(2)*sqrt(c)*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    @test_int [(g*sec(e + f*x))^(3/2)/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))), x, 4, -((g^(3/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(2)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(2)*sqrt(a)*c*f)) + (g*cot(e + f*x)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))/(a*c*f), -((g*sqrt(g*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x)))) + (g^(3/2)*atan((sqrt(2)*sqrt(c)*sqrt(g*sec(e + f*x)))/(sqrt(g)*sqrt(c - c*sec(e + f*x))))*tan(e + f*x))/(sqrt(2)*sqrt(c)*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(g*sec(e + f*x))^(5/2)/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))), x, 8, -((2*g^(5/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*c*f)) + (g^(5/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(2)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(2)*sqrt(a)*c*f) + (g^2*cot(e + f*x)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))/(a*c*f), -((g^2*sqrt(g*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x)))) + (2*g^(5/2)*atan((sqrt(c)*sqrt(g*sec(e + f*x)))/(sqrt(g)*sqrt(c - c*sec(e + f*x))))*tan(e + f*x))/(sqrt(c)*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (g^(5/2)*atan((sqrt(2)*sqrt(c)*sqrt(g*sec(e + f*x)))/(sqrt(g)*sqrt(c - c*sec(e + f*x))))*tan(e + f*x))/(sqrt(2)*sqrt(c)*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

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

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)^2/(sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))), x, 3, (log(tan(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

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

    @test_int [(sec(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c - d*sec(e + f*x)), x, 2, (2*sqrt(a)*atanh((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(c - d)*sqrt(a + a*sec(e + f*x)))))/(sqrt(c - d)*sqrt(d)*f)]

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^4, x, 8, (a*(8*c^4 + 16*c^3*d + 24*c^2*d^2 + 12*c*d^3 + 3*d^4)*atanh(sin(e + f*x)))/(8*f) + (a*(12*c^4 + 95*c^3*d + 112*c^2*d^2 + 80*c*d^3 + 16*d^4)*tan(e + f*x))/(30*f) + (a*d*(24*c^3 + 130*c^2*d + 116*c*d^2 + 45*d^3)*sec(e + f*x)*tan(e + f*x))/(120*f) + (a*(12*c^2 + 35*c*d + 16*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(60*f) + (a*(4*c + 5*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(20*f) + (a*(c + d*sec(e + f*x))^4*tan(e + f*x))/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^3, x, 7, (a*(8*c^3 + 12*c^2*d + 12*c*d^2 + 3*d^3)*atanh(sin(e + f*x)))/(8*f) + (a*(3*c^3 + 16*c^2*d + 12*c*d^2 + 4*d^3)*tan(e + f*x))/(6*f) + (a*d*(6*c^2 + 20*c*d + 9*d^2)*sec(e + f*x)*tan(e + f*x))/(24*f) + (a*(3*c + 4*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(12*f) + (a*(c + d*sec(e + f*x))^3*tan(e + f*x))/(4*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2, x, 6, (a*(2*c^2 + 2*c*d + d^2)*atanh(sin(e + f*x)))/(2*f) + (2*a*(c^2 + 3*c*d + d^2)*tan(e + f*x))/(3*f) + (a*d*(2*c + 3*d)*sec(e + f*x)*tan(e + f*x))/(6*f) + (a*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^1, x, 5, (a*(2*c + d)*atanh(sin(e + f*x)))/(2*f) + (a*(c + d)*tan(e + f*x))/f + (a*d*sec(e + f*x)*tan(e + f*x))/(2*f)]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x)))/(c + d*sec(e + f*x))^1, x, 5, (a*atanh(sin(e + f*x)))/(d*f) - (2*a*sqrt(c - d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(d*sqrt(c + d)*f)]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x)))/(c + d*sec(e + f*x))^2, x, 5, (2*a*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(sqrt(c - d)*(c + d)^(3/2)*f) + (a*tan(e + f*x))/((c + d)*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x)))/(c + d*sec(e + f*x))^3, x, 6, (a*(2*c - d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(3/2)*(c + d)^(5/2)*f) + (a*tan(e + f*x))/(2*(c + d)*f*(c + d*sec(e + f*x))^2) + (a*(c - 2*d)*tan(e + f*x))/(2*(c - d)*(c + d)^2*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x)))/(c + d*sec(e + f*x))^4, x, 7, (a*(2*c^2 - 2*c*d + d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(5/2)*(c + d)^(7/2)*f) + (a*tan(e + f*x))/(3*(c + d)*f*(c + d*sec(e + f*x))^3) + (a*(2*c - 3*d)*tan(e + f*x))/(6*(c - d)*(c + d)^2*f*(c + d*sec(e + f*x))^2) + (a*(c - 4*d)*(2*c - d)*tan(e + f*x))/(6*(c - d)^2*(c + d)^3*f*(c + d*sec(e + f*x)))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^4, x, 9, (a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*atanh(sin(e + f*x)))/(16*f) - (a^2*(4*c^5 - 48*c^4*d - 311*c^3*d^2 - 448*c^2*d^3 - 288*c*d^4 - 64*d^5)*tan(e + f*x))/(60*d*f) - (a^2*(8*c^4 - 96*c^3*d - 438*c^2*d^2 - 464*c*d^3 - 165*d^4)*sec(e + f*x)*tan(e + f*x))/(240*f) - (a^2*(4*c^3 - 48*c^2*d - 123*c*d^2 - 64*d^3)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(120*d*f) - (a^2*(4*c^2 - 48*c*d - 55*d^2)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(120*d*f) - (a^2*(c - 12*d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(30*d*f) + (a^2*(c + d*sec(e + f*x))^5*tan(e + f*x))/(6*d*f), (a^2*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*tan(e + f*x))/(16*f) + (a^3*(24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(8*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((24*c^4 + 64*c^3*d + 84*c^2*d^2 + 48*c*d^3 + 11*d^4)*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(48*f) + (d*(9*c + 2*d)*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2*tan(e + f*x))/(30*f) + (d*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^3*tan(e + f*x))/(6*f) + (1/(120*f))*(d*(a + a*sec(e + f*x))^2*(2*(52*c^3 + 56*c^2*d + 48*c*d^2 + 9*d^3) + d*(48*c^2 + 32*c*d + 19*d^2)*sec(e + f*x))*tan(e + f*x))]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^3, x, 8, (3*a^2*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*atanh(sin(e + f*x)))/(8*f) - (a^2*(c^4 - 10*c^3*d - 44*c^2*d^2 - 40*c*d^3 - 12*d^4)*tan(e + f*x))/(10*d*f) - (a^2*(2*c^3 - 20*c^2*d - 57*c*d^2 - 30*d^3)*sec(e + f*x)*tan(e + f*x))/(40*f) - (a^2*(c^2 - 10*c*d - 12*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(20*d*f) - (a^2*(c - 10*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(20*d*f) + (a^2*(c + d*sec(e + f*x))^4*tan(e + f*x))/(5*d*f), (3*a^2*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*tan(e + f*x))/(8*f) + (3*a^3*(2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(4*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((2*c + d)*(2*c^2 + 3*c*d + 2*d^2)*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(8*f) + (d*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2*tan(e + f*x))/(5*f) + (d*(a + a*sec(e + f*x))^2*(2*(8*c^2 + 5*c*d + 2*d^2) + d*(7*c + 2*d)*sec(e + f*x))*tan(e + f*x))/(20*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2, x, 8, (a^2*(12*c^2 + 16*c*d + 7*d^2)*atanh(sin(e + f*x)))/(8*f) - (a^2*(c^3 - 8*c^2*d - 20*c*d^2 - 8*d^3)*tan(e + f*x))/(6*d*f) - (a^2*(2*c*(c - 8*d) - 21*d^2)*sec(e + f*x)*tan(e + f*x))/(24*f) - (a^2*(c - 8*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(12*d*f) + (a^2*(c + d*sec(e + f*x))^3*tan(e + f*x))/(4*d*f), (a^2*(12*c^2 + 16*c*d + 7*d^2)*tan(e + f*x))/(8*f) + (a^3*(12*c^2 + 16*c*d + 7*d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(4*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (d*(5*c + 2*d)*(a + a*sec(e + f*x))^2*tan(e + f*x))/(12*f) + ((12*c^2 + 16*c*d + 7*d^2)*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(24*f) + (d*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))*tan(e + f*x))/(4*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^1, x, 6, (a^2*(3*c + 2*d)*atanh(sin(e + f*x)))/(2*f) + (2*a^2*(3*c + 2*d)*tan(e + f*x))/(3*f) + (a^2*(3*c + 2*d)*sec(e + f*x)*tan(e + f*x))/(6*f) + (d*(a + a*sec(e + f*x))^2*tan(e + f*x))/(3*f)]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c + d*sec(e + f*x))^1, x, 8, -((a^2*(c - 2*d)*atanh(sin(e + f*x)))/(d^2*f)) + (2*a^2*(c - d)^(3/2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(d^2*sqrt(c + d)*f) + (a^2*tan(e + f*x))/(d*f), (a^2*tan(e + f*x))/(d*f) - (2*a^3*(c - 2*d)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(d^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*a^3*(c - d)^(3/2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(d^2*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c + d*sec(e + f*x))^2, x, 8, (a^2*atanh(sin(e + f*x)))/(d^2*f) - (2*a^2*sqrt(c - d)*(c + 2*d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(d^2*(c + d)^(3/2)*f) - (a^2*(c - d)*tan(e + f*x))/(d*(c + d)*f*(c + d*sec(e + f*x))), (2*a^3*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(d^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*a^3*sqrt(c - d)*(c + 2*d)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(d^2*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a^2*(c - d)*tan(e + f*x))/(d*(c + d)*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c + d*sec(e + f*x))^3, x, 5, (3*a^2*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(sqrt(c - d)*(c + d)^(5/2)*f) + ((a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(2*(c + d)*f*(c + d*sec(e + f*x))^2) + (3*a^2*tan(e + f*x))/(2*(c + d)^2*f*(c + d*sec(e + f*x))), -((3*a^3*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(sqrt(c - d)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) + ((a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(2*(c + d)*f*(c + d*sec(e + f*x))^2) + (3*a^2*tan(e + f*x))/(2*(c + d)^2*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c + d*sec(e + f*x))^4, x, 6, (a^2*(3*c - 2*d)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/((c - d)^(3/2)*(c + d)^(7/2)*f) - (d*(a + a*sec(e + f*x))^2*tan(e + f*x))/(3*(c^2 - d^2)*f*(c + d*sec(e + f*x))^3) + ((3*c - 2*d)*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(6*(c - d)*(c + d)^2*f*(c + d*sec(e + f*x))^2) + (a^2*(3*c - 2*d)*tan(e + f*x))/(2*(c - d)*(c + d)^3*f*(c + d*sec(e + f*x))), -((a^3*(3*c - 2*d)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/((c - d)^(3/2)*(c + d)^(7/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) - (d*(a + a*sec(e + f*x))^2*tan(e + f*x))/(3*(c^2 - d^2)*f*(c + d*sec(e + f*x))^3) + ((3*c - 2*d)*(a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(6*(c - d)*(c + d)^2*f*(c + d*sec(e + f*x))^2) + (a^2*(3*c - 2*d)*tan(e + f*x))/(2*(c - d)*(c + d)^3*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^2)/(c + d*sec(e + f*x))^5, x, 8, (a^2*(12*c^2 - 16*c*d + 7*d^2)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(4*(c - d)^(5/2)*(c + d)^(9/2)*f) - (a^2*(c - d)*tan(e + f*x))/(4*d*(c + d)*f*(c + d*sec(e + f*x))^4) + (a^2*(c + 8*d)*tan(e + f*x))/(12*d*(c + d)^2*f*(c + d*sec(e + f*x))^3) + (a^2*(2*c^2 + 16*c*d - 21*d^2)*tan(e + f*x))/(24*(c - d)*d*(c + d)^3*f*(c + d*sec(e + f*x))^2) + (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*tan(e + f*x))/(24*(c - d)^2*d*(c + d)^4*f*(c + d*sec(e + f*x))), -((a^3*(12*c^2 - 16*c*d + 7*d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(4*(c - d)^(5/2)*(c + d)^(9/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) - (a^2*(c - d)*tan(e + f*x))/(4*d*(c + d)*f*(c + d*sec(e + f*x))^4) + (a^2*(c + 8*d)*tan(e + f*x))/(12*d*(c + d)^2*f*(c + d*sec(e + f*x))^3) + (a^2*(2*c^2 + 16*c*d - 21*d^2)*tan(e + f*x))/(24*(c - d)*d*(c + d)^3*f*(c + d*sec(e + f*x))^2) + (a^2*(2*c^3 + 16*c^2*d - 59*c*d^2 + 32*d^3)*tan(e + f*x))/(24*(c - d)^2*d*(c + d)^4*f*(c + d*sec(e + f*x)))]

    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^3, x, 9, (a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*atanh(sin(e + f*x)))/(16*f) + (a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*tan(e + f*x))/(16*f) + ((40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(48*f) + (a*(3*c + 8*d)*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2*tan(e + f*x))/(30*f) + (a*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^3*tan(e + f*x))/(6*f) + (a*(a + a*sec(e + f*x))^2*(2*(4*c^3 + 74*c^2*d + 66*c*d^2 + 21*d^3) + d*(6*c^2 + 62*c*d + 31*d^2)*sec(e + f*x))*tan(e + f*x))/(120*f), (a^3*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*tan(e + f*x))/(16*f) + (a^4*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(8*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a*(40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a + a*sec(e + f*x))^2*tan(e + f*x))/(120*f) + ((40*c^3 + 90*c^2*d + 78*c*d^2 + 23*d^3)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(48*f) + (d*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^2*tan(e + f*x))/(6*f) + (d*(a + a*sec(e + f*x))^3*(70*c^2 + 54*c*d + 19*d^2 + 4*d*(8*c + 3*d)*sec(e + f*x))*tan(e + f*x))/(120*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^2, x, 9, (a^3*(20*c^2 + 30*c*d + 13*d^2)*atanh(sin(e + f*x)))/(8*f) + (a^3*(2*c^4 - 15*c^3*d + 72*c^2*d^2 + 180*c*d^3 + 76*d^4)*tan(e + f*x))/(30*d^2*f) + (a^3*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*sec(e + f*x)*tan(e + f*x))/(120*d*f) + (a^3*(2*c^2 - 15*c*d + 76*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(60*d^2*f) - (a^3*(2*c - 11*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(20*d^2*f) + ((a^3 + a^3*sec(e + f*x))*(c + d*sec(e + f*x))^3*tan(e + f*x))/(5*d*f), (a^3*(20*c^2 + 30*c*d + 13*d^2)*tan(e + f*x))/(8*f) + (a^4*(20*c^2 + 30*c*d + 13*d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(4*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a*(20*c^2 + 30*c*d + 13*d^2)*(a + a*sec(e + f*x))^2*tan(e + f*x))/(60*f) + (3*d*(2*c + d)*(a + a*sec(e + f*x))^3*tan(e + f*x))/(20*f) + ((20*c^2 + 30*c*d + 13*d^2)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(24*f) + (d*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))*tan(e + f*x))/(5*f)]
    @test_int [sec(e + f*x)*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^1, x, 10, (5*a^3*(4*c + 3*d)*atanh(sin(e + f*x)))/(8*f) + (a^3*(4*c + 3*d)*tan(e + f*x))/f + (3*a^3*(4*c + 3*d)*sec(e + f*x)*tan(e + f*x))/(8*f) + (d*(a + a*sec(e + f*x))^3*tan(e + f*x))/(4*f) + (a^3*(4*c + 3*d)*tan(e + f*x)^3)/(12*f)]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/(c + d*sec(e + f*x))^1, x, 9, (a^3*atanh(sin(e + f*x)))/(2*d*f) + (a^3*(c^2 - 3*c*d + 3*d^2)*atanh(sin(e + f*x)))/(d^3*f) - (2*a^3*(c - d)^(5/2)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(d^3*sqrt(c + d)*f) - (a^3*(c - 3*d)*tan(e + f*x))/(d^2*f) + (a^3*sec(e + f*x)*tan(e + f*x))/(2*d*f), -((a^3*(2*c - 5*d)*tan(e + f*x))/(2*d^2*f)) + (a^4*(2*c^2 - 6*c*d + 7*d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(d^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*a^4*(c - d)^(5/2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(d^3*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(2*d*f)]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/(c + d*sec(e + f*x))^2, x, 9, -((a^3*(2*c - 3*d)*atanh(sin(e + f*x)))/(d^3*f)) + (2*a^3*(c - d)^(3/2)*(2*c + 3*d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(d^3*(c + d)^(3/2)*f) + (2*a^3*c*tan(e + f*x))/(d^2*(c + d)*f) - ((c - d)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(d*(c + d)*f*(c + d*sec(e + f*x))), (2*a^3*c*tan(e + f*x))/(d^2*(c + d)*f) - (2*a^4*(2*c - 3*d)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(d^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*a^4*(c - d)^(3/2)*(2*c + 3*d)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(d^3*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - d)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(d*(c + d)*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/(c + d*sec(e + f*x))^3, x, 9, (a^3*atanh(sin(e + f*x)))/(d^3*f) - (a^3*sqrt(c - d)*(2*c^2 + 6*c*d + 7*d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(d^3*(c + d)^(5/2)*f) - ((c - d)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(2*d*(c + d)*f*(c + d*sec(e + f*x))^2) - (a^3*(c - d)*(2*c + 5*d)*tan(e + f*x))/(2*d^2*(c + d)^2*f*(c + d*sec(e + f*x))), (2*a^4*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(d^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a^4*sqrt(c - d)*(2*c^2 + 6*c*d + 7*d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(d^3*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - d)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(2*d*(c + d)*f*(c + d*sec(e + f*x))^2) - (a^3*(c - d)*(2*c + 5*d)*tan(e + f*x))/(2*d^2*(c + d)^2*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/(c + d*sec(e + f*x))^4, x, 6, (5*a^3*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(sqrt(c - d)*(c + d)^(7/2)*f) + (a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(3*(c + d)*f*(c + d*sec(e + f*x))^3) - (5*a^3*(c - d)*tan(e + f*x))/(6*d*(c + d)^2*f*(c + d*sec(e + f*x))^2) + (5*a^3*(c + 4*d)*tan(e + f*x))/(6*d*(c + d)^3*f*(c + d*sec(e + f*x))), -((5*a^4*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(sqrt(c - d)*(c + d)^(7/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) + (a*(a + a*sec(e + f*x))^2*tan(e + f*x))/(3*(c + d)*f*(c + d*sec(e + f*x))^3) + (5*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(6*(c + d)^2*f*(c + d*sec(e + f*x))^2) + (5*a^3*tan(e + f*x))/(2*(c + d)^3*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + a*sec(e + f*x))^3)/(c + d*sec(e + f*x))^5, x, 7, (5*a^3*(4*c - 3*d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(4*(c - d)^(3/2)*(c + d)^(9/2)*f) - (d*(a + a*sec(e + f*x))^3*tan(e + f*x))/(4*(c^2 - d^2)*f*(c + d*sec(e + f*x))^4) + (a*(4*c - 3*d)*(a + a*sec(e + f*x))^2*tan(e + f*x))/(12*(c - d)*(c + d)^2*f*(c + d*sec(e + f*x))^3) - (5*a^3*(4*c - 3*d)*tan(e + f*x))/(24*d*(c + d)^3*f*(c + d*sec(e + f*x))^2) + (5*a^3*(4*c - 3*d)*(c + 4*d)*tan(e + f*x))/(24*(c - d)*d*(c + d)^4*f*(c + d*sec(e + f*x))), -((5*a^4*(4*c - 3*d)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(4*(c - d)^(3/2)*(c + d)^(9/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) - (d*(a + a*sec(e + f*x))^3*tan(e + f*x))/(4*(c^2 - d^2)*f*(c + d*sec(e + f*x))^4) + (a*(4*c - 3*d)*(a + a*sec(e + f*x))^2*tan(e + f*x))/(12*(c - d)*(c + d)^2*f*(c + d*sec(e + f*x))^3) + (5*(4*c - 3*d)*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(24*(c - d)*(c + d)^3*f*(c + d*sec(e + f*x))^2) + (5*a^3*(4*c - 3*d)*tan(e + f*x))/(8*(c - d)*(c + d)^4*f*(c + d*sec(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^4)/(a + a*sec(e + f*x)), x, 7, (d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*atanh(sin(e + f*x)))/(2*a*f) - ((3*c - 4*d)*d*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*a*f) + ((c - d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(f*(a + a*sec(e + f*x))) - (d*(4*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3) + d*(6*c^2 - 20*c*d + 9*d^2)*sec(e + f*x))*tan(e + f*x))/(6*a*f), (d*(8*c^3 - 12*c^2*d + 12*c*d^2 - 3*d^3)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((3*c - 4*d)*d*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*a*f) + ((c - d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(f*(a + a*sec(e + f*x))) - (d*(4*(3*c^3 - 16*c^2*d + 12*c*d^2 - 4*d^3) + d*(6*c^2 - 20*c*d + 9*d^2)*sec(e + f*x))*tan(e + f*x))/(6*a*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^3)/(a + a*sec(e + f*x)), x, 6, (3*d*(2*c^2 - 2*c*d + d^2)*atanh(sin(e + f*x)))/(2*a*f) + ((c - d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(f*(a + a*sec(e + f*x))) - (d*(4*(c^2 - 3*c*d + d^2) + (2*c - 3*d)*d*sec(e + f*x))*tan(e + f*x))/(2*a*f), (3*d*(2*c^2 - 2*c*d + d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(f*(a + a*sec(e + f*x))) - (d*(4*(c^2 - 3*c*d + d^2) + (2*c - 3*d)*d*sec(e + f*x))*tan(e + f*x))/(2*a*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^2)/(a + a*sec(e + f*x)), x, 6, (d^2*tan(e + f*x))/(a*f) + ((c - d)^2*tan(e + f*x))/(f*(a + a*sec(e + f*x))) + ((2*c - d)*d*atanh(sin(e + f*x)))/(a*f), (d^2*tan(e + f*x))/(a*f) + ((c - d)^2*tan(e + f*x))/(f*(a + a*sec(e + f*x))) + (2*(2*c - d)*d*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^1)/(a + a*sec(e + f*x)), x, 3, (d*atanh(sin(e + f*x)))/(a*f) + ((c - d)*tan(e + f*x))/(f*(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c + d*sec(e + f*x))^1), x, 4, tan(e + f*x)/((c - d)*f*(a + a*sec(e + f*x))) - (2*d*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(a*(c - d)^(3/2)*sqrt(c + d)*f), tan(e + f*x)/((c - d)*f*(a + a*sec(e + f*x))) + (2*d*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/((c - d)^(3/2)*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2), x, 6, ((c + 2*d)*tan(e + f*x))/((c - d)^2*(c + d)*f*(a + a*sec(e + f*x))) - (2*d*(2*c + d)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(a*(c - d)^(5/2)*(c + d)^(3/2)*f) - (d*tan(e + f*x))/((c^2 - d^2)*f*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))), ((c + 2*d)*tan(e + f*x))/((c - d)^2*(c + d)*f*(a + a*sec(e + f*x))) + (2*d*(2*c + d)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/((c - d)^(5/2)*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d*tan(e + f*x))/((c^2 - d^2)*f*(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))*(c + d*sec(e + f*x))^3), x, 7, (-3*d*(2*c^2 + 2*c*d + d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(a*(c - d)^(7/2)*(c + d)^(5/2)*f) + (d*(2*c + 3*d)*tan(e + f*x))/(2*a*(c - d)^2*(c + d)*f*(c + d*sec(e + f*x))^2) + tan(e + f*x)/((c - d)*f*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) + (d*(2*c + d)*(c + 4*d)*tan(e + f*x))/(2*a*(c - d)^3*(c + d)^2*f*(c + d*sec(e + f*x))), If(13>=8, ((2*c + d)*(c + 4*d)*tan(e + f*x))/(2*(c - d)^3*(c + d)^2*f*(a + a*sec(e + f*x))) + (3*d*(2*c^2 + 2*c*d + d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/((c - d)^(7/2)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d*tan(e + f*x))/(2*(c^2 - d^2)*f*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) - (d*(4*c + d)*tan(e + f*x))/(2*(c^2 - d^2)^2*f*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))), ((2*c + d)*(c + 4*d)*tan(e + f*x))/(2*(c - d)^3*(c + d)^2*f*(a + a*sec(e + f*x))) + (3*d*(2*c^2 + 2*c*d + d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/((c - d)^(7/2)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d*tan(e + f*x))/(2*(c^2 - d^2)*f*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) - (d*(4*c + d)*tan(e + f*x))/(2*(c^2 - d^2)^2*f*(a + a*sec(e + f*x))*(c + d*sec(e + f*x))))]

    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^5)/(a + a*sec(e + f*x))^2, x, 8, (5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*atanh(sin(e + f*x)))/(2*a^2*f) - (d*(c^2 + 10*c*d - 12*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*a^2*f) + ((c - d)*(c + 10*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))) + ((c - d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - (d*(4*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4) + d*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*sec(e + f*x))*tan(e + f*x))/(6*a^2*f), (5*(2*c - d)*d^2*(2*c^2 - 3*c*d + 2*d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d*(c^2 + 10*c*d - 12*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*a^2*f) + ((c - d)*(c + 10*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))) + ((c - d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - (d*(4*(c^4 + 10*c^3*d - 44*c^2*d^2 + 40*c*d^3 - 12*d^4) + d*(2*c^3 + 20*c^2*d - 57*c*d^2 + 30*d^3)*sec(e + f*x))*tan(e + f*x))/(6*a^2*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^4)/(a + a*sec(e + f*x))^2, x, 7, (d^2*(12*c^2 - 16*c*d + 7*d^2)*atanh(sin(e + f*x)))/(2*a^2*f) + ((c - d)*(c + 8*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))) + ((c - d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - (d*(4*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3) + d*(2*c^2 + 16*c*d - 21*d^2)*sec(e + f*x))*tan(e + f*x))/(6*a^2*f), (d^2*(12*c^2 - 16*c*d + 7*d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - d)*(c + 8*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))) + ((c - d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) - (d*(4*(c^3 + 8*c^2*d - 20*c*d^2 + 8*d^3) + d*(2*c^2 + 16*c*d - 21*d^2)*sec(e + f*x))*tan(e + f*x))/(6*a^2*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^3)/(a + a*sec(e + f*x))^2, x, 6, ((3*c - 2*d)*d^2*atanh(sin(e + f*x)))/(a^2*f) + ((c - d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) + ((c^3 + 4*c^2*d - 12*c*d^2 + 10*d^3 - (c - 4*d)*d^2*sec(e + f*x))*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))), (2*(3*c - 2*d)*d^2*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) + ((c^3 + 4*c^2*d - 12*c*d^2 + 10*d^3 - (c - 4*d)*d^2*sec(e + f*x))*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^2)/(a + a*sec(e + f*x))^2, x, 6, ((c - d)^2*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) + (d^2*atanh(sin(e + f*x)))/(a^2*f) + ((c - d)*(c + 5*d)*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x))), ((c - d)^2*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) + (2*d^2*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - d)*(c + 5*d)*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^1)/(a + a*sec(e + f*x))^2, x, 2, ((c - d)*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) + ((c + 2*d)*tan(e + f*x))/(3*f*(a^2 + a^2*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^1), x, 6, tan(e + f*x)/(3*(c - d)*f*(a + a*sec(e + f*x))^2) + (2*d^2*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(a^2*(c - d)^(5/2)*sqrt(c + d)*f) + ((c - 4*d)*tan(e + f*x))/(3*(c - d)^2*f*(a^2 + a^2*sec(e + f*x))), tan(e + f*x)/(3*(c - d)*f*(a + a*sec(e + f*x))^2) - (2*d^2*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a*(c - d)^(5/2)*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - 4*d)*tan(e + f*x))/(3*(c - d)^2*f*(a^2 + a^2*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2), x, 7, (2*d^2*(3*c + 2*d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(a^2*(c - d)^(7/2)*(c + d)^(3/2)*f) + (d*(c^2 - 6*c*d - 10*d^2)*tan(e + f*x))/(3*a^2*(c - d)^3*(c + d)*f*(c + d*sec(e + f*x))) + ((c - 6*d)*tan(e + f*x))/(3*a^2*(c - d)^2*f*(1 + sec(e + f*x))*(c + d*sec(e + f*x))) + tan(e + f*x)/(3*(c - d)*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))), ((c + 4*d)*tan(e + f*x))/(3*(c - d)^2*(c + d)*f*(a + a*sec(e + f*x))^2) - (2*d^2*(3*c + 2*d)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a*(c - d)^(7/2)*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c^2 - 6*c*d - 10*d^2)*tan(e + f*x))/(3*(c - d)^3*(c + d)*f*(a^2 + a^2*sec(e + f*x))) - (d*tan(e + f*x))/((c^2 - d^2)*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^3), x, 8, (d^2*(12*c^2 + 16*c*d + 7*d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(a^2*(c - d)^(9/2)*(c + d)^(5/2)*f) + (d*(2*c^2 - 16*c*d - 21*d^2)*tan(e + f*x))/(6*a^2*(c - d)^3*(c + d)*f*(c + d*sec(e + f*x))^2) + ((c - 8*d)*tan(e + f*x))/(3*a^2*(c - d)^2*f*(1 + sec(e + f*x))*(c + d*sec(e + f*x))^2) + tan(e + f*x)/(3*(c - d)*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2) + (d*(2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*tan(e + f*x))/(6*a^2*(c - d)^4*(c + d)^2*f*(c + d*sec(e + f*x))), If(13>=8, ((2*c^2 + 22*c*d + 11*d^2)*tan(e + f*x))/(6*(c - d)^3*(c + d)^2*f*(a + a*sec(e + f*x))^2) - (d^2*(12*c^2 + 16*c*d + 7*d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a*(c - d)^(9/2)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*tan(e + f*x))/(6*(c - d)^4*(c + d)^2*f*(a^2 + a^2*sec(e + f*x))) - (d*tan(e + f*x))/(2*(c^2 - d^2)*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2) - (d*(5*c + 2*d)*tan(e + f*x))/(2*(c^2 - d^2)^2*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))), ((2*c^2 + 22*c*d + 11*d^2)*tan(e + f*x))/(6*(c - d)^3*(c + d)^2*f*(a + a*sec(e + f*x))^2) - (d^2*(12*c^2 + 16*c*d + 7*d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a*(c - d)^(9/2)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((2*c^3 - 16*c^2*d - 59*c*d^2 - 32*d^3)*tan(e + f*x))/(6*(c - d)^4*(c + d)^2*f*(a^2 + a^2*sec(e + f*x))) - (d*tan(e + f*x))/(2*(c^2 - d^2)*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2) - (d*(5*c + 2*d)*tan(e + f*x))/(2*(c^2 - d^2)^2*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))))]

    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^6)/(a + a*sec(e + f*x))^3, x, 9, (d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*atanh(sin(e + f*x)))/(2*a^3*f) - (2*d*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5)*tan(e + f*x))/(15*a^3*f) - (d^2*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*sec(e + f*x)*tan(e + f*x))/(30*a^3*f) - (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(15*a^3*f) + ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x))) + ((c - d)*(2*c + 13*d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((c - d)*(c + d*sec(e + f*x))^5*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3), (d^3*(40*c^3 - 90*c^2*d + 78*c*d^2 - 23*d^3)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d*(2*c^3 + 18*c^2*d + 111*c*d^2 - 136*d^3)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(15*a^3*f) + ((c - d)*(2*c^2 + 18*c*d + 115*d^2)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x))) + ((c - d)*(2*c + 13*d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((c - d)*(c + d*sec(e + f*x))^5*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) - (d*(4*(2*c^5 + 18*c^4*d + 107*c^3*d^2 - 472*c^2*d^3 + 456*c*d^4 - 136*d^5) + d*(4*c^4 + 36*c^3*d + 216*c^2*d^2 - 626*c*d^3 + 345*d^4)*sec(e + f*x))*tan(e + f*x))/(30*a^3*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^5)/(a + a*sec(e + f*x))^3, x, 8, (d^3*(20*c^2 - 30*c*d + 13*d^2)*atanh(sin(e + f*x)))/(2*a^3*f) - (2*d*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4)*tan(e + f*x))/(15*a^3*f) - (d^2*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*sec(e + f*x)*tan(e + f*x))/(30*a^3*f) + ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x))) + ((c - d)*(2*c + 11*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((c - d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3), (d^3*(20*c^2 - 30*c*d + 13*d^2)*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - d)*(2*c^2 + 15*c*d + 76*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x))) + ((c - d)*(2*c + 11*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((c - d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) - (d*(4*(2*c^4 + 15*c^3*d + 72*c^2*d^2 - 180*c*d^3 + 76*d^4) + d*(4*c^3 + 30*c^2*d + 146*c*d^2 - 195*d^3)*sec(e + f*x))*tan(e + f*x))/(30*a^3*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^4)/(a + a*sec(e + f*x))^3, x, 7, ((4*c - 3*d)*d^3*atanh(sin(e + f*x)))/(a^3*f) + ((c - d)*(2*c + 9*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((c - d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + ((2*c^4 + 8*c^3*d + 21*c^2*d^2 - 88*c*d^3 + 72*d^4 - d^2*(2*c^2 + 10*c*d - 27*d^2)*sec(e + f*x))*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x))), (2*(4*c - 3*d)*d^3*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - d)*(2*c + 9*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((c - d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + ((2*c^4 + 8*c^3*d + 21*c^2*d^2 - 88*c*d^3 + 72*d^4 - d^2*(2*c^2 + 10*c*d - 27*d^2)*sec(e + f*x))*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^3)/(a + a*sec(e + f*x))^3, x, 6, (d^3*atanh(sin(e + f*x)))/(a^3*f) + ((c - d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + ((c - d)*(2*(2*c^2 + 8*c*d + 11*d^2) + (2*c^2 + 11*c*d + 29*d^2)*sec(e + f*x))*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2), (2*d^3*atan(sqrt(a - a*sec(e + f*x))/sqrt(a*(1 + sec(e + f*x))))*tan(e + f*x))/(a^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((c - d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + ((c - d)*(2*(2*c^2 + 8*c*d + 11*d^2) + (2*c^2 + 11*c*d + 29*d^2)*sec(e + f*x))*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^2)/(a + a*sec(e + f*x))^3, x, 4, ((c - d)^2*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + (2*(c - d)*(c + 4*d)*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((2*c^2 + 6*c*d + 7*d^2)*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^1)/(a + a*sec(e + f*x))^3, x, 3, ((c - d)*tan(e + f*x))/(5*f*(a + a*sec(e + f*x))^3) + ((2*c + 3*d)*tan(e + f*x))/(15*a*f*(a + a*sec(e + f*x))^2) + ((2*c + 3*d)*tan(e + f*x))/(15*f*(a^3 + a^3*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^1), x, 7, (-2*d^3*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(a^3*(c - d)^(7/2)*sqrt(c + d)*f) + tan(e + f*x)/(5*(c - d)*f*(a + a*sec(e + f*x))^3) + ((2*c - 7*d)*tan(e + f*x))/(15*a*(c - d)^2*f*(a + a*sec(e + f*x))^2) + ((2*c^2 - 9*c*d + 22*d^2)*tan(e + f*x))/(15*(c - d)^3*f*(a^3 + a^3*sec(e + f*x))), tan(e + f*x)/(5*(c - d)*f*(a + a*sec(e + f*x))^3) + ((2*c - 7*d)*tan(e + f*x))/(15*a*(c - d)^2*f*(a + a*sec(e + f*x))^2) + (2*d^3*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a^2*(c - d)^(7/2)*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((2*c^2 - 9*c*d + 22*d^2)*tan(e + f*x))/(15*(c - d)^3*f*(a^3 + a^3*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^2), x, 8, (-2*d^3*(4*c + 3*d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(a^3*(c - d)^(9/2)*(c + d)^(3/2)*f) + (d*(2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*tan(e + f*x))/(15*a^3*(c - d)^4*(c + d)*f*(c + d*sec(e + f*x))) + tan(e + f*x)/(5*(c - d)*f*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))) + ((2*c - 9*d)*tan(e + f*x))/(15*a*(c - d)^2*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))) + ((2*c^2 - 12*c*d + 45*d^2)*tan(e + f*x))/(15*(c - d)^3*f*(a^3 + a^3*sec(e + f*x))*(c + d*sec(e + f*x))), ((c + 6*d)*tan(e + f*x))/(5*(c - d)^2*(c + d)*f*(a + a*sec(e + f*x))^3) + ((2*c^2 - 10*c*d - 27*d^2)*tan(e + f*x))/(15*a*(c - d)^3*(c + d)*f*(a + a*sec(e + f*x))^2) + (2*d^3*(4*c + 3*d)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a^2*(c - d)^(9/2)*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((2*c^3 - 12*c^2*d + 43*c*d^2 + 72*d^3)*tan(e + f*x))/(15*(c - d)^4*(c + d)*f*(a^3 + a^3*sec(e + f*x))) - (d*tan(e + f*x))/((c^2 - d^2)*f*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^3), x, 9, -((d^3*(20*c^2 + 30*c*d + 13*d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(a^3*(c - d)^(11/2)*(c + d)^(5/2)*f)) + (d*(4*c^3 - 30*c^2*d + 146*c*d^2 + 195*d^3)*tan(e + f*x))/(30*a^3*(c - d)^4*(c + d)*f*(c + d*sec(e + f*x))^2) + tan(e + f*x)/(5*(c - d)*f*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^2) + ((2*c - 11*d)*tan(e + f*x))/(15*a*(c - d)^2*f*(a + a*sec(e + f*x))^2*(c + d*sec(e + f*x))^2) + ((2*c^2 - 15*c*d + 76*d^2)*tan(e + f*x))/(15*(c - d)^3*f*(a^3 + a^3*sec(e + f*x))*(c + d*sec(e + f*x))^2) + (d*(4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*tan(e + f*x))/(30*a^3*(c - d)^5*(c + d)^2*f*(c + d*sec(e + f*x))), If(13>=8, ((2*c^2 + 39*c*d + 22*d^2)*tan(e + f*x))/(10*(c - d)^3*(c + d)^2*f*(a + a*sec(e + f*x))^3) + ((4*c^3 - 26*c^2*d - 184*c*d^2 - 109*d^3)*tan(e + f*x))/(30*a*(c - d)^4*(c + d)^2*f*(a + a*sec(e + f*x))^2) + (d^3*(20*c^2 + 30*c*d + 13*d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a^2*(c - d)^(11/2)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*tan(e + f*x))/(30*(c - d)^5*(c + d)^2*f*(a^3 + a^3*sec(e + f*x))) - (d*tan(e + f*x))/(2*(c^2 - d^2)*f*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^2) - (3*d*(2*c + d)*tan(e + f*x))/(2*(c^2 - d^2)^2*f*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))), ((2*c^2 + 39*c*d + 22*d^2)*tan(e + f*x))/(10*(c - d)^3*(c + d)^2*f*(a + a*sec(e + f*x))^3) + ((4*c^3 - 26*c^2*d - 184*c*d^2 - 109*d^3)*tan(e + f*x))/(30*a*(c - d)^4*(c + d)^2*f*(a + a*sec(e + f*x))^2) + (d^3*(20*c^2 + 30*c*d + 13*d^2)*atan((sqrt(c + d)*sqrt(a + a*sec(e + f*x)))/(sqrt(c - d)*sqrt(a - a*sec(e + f*x))))*tan(e + f*x))/(a^2*(c - d)^(11/2)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((4*c^4 - 30*c^3*d + 142*c^2*d^2 + 525*c*d^3 + 304*d^4)*tan(e + f*x))/(30*(c - d)^5*(c + d)^2*f*(a^3 + a^3*sec(e + f*x))) - (d*tan(e + f*x))/(2*(c^2 - d^2)*f*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))^2) - (3*d*(2*c + d)*tan(e + f*x))/(2*(c^2 - d^2)^2*f*(a + a*sec(e + f*x))^3*(c + d*sec(e + f*x))))]

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(e + f*x)*sqrt(a + a*sec(e + f*x))/sqrt(c + d*sec(e + f*x)), x, 2, (2*sqrt(a)*atanh((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))))/(sqrt(d)*f)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)*sqrt(c + d*sec(e + f*x))/sqrt(a + a*sec(e + f*x)), x, 5, (sqrt(2)*sqrt(c - d)*atan((sqrt(a)*sqrt(c - d)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))))/(sqrt(a)*f) + (2*sqrt(d)*atanh((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))))/(sqrt(a)*f)]
    @test_int [sec(e + f*x)/(sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x))), x, 2, (sqrt(2)*atan((sqrt(a)*sqrt(c - d)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))))/(sqrt(a)*sqrt(c - d)*f)]

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

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)^2/(sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x))), x, 5, -((sqrt(2)*atan((sqrt(a)*sqrt(c - d)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))))/(sqrt(a)*sqrt(c - d)*f)) + (2*atanh((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(a + a*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))))/(sqrt(a)*sqrt(d)*f)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(g*sec(e+f*x))^p*(a+a*sec(e+f*x))^(m/2) / (c+d*sec(e+f*x))=#

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(sec(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c + d*sec(e + f*x)), x, 2, (2*sqrt(a)*atan((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(c + d)*sqrt(a + a*sec(e + f*x)))))/(sqrt(d)*sqrt(c + d)*f)]

    @test_int [(g*sec(e + f*x))^(3/2)*sqrt(a + a*sec(e + f*x))/(c + d*sec(e + f*x)), x, 5, (2*sqrt(a)*g^(3/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(d*f) - (2*sqrt(a)*sqrt(c)*g^(3/2)*atanh((sqrt(a)*sqrt(c)*sqrt(g)*tan(e + f*x))/(sqrt(c + d)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(d*sqrt(c + d)*f)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)^1/(sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))), x, 5, (sqrt(2)*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*f) - (2*sqrt(d)*atan((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(c + d)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*sqrt(c + d)*f)]
    @test_int [sec(e + f*x)^2/(sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))), x, 5, -((sqrt(2)*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*f)) + (2*c*atan((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(c + d)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*sqrt(d)*sqrt(c + d)*f)]

    @test_int [(g*sec(e + f*x))^(3/2)/(sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))), x, 5, -((sqrt(2)*g^(3/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(2)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*f)) + (2*sqrt(c)*g^(3/2)*atanh((sqrt(a)*sqrt(c)*sqrt(g)*tan(e + f*x))/(sqrt(c + d)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*sqrt(c + d)*f)]
    @test_int [(g*sec(e + f*x))^(5/2)/(sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))), x, 8, (2*g^(5/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*d*f) + (sqrt(2)*g^(5/2)*atanh((sqrt(a)*sqrt(g)*tan(e + f*x))/(sqrt(2)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*f) - (2*c^(3/2)*g^(5/2)*atanh((sqrt(a)*sqrt(c)*sqrt(g)*tan(e + f*x))/(sqrt(c + d)*sqrt(g*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*(c - d)*d*sqrt(c + d)*f)]

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

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^4, x, 8, ((8*a*c^4 + 16*b*c^3*d + 24*a*c^2*d^2 + 12*b*c*d^3 + 3*a*d^4)*atanh(sin(e + f*x)))/(8*f) + ((12*b*c^4 + 95*a*c^3*d + 112*b*c^2*d^2 + 80*a*c*d^3 + 16*b*d^4)*tan(e + f*x))/(30*f) + (d*(24*b*c^3 + 130*a*c^2*d + 116*b*c*d^2 + 45*a*d^3)*sec(e + f*x)*tan(e + f*x))/(120*f) + ((12*b*c^2 + 35*a*c*d + 16*b*d^2)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(60*f) + ((4*b*c + 5*a*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(20*f) + (b*(c + d*sec(e + f*x))^4*tan(e + f*x))/(5*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^3, x, 7, ((8*a*c^3 + 12*b*c^2*d + 12*a*c*d^2 + 3*b*d^3)*atanh(sin(e + f*x)))/(8*f) + ((4*a*d*(4*c^2 + d^2) + 3*b*(c^3 + 4*c*d^2))*tan(e + f*x))/(6*f) + (d*(6*b*c^2 + 20*a*c*d + 9*b*d^2)*sec(e + f*x)*tan(e + f*x))/(24*f) + ((3*b*c + 4*a*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(12*f) + (b*(c + d*sec(e + f*x))^3*tan(e + f*x))/(4*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^2, x, 6, ((2*b*c*d + a*(2*c^2 + d^2))*atanh(sin(e + f*x)))/(2*f) + (2*(3*a*c*d + b*(c^2 + d^2))*tan(e + f*x))/(3*f) + (d*(2*b*c + 3*a*d)*sec(e + f*x)*tan(e + f*x))/(6*f) + (b*(c + d*sec(e + f*x))^2*tan(e + f*x))/(3*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^1, x, 5, ((2*a*c + b*d)*atanh(sin(e + f*x)))/(2*f) + ((b*c + a*d)*tan(e + f*x))/f + (b*d*sec(e + f*x)*tan(e + f*x))/(2*f)]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x)))/(c + d*sec(e + f*x))^1, x, 5, (b*atanh(sin(e + f*x)))/(d*f) - (2*(b*c - a*d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(sqrt(c - d)*d*sqrt(c + d)*f)]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x)))/(c + d*sec(e + f*x))^2, x, 5, (2*(a*c - b*d)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(3/2)*(c + d)^(3/2)*f) + ((b*c - a*d)*tan(e + f*x))/((c^2 - d^2)*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x)))/(c + d*sec(e + f*x))^3, x, 6, -(((3*b*c*d - a*(2*c^2 + d^2))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(5/2)*(c + d)^(5/2)*f)) + ((b*c - a*d)*tan(e + f*x))/(2*(c^2 - d^2)*f*(c + d*sec(e + f*x))^2) - ((3*a*c*d - b*(c^2 + 2*d^2))*tan(e + f*x))/(2*(c^2 - d^2)^2*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x)))/(c + d*sec(e + f*x))^4, x, 7, ((2*a*c^3 - 4*b*c^2*d + 3*a*c*d^2 - b*d^3)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(7/2)*(c + d)^(7/2)*f) + ((b*c - a*d)*tan(e + f*x))/(3*(c^2 - d^2)*f*(c + d*sec(e + f*x))^3) + ((2*b*c^2 - 5*a*c*d + 3*b*d^2)*tan(e + f*x))/(6*(c^2 - d^2)^2*f*(c + d*sec(e + f*x))^2) + ((2*b*c^3 - 11*a*c^2*d + 13*b*c*d^2 - 4*a*d^3)*tan(e + f*x))/(6*(c^2 - d^2)^3*f*(c + d*sec(e + f*x)))]

    #= [sec(e + f*x)*(a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^4, x, 8, ((16*a*b*d*(4*c^3 + 3*c*d^2) + 2*a^2*(8*c^4 + 24*c^2*d^2 + 3*d^4) + b^2*(8*c^4 + 36*c^2*d^2 + 5*d^4))*atanh(sin(e + f*x)))/(16*f) + ((10*a^2*c*d^2*(19*c^2 + 16*d^2) + 16*a*b*d*(3*c^4 + 28*c^2*d^2 + 4*d^4) - b^2*(4*c^5 - 121*c^3*d^2 - 128*c*d^4))*tan(e + f*x))/(60*d*f) + ((10*a^2*d^2*(26*c^2 + 9*d^2) + 16*a*b*d*(6*c^3 + 29*c*d^2) - b^2*(8*c^4 - 178*c^2*d^2 - 75*d^4))*sec(e + f*x)*tan(e + f*x))/(240*f) + ((70*a^2*c*d^2 + 16*a*b*d*(3*c^2 + 4*d^2) - b^2*(4*c^3 - 53*c*d^2))*(c + d*sec(e + f*x))^2*tan(e + f*x))/(120*d*f) + ((5*(6*a^2 + 5*b^2)*d^2 - 4*b*c*(b*c - 12*a*d))*(c + d*sec(e + f*x))^3*tan(e + f*x))/(120*d*f) - (b*(b*c - 12*a*d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(30*d*f) + (b^2*(c + d*sec(e + f*x))^5*tan(e + f*x))/(6*d*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^3, x, 7, ((6*a*b*d*(4*c^2 + d^2) + b^2*c*(4*c^2 + 9*d^2) + 4*a^2*(2*c^3 + 3*c*d^2))*atanh(sin(e + f*x)))/(8*f) + ((20*a^2*d^2*(4*c^2 + d^2) + 30*a*b*c*d*(c^2 + 4*d^2) - b^2*(3*c^4 - 52*c^2*d^2 - 16*d^4))*tan(e + f*x))/(30*d*f) + ((100*a^2*c*d^2 + 30*a*b*d*(2*c^2 + 3*d^2) - b^2*(6*c^3 - 71*c*d^2))*sec(e + f*x)*tan(e + f*x))/(120*f) + ((4*(5*a^2 + 4*b^2)*d^2 - 3*b*c*(b*c - 10*a*d))*(c + d*sec(e + f*x))^2*tan(e + f*x))/(60*d*f) - (b*(b*c - 10*a*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(20*d*f) + (b^2*(c + d*sec(e + f*x))^4*tan(e + f*x))/(5*d*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^2, x, 6, ((16*a*b*c*d + 4*a^2*(2*c^2 + d^2) + b^2*(4*c^2 + 3*d^2))*atanh(sin(e + f*x)))/(8*f) + ((8*a^2*b*c*d + 8*b^3*c*d - a^3*d^2 + 4*a*b^2*(3*c^2 + 2*d^2))*tan(e + f*x))/(6*b*f) + ((2*a*d*(8*b*c - a*d) + 3*b^2*(4*c^2 + 3*d^2))*sec(e + f*x)*tan(e + f*x))/(24*f) + (d*(8*b*c - a*d)*(a + b*sec(e + f*x))^2*tan(e + f*x))/(12*b*f) + (d^2*(a + b*sec(e + f*x))^3*tan(e + f*x))/(4*b*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^1, x, 5, ((2*a^2*c + b^2*c + 2*a*b*d)*atanh(sin(e + f*x)))/(2*f) + (2*(3*a*b*c + a^2*d + b^2*d)*tan(e + f*x))/(3*f) + (b*(3*b*c + 2*a*d)*sec(e + f*x)*tan(e + f*x))/(6*f) + (d*(a + b*sec(e + f*x))^2*tan(e + f*x))/(3*f)]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^2)/(c + d*sec(e + f*x))^1, x, 6, -((b*(b*c - 2*a*d)*atanh(sin(e + f*x)))/(d^2*f)) + (2*(b*c - a*d)^2*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(sqrt(c - d)*d^2*sqrt(c + d)*f) + (b^2*tan(e + f*x))/(d*f)]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^2)/(c + d*sec(e + f*x))^2, x, 6, (b^2*atanh(sin(e + f*x)))/(d^2*f) - (2*(b*c - a*d)*(b*c^2 + a*c*d - 2*b*d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(3/2)*d^2*(c + d)^(3/2)*f) - ((b*c - a*d)^2*tan(e + f*x))/(d*(c^2 - d^2)*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^2)/(c + d*sec(e + f*x))^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(5/2)*(c + d)^(5/2)*f)) - ((b*c - a*d)^2*tan(e + f*x))/(2*d*(c^2 - d^2)*f*(c + d*sec(e + f*x))^2) + ((b*c - a*d)*(3*a*c*d + b*(c^2 - 4*d^2))*tan(e + f*x))/(2*(c - d)^2*d*(c + d)^2*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^2)/(c + d*sec(e + f*x))^4, x, 7, -(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(7/2)*(c + d)^(7/2)*f)) - ((b*c - a*d)^2*tan(e + f*x))/(3*d*(c^2 - d^2)*f*(c + d*sec(e + f*x))^3) + ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*tan(e + f*x))/(6*(c - d)^2*d*(c + d)^2*f*(c + d*sec(e + f*x))^2) - ((a^2*d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*tan(e + f*x))/(6*d*(c^2 - d^2)^3*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^2)/(c + d*sec(e + f*x))^5, x, 8, -((10*a*b*d*(4*c^3 + 3*c*d^2) - a^2*(8*c^4 + 24*c^2*d^2 + 3*d^4) - b^2*(4*c^4 + 27*c^2*d^2 + 4*d^4))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(4*(c - d)^(9/2)*(c + d)^(9/2)*f) - ((b*c - a*d)^2*tan(e + f*x))/(4*d*(c^2 - d^2)*f*(c + d*sec(e + f*x))^4) + ((b*c - a*d)*(7*a*c*d + b*(c^2 - 8*d^2))*tan(e + f*x))/(12*(c - d)^2*d*(c + d)^2*f*(c + d*sec(e + f*x))^3) - ((a^2*d^2*(26*c^2 + 9*d^2) - 2*a*b*d*(6*c^3 + 29*c*d^2) - b^2*(2*c^4 - 25*c^2*d^2 - 12*d^4))*tan(e + f*x))/(24*d*(c^2 - d^2)^3*f*(c + d*sec(e + f*x))^2) - ((5*a^2*c*d^2*(10*c^2 + 11*d^2) - 2*a*b*d*(6*c^4 + 83*c^2*d^2 + 16*d^4) - b^2*(2*c^5 - 39*c^3*d^2 - 68*c*d^4))*tan(e + f*x))/(24*d*(c^2 - d^2)^4*f*(c + d*sec(e + f*x)))] =#

    #= [sec(e + f*x)*(a + b*sec(e + f*x))^3*(c + d*sec(e + f*x))^3, x, 8, ((18*a^2*b*d*(4*c^2 + d^2) + b^3*d*(18*c^2 + 5*d^2) + 6*a*b^2*c*(4*c^2 + 9*d^2) + 8*a^3*(2*c^3 + 3*c*d^2))*atanh(sin(e + f*x)))/(16*f) + ((40*a^3*d^3*(4*c^2 + d^2) + 90*a^2*b*c*d^2*(c^2 + 4*d^2) - 6*a*b^2*d*(3*c^4 - 52*c^2*d^2 - 16*d^4) + b^3*(2*c^5 + 17*c^3*d^2 + 96*c*d^4))*tan(e + f*x))/(60*d^2*f) + ((200*a^3*c*d^3 + 90*a^2*b*d^2*(2*c^2 + 3*d^2) - 6*a*b^2*d*(6*c^3 - 71*c*d^2) + b^3*(4*c^4 + 36*c^2*d^2 + 75*d^4))*sec(e + f*x)*tan(e + f*x))/(240*d*f) + ((90*a^2*b*c*d^2 + 40*a^3*d^3 + b^3*(2*c^3 + 21*c*d^2) - a*b^2*(18*c^2*d - 96*d^3))*(c + d*sec(e + f*x))^2*tan(e + f*x))/(120*d^2*f) - (b*(18*a*b*c*d - 90*a^2*d^2 - b^2*(2*c^2 + 25*d^2))*(c + d*sec(e + f*x))^3*tan(e + f*x))/(120*d^2*f) - (b^2*(2*b*c - 13*a*d)*(c + d*sec(e + f*x))^4*tan(e + f*x))/(30*d^2*f) + (b^2*(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^4*tan(e + f*x))/(6*d*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))^3*(c + d*sec(e + f*x))^2, x, 7, ((24*a^2*b*c*d + 6*b^3*c*d + 4*a^3*(2*c^2 + d^2) + 3*a*b^2*(4*c^2 + 3*d^2))*atanh(sin(e + f*x)))/(8*f) + ((30*a^3*b*c*d + 120*a*b^3*c*d - 3*a^4*d^2 + 4*b^4*(5*c^2 + 4*d^2) + 4*a^2*b^2*(20*c^2 + 13*d^2))*tan(e + f*x))/(30*b*f) + ((60*a^2*b*c*d + 90*b^3*c*d - 6*a^3*d^2 + a*b^2*(100*c^2 + 71*d^2))*sec(e + f*x)*tan(e + f*x))/(120*f) + ((3*a*d*(10*b*c - a*d) + 4*b^2*(5*c^2 + 4*d^2))*(a + b*sec(e + f*x))^2*tan(e + f*x))/(60*b*f) + (d*(10*b*c - a*d)*(a + b*sec(e + f*x))^3*tan(e + f*x))/(20*b*f) + (d^2*(a + b*sec(e + f*x))^4*tan(e + f*x))/(5*b*f)]
    @test_int [sec(e + f*x)*(a + b*sec(e + f*x))^3*(c + d*sec(e + f*x))^1, x, 6, ((8*a^3*c + 12*a*b^2*c + 12*a^2*b*d + 3*b^3*d)*atanh(sin(e + f*x)))/(8*f) + ((16*a^2*b*c + 4*b^3*c + 3*a^3*d + 12*a*b^2*d)*tan(e + f*x))/(6*f) + (b*(20*a*b*c + 6*a^2*d + 9*b^2*d)*sec(e + f*x)*tan(e + f*x))/(24*f) + ((4*b*c + 3*a*d)*(a + b*sec(e + f*x))^2*tan(e + f*x))/(12*f) + (d*(a + b*sec(e + f*x))^3*tan(e + f*x))/(4*f)]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^3)/(c + d*sec(e + f*x))^1, x, 7, -(b*(6*a*b*c*d - 6*a^2*d^2 - b^2*(2*c^2 + d^2))*atanh(sin(e + f*x)))/(2*d^3*f) - (2*(b*c - a*d)^3*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(sqrt(c - d)*d^3*sqrt(c + d)*f) - (b^2*(2*b*c - 5*a*d)*tan(e + f*x))/(2*d^2*f) + (b^2*(a + b*sec(e + f*x))*tan(e + f*x))/(2*d*f)]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^3)/(c + d*sec(e + f*x))^2, x, 7, -((b^2*(2*b*c - 3*a*d)*atanh(sin(e + f*x)))/(d^3*f)) + (2*(b*c - a*d)^2*(2*b*c^2 + a*c*d - 3*b*d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(3/2)*d^3*(c + d)^(3/2)*f) - (b*(2*a*b*c*d - a^2*d^2 - b^2*(2*c^2 - d^2))*tan(e + f*x))/(d^2*(c^2 - d^2)*f) - ((b*c - a*d)^2*(a + b*sec(e + f*x))*tan(e + f*x))/(d*(c^2 - d^2)*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^3)/(c + d*sec(e + f*x))^3, x, 7, (b^3*atanh(sin(e + f*x)))/(d^3*f) - ((b*c - a*d)*(2*a*b*c*d*(c^2 - 4*d^2) + a^2*d^2*(2*c^2 + d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 6*d^4))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(5/2)*d^3*(c + d)^(5/2)*f) - ((b*c - a*d)^2*(a + b*sec(e + f*x))*tan(e + f*x))/(2*d*(c^2 - d^2)*f*(c + d*sec(e + f*x))^2) - ((b*c - a*d)^2*(2*b*c^2 + 3*a*c*d - 5*b*d^2)*tan(e + f*x))/(2*d^2*(c^2 - d^2)^2*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^3)/(c + d*sec(e + f*x))^4, x, 7, -(((a*c - b*d)*(10*a*b*c*d - b^2*(3*c^2 + 2*d^2) - a^2*(2*c^2 + 3*d^2))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(7/2)*(c + d)^(7/2)*f)) - ((b*c - a*d)^2*(a + b*sec(e + f*x))*tan(e + f*x))/(3*d*(c^2 - d^2)*f*(c + d*sec(e + f*x))^3) - ((b*c - a*d)^2*(2*b*c^2 + 5*a*c*d - 7*b*d^2)*tan(e + f*x))/(6*d^2*(c^2 - d^2)^2*f*(c + d*sec(e + f*x))^2) + ((b*c - a*d)*(5*a*b*c*d*(c^2 - 7*d^2) + a^2*d^2*(11*c^2 + 4*d^2) + b^2*(2*c^4 - 5*c^2*d^2 + 18*d^4))*tan(e + f*x))/(6*(c - d)^3*d^2*(c + d)^3*f*(c + d*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(a + b*sec(e + f*x))^3)/(c + d*sec(e + f*x))^5, x, 8, -((5*b^3*c*d*(3*c^2 + 4*d^2) + 15*a^2*b*d*(4*c^3 + 3*c*d^2) - a^3*(8*c^4 + 24*c^2*d^2 + 3*d^4) - 3*a*b^2*(4*c^4 + 27*c^2*d^2 + 4*d^4))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(4*(c - d)^(9/2)*(c + d)^(9/2)*f) - ((b*c - a*d)^2*(a + b*sec(e + f*x))*tan(e + f*x))/(4*d*(c^2 - d^2)*f*(c + d*sec(e + f*x))^4) - ((b*c - a*d)^2*(2*b*c^2 + 7*a*c*d - 9*b*d^2)*tan(e + f*x))/(12*d^2*(c^2 - d^2)^2*f*(c + d*sec(e + f*x))^3) + ((b*c - a*d)*(a^2*d^2*(26*c^2 + 9*d^2) + a*b*(8*c^3*d - 78*c*d^3) + b^2*(2*c^4 - 3*c^2*d^2 + 36*d^4))*tan(e + f*x))/(24*(c - d)^3*d^2*(c + d)^3*f*(c + d*sec(e + f*x))^2) - ((5*a^3*c*d^3*(10*c^2 + 11*d^2) - 3*a*b^2*c*d*(2*c^4 - 39*c^2*d^2 - 68*d^4) - 3*a^2*b*d^2*(6*c^4 + 83*c^2*d^2 + 16*d^4) - b^3*(2*c^6 - 5*c^4*d^2 + 84*c^2*d^4 + 24*d^6))*tan(e + f*x))/(24*d^2*(c^2 - d^2)^4*f*(c + d*sec(e + f*x)))] =#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^4)/(a + b*sec(e + f*x)), x, 12, (d^3*(4*b*c - a*d)*atanh(sin(e + f*x)))/(2*b^2*f) + (d*(2*b*c - a*d)*(2*b^2*c^2 - 2*a*b*c*d + a^2*d^2)*atanh(sin(e + f*x)))/(b^4*f) + (2*(b*c - a*d)^4*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(sqrt(a - b)*b^4*sqrt(a + b)*f) + (d^4*tan(e + f*x))/(b*f) + (d^2*(6*b^2*c^2 - 4*a*b*c*d + a^2*d^2)*tan(e + f*x))/(b^3*f) + (d^3*(4*b*c - a*d)*sec(e + f*x)*tan(e + f*x))/(2*b^2*f) + (d^4*tan(e + f*x)^3)/(3*b*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^3)/(a + b*sec(e + f*x)), x, 10, (d^3*atanh(sin(e + f*x)))/(2*b*f) + (d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*atanh(sin(e + f*x)))/(b^3*f) + (2*(b*c - a*d)^3*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(sqrt(a - b)*b^3*sqrt(a + b)*f) + (d^2*(3*b*c - a*d)*tan(e + f*x))/(b^2*f) + (d^3*sec(e + f*x)*tan(e + f*x))/(2*b*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^2)/(a + b*sec(e + f*x)), x, 8, (d*(2*b*c - a*d)*atanh(sin(e + f*x)))/(b^2*f) + (2*(b*c - a*d)^2*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/(sqrt(a - b)*b^2*sqrt(a + b)*f) + (d^2*tan(e + f*x))/(b*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^1)/(a + b*sec(e + f*x)), x, 5, (d*atanh(sin(e + f*x)))/(b*f) + (2*(b*c - a*d)*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/(sqrt(a - b)*b*sqrt(a + b)*f)]
    @test_int [sec(e + f*x)/((a + b*sec(e + f*x))*(c + d*sec(e + f*x))^1), x, 6, (2*b*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/(sqrt(a - b)*sqrt(a + b)*(b*c - a*d)*f) - (2*d*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(sqrt(c - d)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [sec(e + f*x)/((a + b*sec(e + f*x))*(c + d*sec(e + f*x))^2), x, 7, (2*b^2*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(sqrt(a - b)*sqrt(a + b)*(b*c - a*d)^2*f) - (2*d*(2*b*c^2 - a*c*d - b*d^2)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^2*f) + (d^2*sin(e + f*x))/((b*c - a*d)*(c^2 - d^2)*f*(d + c*cos(e + f*x)))]
    #= [sec(e + f*x)/((a + b*sec(e + f*x))*(c + d*sec(e + f*x))^3), x, 9, (2*b^3*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/(sqrt(a - b)*sqrt(a + b)*(b*c - a*d)^3*f) + (d*(6*a*b*c^3*d - a^2*d^2*(2*c^2 + d^2) - b^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^3*f) + (d^2*tan(e + f*x))/(2*(b*c - a*d)*(c^2 - d^2)*f*(c + d*sec(e + f*x))^2) + (d^2*(5*b*c^2 - 3*a*c*d - 2*b*d^2)*tan(e + f*x))/(2*(b*c - a*d)^2*(c^2 - d^2)^2*f*(c + d*sec(e + f*x)))] =#

    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^5)/(a + b*sec(e + f*x))^2, x, 16, (d^4*(5*b*c - 2*a*d)*atanh(sin(e + f*x)))/(2*b^3*f) + (d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*atanh(sin(e + f*x)))/(b^5*f) + (2*(b*c - a*d)^5*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*(a - b)^(3/2)*b^3*(a + b)^(3/2)*f) + (2*(b*c - a*d)^4*(b*c + 4*a*d)*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*sqrt(a - b)*b^5*sqrt(a + b)*f) - ((b*c - a*d)^5*sin(e + f*x))/(b^4*(a^2 - b^2)*f*(b + a*cos(e + f*x))) + (d^5*tan(e + f*x))/(b^2*f) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*tan(e + f*x))/(b^4*f) + (d^4*(5*b*c - 2*a*d)*sec(e + f*x)*tan(e + f*x))/(2*b^3*f) + (d^5*tan(e + f*x)^3)/(3*b^2*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^4)/(a + b*sec(e + f*x))^2, x, 14, (d^4*atanh(sin(e + f*x)))/(2*b^2*f) + (d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*atanh(sin(e + f*x)))/(b^4*f) + (2*(b*c - a*d)^4*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*(a - b)^(3/2)*b^2*(a + b)^(3/2)*f) + (2*(b*c - a*d)^3*(b*c + 3*a*d)*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*sqrt(a - b)*b^4*sqrt(a + b)*f) - ((b*c - a*d)^4*sin(e + f*x))/(b^3*(a^2 - b^2)*f*(b + a*cos(e + f*x))) + (2*d^3*(2*b*c - a*d)*tan(e + f*x))/(b^3*f) + (d^4*sec(e + f*x)*tan(e + f*x))/(2*b^2*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^3)/(a + b*sec(e + f*x))^2, x, 12, (d^2*(3*b*c - 2*a*d)*atanh(sin(e + f*x)))/(b^3*f) + (2*(b*c - a*d)^3*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*(a - b)^(3/2)*b*(a + b)^(3/2)*f) + (2*(b*c - a*d)^2*(b*c + 2*a*d)*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*sqrt(a - b)*b^3*sqrt(a + b)*f) - ((b*c - a*d)^3*sin(e + f*x))/(b^2*(a^2 - b^2)*f*(b + a*cos(e + f*x))) + (d^3*tan(e + f*x))/(b^2*f)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^2)/(a + b*sec(e + f*x))^2, x, 10, (d^2*atanh(sin(e + f*x)))/(b^2*f) + (2*(b*c - a*d)^2*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*(a - b)^(3/2)*(a + b)^(3/2)*f) + (2*(b^2*c^2 - a^2*d^2)*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/(a*sqrt(a - b)*b^2*sqrt(a + b)*f) - ((b*c - a*d)^2*sin(e + f*x))/(b*(a^2 - b^2)*f*(b + a*cos(e + f*x)))]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^1)/(a + b*sec(e + f*x))^2, x, 5, (2*(a*c - b*d)*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*f) - ((b*c - a*d)*tan(e + f*x))/((a^2 - b^2)*f*(a + b*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^1), x, 7, (2*b*(a*b*c - 2*a^2*d + b^2*d)*atanh((sqrt(a - b)*tan((1/2)*(e + f*x)))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^2*f) + (2*d^2*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(sqrt(c - d)*sqrt(c + d)*(b*c - a*d)^2*f) - (b^2*sin(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(b + a*cos(e + f*x)))]
    #= [sec(e + f*x)/((a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^2), x, 9, (2*b^2*(a*b*c - 3*a^2*d + 2*b^2*d)*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^3*f) + (2*d^2*(3*b*c^2 - a*c*d - 2*b*d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^3*f) - (d*(a^2*d^2 + b^2*(c^2 - 2*d^2))*tan(e + f*x))/((a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*sec(e + f*x))) - (b^2*tan(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^3), x, 10, (2*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(3/2)*(a + b)^(3/2)*(b*c - a*d)^4*f) - (d^2*(2*a*b*c*d*(4*c^2 - d^2) - a^2*d^2*(2*c^2 + d^2) - 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^4*f) - (d*(a^2*d^2 + b^2*(2*c^2 - 3*d^2))*tan(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*sec(e + f*x))^2) - (b^2*tan(e + f*x))/((a^2 - b^2)*(b*c - a*d)*f*(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^2) + (d*(3*a^3*c*d^3 - 3*a*b^2*c*d^3 - a^2*b*d^2*(7*c^2 - 4*d^2) - b^3*(2*c^4 - 11*c^2*d^2 + 6*d^4))*tan(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)^3*(c^2 - d^2)^2*f*(c + d*sec(e + f*x)))] =#

    #= [(sec(e + f*x)*(c + d*sec(e + f*x))^6)/(a + b*sec(e + f*x))^3, x, 10, (d^3*(72*a^2*b*c*d^2 - 20*a^3*d^3 - 3*a*b^2*d*(30*c^2 + d^2) + b^3*(40*c^3 + 6*c*d^2))*atanh(sin(e + f*x)))/(2*b^6*f) + ((b*c - a*d)^4*(8*a^3*b*c*d - 14*a*b^3*c*d + 20*a^4*d^2 + a^2*b^2*(2*c^2 - 47*d^2) + b^4*(c^2 + 30*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*b^6*(a + b)^(5/2)*f) - (d*(216*a^5*b*c*d^4 - 60*a^6*d^5 - a^4*b^2*d^3*(250*c^2 - 101*d^2) + 6*a^3*b^3*c*d^2*(13*c^2 - 63*d^2) - 6*a*b^5*c*(3*c^4 + 37*c^2*d^2 - 18*d^4) + 2*a^2*b^4*d*(9*c^4 + 232*c^2*d^2 - 14*d^4) + b^6*d*(63*c^4 - 88*c^2*d^2 - 4*d^4))*tan(e + f*x))/(6*b^5*(a^2 - b^2)^2*f) + (d^2*(68*a^4*b*c*d^3 - 30*a^5*d^4 - 2*b^5*c*d*(15*c^2 - 7*d^2) - 12*a^3*b^2*d^2*(3*c^2 - 4*d^2) - 2*a^2*b^3*c*d*(3*c^2 + 59*d^2) + 9*a*b^4*(c^4 + 10*c^2*d^2 - d^4))*sec(e + f*x)*tan(e + f*x))/(6*b^4*(a^2 - b^2)^2*f) - (d*(27*a^3*b*c*d^2 - 20*a^4*d^3 + 31*a^2*b^2*d^3 + b^4*d*(27*c^2 - 2*d^2) - 9*a*b^3*c*(c^2 + 6*d^2))*(c + d*sec(e + f*x))^2*tan(e + f*x))/(6*b^3*(a^2 - b^2)^2*f) - ((b*c - a*d)^2*(3*a*b*c + 5*a^2*d - 8*b^2*d)*(c + d*sec(e + f*x))^3*tan(e + f*x))/(2*b^2*(a^2 - b^2)^2*f*(a + b*sec(e + f*x))) - ((b*c - a*d)^2*(c + d*sec(e + f*x))^4*tan(e + f*x))/(2*b*(a^2 - b^2)*f*(a + b*sec(e + f*x))^2)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^5)/(a + b*sec(e + f*x))^3, x, 9, -(d^3*(30*a*b*c*d - 12*a^2*d^2 - b^2*(20*c^2 + d^2))*atanh(sin(e + f*x)))/(2*b^5*f) + ((b*c - a*d)^3*(6*a^3*b*c*d - 12*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 29*d^2) + b^4*(c^2 + 20*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*b^5*(a + b)^(5/2)*f) + (d*(30*a^4*b*c*d^3 - 12*a^5*d^4 - a^3*b^2*d^2*(16*c^2 - 21*d^2) - b^5*c*d*(17*c^2 - 10*d^2) - a^2*b^3*c*d*(4*c^2 + 55*d^2) + a*b^4*(6*c^4 + 43*c^2*d^2 - 6*d^4))*tan(e + f*x))/(2*b^4*(a^2 - b^2)^2*f) - (d^2*(7*a^3*b*c*d^2 - 6*a^4*d^3 + b^4*d*(8*c^2 - d^2) + a^2*b^2*d*(c^2 + 10*d^2) - a*b^3*c*(3*c^2 + 16*d^2))*sec(e + f*x)*tan(e + f*x))/(2*b^3*(a^2 - b^2)^2*f) - ((b*c - a*d)^2*(3*a*b*c + 4*a^2*d - 7*b^2*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(2*b^2*(a^2 - b^2)^2*f*(a + b*sec(e + f*x))) - ((b*c - a*d)^2*(c + d*sec(e + f*x))^3*tan(e + f*x))/(2*b*(a^2 - b^2)*f*(a + b*sec(e + f*x))^2)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^4)/(a + b*sec(e + f*x))^3, x, 8, (d^3*(4*b*c - 3*a*d)*atanh(sin(e + f*x)))/(b^4*f) + ((b*c - a*d)^2*(4*a^3*b*c*d - 10*a*b^3*c*d + 6*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 12*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*b^4*(a + b)^(5/2)*f) - (d^2*(2*a*b*c*d - 3*a^2*d^2 - b^2*(c^2 - 2*d^2))*tan(e + f*x))/(2*b^3*(a^2 - b^2)*f) - (3*(b*c - a*d)^3*(a*b*c + a^2*d - 2*b^2*d)*tan(e + f*x))/(2*b^3*(a^2 - b^2)^2*f*(a + b*sec(e + f*x))) - ((b*c - a*d)^2*(c + d*sec(e + f*x))^2*tan(e + f*x))/(2*b*(a^2 - b^2)*f*(a + b*sec(e + f*x))^2)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^3)/(a + b*sec(e + f*x))^3, x, 7, (d^3*atanh(sin(e + f*x)))/(b^3*f) + ((b*c - a*d)*(2*a^3*b*c*d - 8*a*b^3*c*d + 2*a^4*d^2 + a^2*b^2*(2*c^2 - 5*d^2) + b^4*(c^2 + 6*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*b^3*(a + b)^(5/2)*f) - ((b*c - a*d)^2*(3*a*b*c + 2*a^2*d - 5*b^2*d)*tan(e + f*x))/(2*b^2*(a^2 - b^2)^2*f*(a + b*sec(e + f*x))) - ((b*c - a*d)^2*(c + d*sec(e + f*x))*tan(e + f*x))/(2*b*(a^2 - b^2)*f*(a + b*sec(e + f*x))^2)]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^2)/(a + b*sec(e + f*x))^3, x, 6, -(((6*a*b*c*d - a^2*(2*c^2 + d^2) - b^2*(c^2 + 2*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*f)) - ((b*c - a*d)^2*tan(e + f*x))/(2*b*(a^2 - b^2)*f*(a + b*sec(e + f*x))^2) - ((b*c - a*d)*(3*a*b*c + a^2*d - 4*b^2*d)*tan(e + f*x))/(2*b*(a^2 - b^2)^2*f*(a + b*sec(e + f*x)))]
    @test_int [(sec(e + f*x)*(c + d*sec(e + f*x))^1)/(a + b*sec(e + f*x))^3, x, 6, ((2*a^2*c + b^2*c - 3*a*b*d)*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*f) - ((b*c - a*d)*tan(e + f*x))/(2*(a^2 - b^2)*f*(a + b*sec(e + f*x))^2) - ((3*a*b*c - a^2*d - 2*b^2*d)*tan(e + f*x))/(2*(a^2 - b^2)^2*f*(a + b*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + b*sec(e + f*x))^3*(c + d*sec(e + f*x))^1), x, 9, -((b*(6*a^3*b*c*d - 6*a^4*d^2 - a^2*b^2*(2*c^2 - 5*d^2) - b^4*(c^2 + 2*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^3*f)) - (2*d^3*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/(sqrt(c - d)*sqrt(c + d)*(b*c - a*d)^3*f) - (b^2*tan(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sec(e + f*x))^2) - (b^2*(3*a*b*c - 5*a^2*d + 2*b^2*d)*tan(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + b*sec(e + f*x))^3*(c + d*sec(e + f*x))^2), x, 10, -((b^2*(8*a^3*b*c*d - 2*a*b^3*c*d - 12*a^4*d^2 - a^2*b^2*(2*c^2 - 15*d^2) - b^4*(c^2 + 6*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^4*f)) - (2*d^3*(4*b*c^2 - a*c*d - 3*b*d^2)*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(3/2)*(c + d)^(3/2)*(b*c - a*d)^4*f) + (d*(2*a^4*d^3 + a^2*b^2*d*(7*c^2 - 11*d^2) - 2*b^4*d*(2*c^2 - 3*d^2) - 3*a*b^3*c*(c^2 - d^2))*tan(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*sec(e + f*x))) - (b^2*tan(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))) - (3*b^2*(a*b*c - 2*a^2*d + b^2*d)*tan(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [sec(e + f*x)/((a + b*sec(e + f*x))^3*(c + d*sec(e + f*x))^3), x, 11, -((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*atanh((sqrt(a - b)*tan((e + f*x)/2))/sqrt(a + b)))/((a - b)^(5/2)*(a + b)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*atanh((sqrt(c - d)*tan((e + f*x)/2))/sqrt(c + d)))/((c - d)^(5/2)*(c + d)^(5/2)*(b*c - a*d)^5*f) + (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b^3*c*(c^2 - d^2))*tan(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*sec(e + f*x))^2) - (b^2*tan(e + f*x))/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*sec(e + f*x))^2*(c + d*sec(e + f*x))^2) - (b^2*(3*a*b*c - 7*a^2*d + 4*b^2*d)*tan(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^2) - (3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*tan(e + f*x))/(2*(a^2 - b^2)^2*(b*c - a*d)^4*(c^2 - d^2)^2*f*(c + d*sec(e + f*x)))] =#

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

    @test_int [(sec(e + f*x)*sqrt(a + b*sec(e + f*x)))/(c + d*sec(e + f*x)), x, 3, (2*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(d*f) - (2*(b*c - a*d)*Elliptic.Pi((2*d)/(c + d), asin(sqrt(1 - sec(e + f*x))/sqrt(2)), (2*b)/(a + b))*sqrt((a + b*sec(e + f*x))/(a + b))*tan(e + f*x))/(d*(c + d)*f*sqrt(a + b*sec(e + f*x))*sqrt(-tan(e + f*x)^2))]

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [sec(e + f*x)*sqrt(a + b*sec(e + f*x))/sqrt(c + d*sec(e + f*x)), x, 1, (2*cot(e + f*x)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt((a + b)/(c + d))*sqrt(c + d*sec(e + f*x)))/sqrt(a + b*sec(e + f*x))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(-(((b*c - a*d)*(1 - sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x)))/(d*sqrt((a + b)/(c + d))*f)]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)/(sqrt(a + b*sec(e + f*x))*sqrt(c + d*sec(e + f*x))), x, 1, (2*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(sqrt(c + d)*(b*c - a*d)*f)]

    @test_int [sec(e + f*x)/(sqrt(2 + 3*sec(e + f*x))*sqrt(-4 + 5*sec(e + f*x))), x, 1, (2*cot(e + f*x)*Elliptic.F(asin(sqrt(2 + 3*sec(e + f*x))/(sqrt(5)*sqrt(-4 + 5*sec(e + f*x)))), 45)*(4 - 5*sec(e + f*x))*sqrt((1 - sec(e + f*x))/(4 - 5*sec(e + f*x)))*sqrt((1 + sec(e + f*x))/(4 - 5*sec(e + f*x))))/f]
    @test_int [sec(e + f*x)/(sqrt(2 + 3*sec(e + f*x))*sqrt(4 - 5*sec(e + f*x))), x, 1, (2*im*cot(e + f*x)*Elliptic.F(im*asinh((sqrt(5)*sqrt(4 - 5*sec(e + f*x)))/sqrt(2 + 3*sec(e + f*x))), 1/45)*sqrt((1 - sec(e + f*x))/(2 + 3*sec(e + f*x)))*sqrt((1 + sec(e + f*x))/(2 + 3*sec(e + f*x)))*(2 + 3*sec(e + f*x)))/(3*sqrt(5)*f)]

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

    #= ::Subsubsection:: =#
    #=m>0=#

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)^2/(sqrt(a + b*sec(e + f*x))*sqrt(c + d*sec(e + f*x))), x, 3, (2*cot(e + f*x)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt((a + b)/(c + d))*sqrt(c + d*sec(e + f*x)))/sqrt(a + b*sec(e + f*x))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(-(((b*c - a*d)*(1 - sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x)))/(b*d*sqrt((a + b)/(c + d))*f) - (2*a*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(b*sqrt(c + d)*(b*c - a*d)*f)]

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

    #= [(g*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^(5/2)/(a + b*sec(e + f*x)), x, 0, 0]
    @test_int [(g*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^(3/2)/(a + b*sec(e + f*x)), x, 0, 0] =#
    @test_int [(g*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^(1/2)/(a + b*sec(e + f*x)), x, 7, (2*d*g*sqrt((d + c*cos(e + f*x))/(c + d))*Elliptic.Pi(2, (1/2)*(e + f*x), (2*c)/(c + d))*sqrt(g*sec(e + f*x)))/(b*f*sqrt(c + d*sec(e + f*x))) + (2*(b*c - a*d)*g*sqrt((d + c*cos(e + f*x))/(c + d))*Elliptic.Pi((2*a)/(a + b), (1/2)*(e + f*x), (2*c)/(c + d))*sqrt(g*sec(e + f*x)))/(b*(a + b)*f*sqrt(c + d*sec(e + f*x)))]
    @test_int [(g*sec(e + f*x))^(3/2)/((c + d*sec(e + f*x))^(1/2)*(a + b*sec(e + f*x))), x, 3, (2*g*sqrt((d + c*cos(e + f*x))/(c + d))*Elliptic.Pi((2*a)/(a + b), (1/2)*(e + f*x), (2*c)/(c + d))*sqrt(g*sec(e + f*x)))/((a + b)*f*sqrt(c + d*sec(e + f*x)))]
    #= [(g*sec(e + f*x))^(3/2)/((c + d*sec(e + f*x))^(3/2)*(a + b*sec(e + f*x))), x, 0, 0]
    @test_int [(g*sec(e + f*x))^(3/2)/((c + d*sec(e + f*x))^(5/2)*(a + b*sec(e + f*x))), x, 0, 0] =#

    @test_int [(sqrt(g*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))/(a + b*cos(e + f*x)), x, 8, (2*d*sqrt((d + c*cos(e + f*x))/(c + d))*Elliptic.Pi(2, (1/2)*(e + f*x), (2*c)/(c + d))*sqrt(g*sec(e + f*x)))/(a*f*sqrt(c + d*sec(e + f*x))) + (2*(a*c - b*d)*sqrt((d + c*cos(e + f*x))/(c + d))*Elliptic.Pi((2*b)/(a + b), (1/2)*(e + f*x), (2*c)/(c + d))*sqrt(g*sec(e + f*x)))/(a*(a + b)*f*sqrt(c + d*sec(e + f*x)))]

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(sec(e + f*x)*sqrt(a + b*sec(e + f*x)))/(c + c*sec(e + f*x)), x, 1, (Elliptic.E(asin(tan(e + f*x)/(1 + sec(e + f*x))), (a - b)/(a + b))*sqrt(1/(1 + sec(e + f*x)))*sqrt(a + b*sec(e + f*x)))/(c*f*sqrt((a + b*sec(e + f*x))/((a + b)*(1 + sec(e + f*x)))))]

    @test_int [(g*sec(e + f*x))^(3/2)*sqrt(a + b*sec(e + f*x))/(c + c*sec(e + f*x)), x, 11, (g*(b + a*cos(e + f*x))*Elliptic.E((1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/(c*f*sqrt((b + a*cos(e + f*x))/(a + b))*sqrt(a + b*sec(e + f*x))) + ((a - b)*g*sqrt((b + a*cos(e + f*x))/(a + b))*Elliptic.F((1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/(c*f*sqrt(a + b*sec(e + f*x))) + (2*b*g*sqrt((b + a*cos(e + f*x))/(a + b))*Elliptic.Pi(2, (1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/(c*f*sqrt(a + b*sec(e + f*x))) - (g*(b + a*cos(e + f*x))*sqrt(g*sec(e + f*x))*sin(e + f*x))/(f*(c + c*cos(e + f*x))*sqrt(a + b*sec(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)^1/(sqrt(a + b*sec(e + f*x))*(c + c*sec(e + f*x))), x, 3, -((2*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/((a - b)*c*f)) + (Elliptic.E(asin(tan(e + f*x)/(1 + sec(e + f*x))), (a - b)/(a + b))*sqrt(1/(1 + sec(e + f*x)))*sqrt(a + b*sec(e + f*x)))/((a - b)*c*f*sqrt((a + b*sec(e + f*x))/((a + b)*(1 + sec(e + f*x)))))]
    @test_int [sec(e + f*x)^2/(sqrt(a + b*sec(e + f*x))*(c + c*sec(e + f*x))), x, 3, (2*a*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/((a - b)*b*c*f) - (Elliptic.E(asin(tan(e + f*x)/(1 + sec(e + f*x))), (a - b)/(a + b))*sqrt(1/(1 + sec(e + f*x)))*sqrt(a + b*sec(e + f*x)))/((a - b)*c*f*sqrt((a + b*sec(e + f*x))/((a + b)*(1 + sec(e + f*x)))))]

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

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

    #= ::Subsubsection::Closed:: =#
    #=m>0=#

    @test_int [(sec(e + f*x)*sqrt(a + b*sec(e + f*x)))/(c + d*sec(e + f*x)), x, 3, (2*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(d*f) - (2*(b*c - a*d)*Elliptic.Pi((2*d)/(c + d), asin(sqrt(1 - sec(e + f*x))/sqrt(2)), (2*b)/(a + b))*sqrt((a + b*sec(e + f*x))/(a + b))*tan(e + f*x))/(d*(c + d)*f*sqrt(a + b*sec(e + f*x))*sqrt(-tan(e + f*x)^2))]

    @test_int [(g*sec(e + f*x))^(3/2)*sqrt(a + b*sec(e + f*x))/(c + d*sec(e + f*x)), x, 7, (2*b*g*sqrt((b + a*cos(e + f*x))/(a + b))*Elliptic.Pi(2, (1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/(d*f*sqrt(a + b*sec(e + f*x))) - (2*(b*c - a*d)*g*sqrt((b + a*cos(e + f*x))/(a + b))*Elliptic.Pi((2*c)/(c + d), (1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/(d*(c + d)*f*sqrt(a + b*sec(e + f*x)))]

    #= ::Subsubsection::Closed:: =#
    #=m<0=#

    @test_int [sec(e + f*x)^1/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))), x, 1, (2*Elliptic.Pi((2*d)/(c + d), asin(sqrt(1 - sec(e + f*x))/sqrt(2)), (2*b)/(a + b))*sqrt((a + b*sec(e + f*x))/(a + b))*tan(e + f*x))/((c + d)*f*sqrt(a + b*sec(e + f*x))*sqrt(-tan(e + f*x)^2))]
    @test_int [sec(e + f*x)^2/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))), x, 3, (2*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(b*d*f) - (2*c*Elliptic.Pi((2*d)/(c + d), asin(sqrt(1 - sec(e + f*x))/sqrt(2)), (2*b)/(a + b))*sqrt((a + b*sec(e + f*x))/(a + b))*tan(e + f*x))/(d*(c + d)*f*sqrt(a + b*sec(e + f*x))*sqrt(-tan(e + f*x)^2))]

    @test_int [(g*sec(e + f*x))^(3/2)/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))), x, 3, (2*g*sqrt((b + a*cos(e + f*x))/(a + b))*Elliptic.Pi((2*c)/(c + d), (1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/((c + d)*f*sqrt(a + b*sec(e + f*x)))]
    @test_int [(g*sec(e + f*x))^(5/2)/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))), x, 7, (2*g^2*sqrt((b + a*cos(e + f*x))/(a + b))*Elliptic.Pi(2, (1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/(d*f*sqrt(a + b*sec(e + f*x))) - (2*c*g^2*sqrt((b + a*cos(e + f*x))/(a + b))*Elliptic.Pi((2*c)/(c + d), (1/2)*(e + f*x), (2*a)/(a + b))*sqrt(g*sec(e + f*x)))/(d*(c + d)*f*sqrt(a + b*sec(e + f*x)))]

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

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

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

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

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

    @test_int [sec(e + f*x)*tan(e + f*x)^4/(c - c*sec(e + f*x))^7, x, 4, cot((1/2)*(e + f*x))^5/(20*c^7*f) - cot((1/2)*(e + f*x))^7/(14*c^7*f) + cot((1/2)*(e + f*x))^9/(36*c^7*f)]
    @test_int [sec(e + f*x)*tan(e + f*x)^4/(c - c*sec(e + f*x))^8, x, 4, cot((1/2)*(e + f*x))^5/(40*c^8*f) - (3*cot((1/2)*(e + f*x))^7)/(56*c^8*f) + cot((1/2)*(e + f*x))^9/(24*c^8*f) - cot((1/2)*(e + f*x))^11/(88*c^8*f)]
    end

max_stars_repo_path: test/4 Trig functions/4.5 Secant/4.5.2.3 (g sec)^p (a+b sec)^m (c+d sec)^n.jl