text:
@testset "4.1.10 (c+d x)^m (a+b sin)^n" begin
    (a, b, c, d, e, f, m, n, x, ) = @variables a b c d e f m n x

    #= ::Package:: =#

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

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

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

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

    @test_int [sin(a + b*x)*(c + d*x)^4, x, 5, -((24*d^4*cos(a + b*x))/b^5) + (12*d^2*(c + d*x)^2*cos(a + b*x))/b^3 - ((c + d*x)^4*cos(a + b*x))/b - (24*d^3*(c + d*x)*sin(a + b*x))/b^4 + (4*d*(c + d*x)^3*sin(a + b*x))/b^2]
    @test_int [sin(a + b*x)*(c + d*x)^3, x, 4, (6*d^2*(c + d*x)*cos(a + b*x))/b^3 - ((c + d*x)^3*cos(a + b*x))/b - (6*d^3*sin(a + b*x))/b^4 + (3*d*(c + d*x)^2*sin(a + b*x))/b^2]
    @test_int [sin(a + b*x)*(c + d*x)^2, x, 3, (2*d^2*cos(a + b*x))/b^3 - ((c + d*x)^2*cos(a + b*x))/b + (2*d*(c + d*x)*sin(a + b*x))/b^2]
    @test_int [sin(a + b*x)*(c + d*x)^1, x, 2, -(((c + d*x)*cos(a + b*x))/b) + (d*sin(a + b*x))/b^2]
    @test_int [sin(a + b*x)/(c + d*x)^1, x, 3, (CosIntegral((b*c)/d + b*x)*sin(a - (b*c)/d))/d + (cos(a - (b*c)/d)*SinIntegral((b*c)/d + b*x))/d]
    @test_int [sin(a + b*x)/(c + d*x)^2, x, 4, (b*cos(a - (b*c)/d)*CosIntegral((b*c)/d + b*x))/d^2 - sin(a + b*x)/(d*(c + d*x)) - (b*sin(a - (b*c)/d)*SinIntegral((b*c)/d + b*x))/d^2]
    @test_int [sin(a + b*x)/(c + d*x)^3, x, 5, -((b*cos(a + b*x))/(2*d^2*(c + d*x))) - (b^2*CosIntegral((b*c)/d + b*x)*sin(a - (b*c)/d))/(2*d^3) - sin(a + b*x)/(2*d*(c + d*x)^2) - (b^2*cos(a - (b*c)/d)*SinIntegral((b*c)/d + b*x))/(2*d^3)]

    @test_int [sin(a + b*x)^2*(c + d*x)^4, x, 6, (3*d^4*x)/(4*b^4) - (d*(c + d*x)^3)/(2*b^2) + (c + d*x)^5/(10*d) - (3*d^4*cos(a + b*x)*sin(a + b*x))/(4*b^5) + (3*d^2*(c + d*x)^2*cos(a + b*x)*sin(a + b*x))/(2*b^3) - ((c + d*x)^4*cos(a + b*x)*sin(a + b*x))/(2*b) - (3*d^3*(c + d*x)*sin(a + b*x)^2)/(2*b^4) + (d*(c + d*x)^3*sin(a + b*x)^2)/b^2]
    @test_int [sin(a + b*x)^2*(c + d*x)^3, x, 4, -((3*c*d^2*x)/(4*b^2)) - (3*d^3*x^2)/(8*b^2) + (c + d*x)^4/(8*d) + (3*d^2*(c + d*x)*cos(a + b*x)*sin(a + b*x))/(4*b^3) - ((c + d*x)^3*cos(a + b*x)*sin(a + b*x))/(2*b) - (3*d^3*sin(a + b*x)^2)/(8*b^4) + (3*d*(c + d*x)^2*sin(a + b*x)^2)/(4*b^2)]
    @test_int [sin(a + b*x)^2*(c + d*x)^2, x, 4, -((d^2*x)/(4*b^2)) + (c + d*x)^3/(6*d) + (d^2*cos(a + b*x)*sin(a + b*x))/(4*b^3) - ((c + d*x)^2*cos(a + b*x)*sin(a + b*x))/(2*b) + (d*(c + d*x)*sin(a + b*x)^2)/(2*b^2)]
    @test_int [sin(a + b*x)^2*(c + d*x)^1, x, 2, (c*x)/2 + (d*x^2)/4 - ((c + d*x)*cos(a + b*x)*sin(a + b*x))/(2*b) + (d*sin(a + b*x)^2)/(4*b^2)]
    @test_int [sin(a + b*x)^2/(c + d*x)^1, x, 5, -((cos(2*a - (2*b*c)/d)*CosIntegral((2*b*c)/d + 2*b*x))/(2*d)) + log(c + d*x)/(2*d) + (sin(2*a - (2*b*c)/d)*SinIntegral((2*b*c)/d + 2*b*x))/(2*d)]
    @test_int [sin(a + b*x)^2/(c + d*x)^2, x, 5, (b*CosIntegral((2*b*c)/d + 2*b*x)*sin(2*a - (2*b*c)/d))/d^2 - sin(a + b*x)^2/(d*(c + d*x)) + (b*cos(2*a - (2*b*c)/d)*SinIntegral((2*b*c)/d + 2*b*x))/d^2]
    @test_int [sin(a + b*x)^2/(c + d*x)^3, x, 7, (b^2*cos(2*a - (2*b*c)/d)*CosIntegral((2*b*c)/d + 2*b*x))/d^3 - (b*cos(a + b*x)*sin(a + b*x))/(d^2*(c + d*x)) - sin(a + b*x)^2/(2*d*(c + d*x)^2) - (b^2*sin(2*a - (2*b*c)/d)*SinIntegral((2*b*c)/d + 2*b*x))/d^3]
    @test_int [sin(a + b*x)^2/(c + d*x)^4, x, 7, -(b^2/(3*d^3*(c + d*x))) - (2*b^3*CosIntegral((2*b*c)/d + 2*b*x)*sin(2*a - (2*b*c)/d))/(3*d^4) - (b*cos(a + b*x)*sin(a + b*x))/(3*d^2*(c + d*x)^2) - sin(a + b*x)^2/(3*d*(c + d*x)^3) + (2*b^2*sin(a + b*x)^2)/(3*d^3*(c + d*x)) - (2*b^3*cos(2*a - (2*b*c)/d)*SinIntegral((2*b*c)/d + 2*b*x))/(3*d^4)]

    @test_int [sin(a + b*x)^3*(c + d*x)^4, x, 12, -((488*d^4*cos(a + b*x))/(27*b^5)) + (80*d^2*(c + d*x)^2*cos(a + b*x))/(9*b^3) - (2*(c + d*x)^4*cos(a + b*x))/(3*b) + (8*d^4*cos(a + b*x)^3)/(81*b^5) - (160*d^3*(c + d*x)*sin(a + b*x))/(9*b^4) + (8*d*(c + d*x)^3*sin(a + b*x))/(3*b^2) + (4*d^2*(c + d*x)^2*cos(a + b*x)*sin(a + b*x)^2)/(9*b^3) - ((c + d*x)^4*cos(a + b*x)*sin(a + b*x)^2)/(3*b) - (8*d^3*(c + d*x)*sin(a + b*x)^3)/(27*b^4) + (4*d*(c + d*x)^3*sin(a + b*x)^3)/(9*b^2)]
    @test_int [sin(a + b*x)^3*(c + d*x)^3, x, 8, (40*d^2*(c + d*x)*cos(a + b*x))/(9*b^3) - (2*(c + d*x)^3*cos(a + b*x))/(3*b) - (40*d^3*sin(a + b*x))/(9*b^4) + (2*d*(c + d*x)^2*sin(a + b*x))/b^2 + (2*d^2*(c + d*x)*cos(a + b*x)*sin(a + b*x)^2)/(9*b^3) - ((c + d*x)^3*cos(a + b*x)*sin(a + b*x)^2)/(3*b) - (2*d^3*sin(a + b*x)^3)/(27*b^4) + (d*(c + d*x)^2*sin(a + b*x)^3)/(3*b^2)]
    @test_int [sin(a + b*x)^3*(c + d*x)^2, x, 6, (14*d^2*cos(a + b*x))/(9*b^3) - (2*(c + d*x)^2*cos(a + b*x))/(3*b) - (2*d^2*cos(a + b*x)^3)/(27*b^3) + (4*d*(c + d*x)*sin(a + b*x))/(3*b^2) - ((c + d*x)^2*cos(a + b*x)*sin(a + b*x)^2)/(3*b) + (2*d*(c + d*x)*sin(a + b*x)^3)/(9*b^2)]
    @test_int [sin(a + b*x)^3*(c + d*x)^1, x, 3, -((2*(c + d*x)*cos(a + b*x))/(3*b)) + (2*d*sin(a + b*x))/(3*b^2) - ((c + d*x)*cos(a + b*x)*sin(a + b*x)^2)/(3*b) + (d*sin(a + b*x)^3)/(9*b^2)]
    @test_int [sin(a + b*x)^3/(c + d*x)^1, x, 8, -((CosIntegral((3*b*c)/d + 3*b*x)*sin(3*a - (3*b*c)/d))/(4*d)) + (3*CosIntegral((b*c)/d + b*x)*sin(a - (b*c)/d))/(4*d) + (3*cos(a - (b*c)/d)*SinIntegral((b*c)/d + b*x))/(4*d) - (cos(3*a - (3*b*c)/d)*SinIntegral((3*b*c)/d + 3*b*x))/(4*d)]
    @test_int [sin(a + b*x)^3/(c + d*x)^2, x, 8, (3*b*cos(a - (b*c)/d)*CosIntegral((b*c)/d + b*x))/(4*d^2) - (3*b*cos(3*a - (3*b*c)/d)*CosIntegral((3*b*c)/d + 3*b*x))/(4*d^2) - sin(a + b*x)^3/(d*(c + d*x)) - (3*b*sin(a - (b*c)/d)*SinIntegral((b*c)/d + b*x))/(4*d^2) + (3*b*sin(3*a - (3*b*c)/d)*SinIntegral((3*b*c)/d + 3*b*x))/(4*d^2)]
    @test_int [sin(a + b*x)^3/(c + d*x)^3, x, 12, (9*b^2*CosIntegral((3*b*c)/d + 3*b*x)*sin(3*a - (3*b*c)/d))/(8*d^3) - (3*b^2*CosIntegral((b*c)/d + b*x)*sin(a - (b*c)/d))/(8*d^3) - (3*b*cos(a + b*x)*sin(a + b*x)^2)/(2*d^2*(c + d*x)) - sin(a + b*x)^3/(2*d*(c + d*x)^2) - (3*b^2*cos(a - (b*c)/d)*SinIntegral((b*c)/d + b*x))/(8*d^3) + (9*b^2*cos(3*a - (3*b*c)/d)*SinIntegral((3*b*c)/d + 3*b*x))/(8*d^3)]

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

    @test_int [csc(a + b*x)*(c + d*x)^3, x, 9, -((2*(c + d*x)^3*atanh(ℯ^(im*(a + b*x))))/b) + (3*im*d*(c + d*x)^2*PolyLog(2, -ℯ^(im*(a + b*x))))/b^2 - (3*im*d*(c + d*x)^2*PolyLog(2, ℯ^(im*(a + b*x))))/b^2 - (6*d^2*(c + d*x)*PolyLog(3, -ℯ^(im*(a + b*x))))/b^3 + (6*d^2*(c + d*x)*PolyLog(3, ℯ^(im*(a + b*x))))/b^3 - (6*im*d^3*PolyLog(4, -ℯ^(im*(a + b*x))))/b^4 + (6*im*d^3*PolyLog(4, ℯ^(im*(a + b*x))))/b^4]
    @test_int [csc(a + b*x)*(c + d*x)^2, x, 7, -((2*(c + d*x)^2*atanh(ℯ^(im*(a + b*x))))/b) + (2*im*d*(c + d*x)*PolyLog(2, -ℯ^(im*(a + b*x))))/b^2 - (2*im*d*(c + d*x)*PolyLog(2, ℯ^(im*(a + b*x))))/b^2 - (2*d^2*PolyLog(3, -ℯ^(im*(a + b*x))))/b^3 + (2*d^2*PolyLog(3, ℯ^(im*(a + b*x))))/b^3]
    @test_int [csc(a + b*x)*(c + d*x)^1, x, 5, -((2*(c + d*x)*atanh(ℯ^(im*(a + b*x))))/b) + (im*d*PolyLog(2, -ℯ^(im*(a + b*x))))/b^2 - (im*d*PolyLog(2, ℯ^(im*(a + b*x))))/b^2]
    @test_int [csc(a + b*x)/(c + d*x)^1, x, 0, Unintegrable(csc(a + b*x)/(c + d*x), x)]
    @test_int [csc(a + b*x)/(c + d*x)^2, x, 0, Unintegrable(csc(a + b*x)/(c + d*x)^2, x)]

    @test_int [csc(a + b*x)^2*(c + d*x)^3, x, 6, -((im*(c + d*x)^3)/b) - ((c + d*x)^3*cot(a + b*x))/b + (3*d*(c + d*x)^2*log(1 - ℯ^(2*im*(a + b*x))))/b^2 - (3*im*d^2*(c + d*x)*PolyLog(2, ℯ^(2*im*(a + b*x))))/b^3 + (3*d^3*PolyLog(3, ℯ^(2*im*(a + b*x))))/(2*b^4)]
    @test_int [csc(a + b*x)^2*(c + d*x)^2, x, 5, -((im*(c + d*x)^2)/b) - ((c + d*x)^2*cot(a + b*x))/b + (2*d*(c + d*x)*log(1 - ℯ^(2*im*(a + b*x))))/b^2 - (im*d^2*PolyLog(2, ℯ^(2*im*(a + b*x))))/b^3]
    @test_int [csc(a + b*x)^2*(c + d*x)^1, x, 2, -(((c + d*x)*cot(a + b*x))/b) + (d*log(sin(a + b*x)))/b^2]
    @test_int [csc(a + b*x)^2/(c + d*x)^1, x, 0, Unintegrable(csc(a + b*x)^2/(c + d*x), x)]
    @test_int [csc(a + b*x)^2/(c + d*x)^2, x, 0, Unintegrable(csc(a + b*x)^2/(c + d*x)^2, x)]

    @test_int [csc(a + b*x)^3*(c + d*x)^3, x, 15, -((6*d^2*(c + d*x)*atanh(ℯ^(im*(a + b*x))))/b^3) - ((c + d*x)^3*atanh(ℯ^(im*(a + b*x))))/b - (3*d*(c + d*x)^2*csc(a + b*x))/(2*b^2) - ((c + d*x)^3*cot(a + b*x)*csc(a + b*x))/(2*b) + (3*im*d^3*PolyLog(2, -ℯ^(im*(a + b*x))))/b^4 + (3*im*d*(c + d*x)^2*PolyLog(2, -ℯ^(im*(a + b*x))))/(2*b^2) - (3*im*d^3*PolyLog(2, ℯ^(im*(a + b*x))))/b^4 - (3*im*d*(c + d*x)^2*PolyLog(2, ℯ^(im*(a + b*x))))/(2*b^2) - (3*d^2*(c + d*x)*PolyLog(3, -ℯ^(im*(a + b*x))))/b^3 + (3*d^2*(c + d*x)*PolyLog(3, ℯ^(im*(a + b*x))))/b^3 - (3*im*d^3*PolyLog(4, -ℯ^(im*(a + b*x))))/b^4 + (3*im*d^3*PolyLog(4, ℯ^(im*(a + b*x))))/b^4]
    @test_int [csc(a + b*x)^3*(c + d*x)^2, x, 9, -(((c + d*x)^2*atanh(ℯ^(im*(a + b*x))))/b) - (d^2*atanh(cos(a + b*x)))/b^3 - (d*(c + d*x)*csc(a + b*x))/b^2 - ((c + d*x)^2*cot(a + b*x)*csc(a + b*x))/(2*b) + (im*d*(c + d*x)*PolyLog(2, -ℯ^(im*(a + b*x))))/b^2 - (im*d*(c + d*x)*PolyLog(2, ℯ^(im*(a + b*x))))/b^2 - (d^2*PolyLog(3, -ℯ^(im*(a + b*x))))/b^3 + (d^2*PolyLog(3, ℯ^(im*(a + b*x))))/b^3]
    @test_int [csc(a + b*x)^3*(c + d*x)^1, x, 6, -(((c + d*x)*atanh(ℯ^(im*(a + b*x))))/b) - (d*csc(a + b*x))/(2*b^2) - ((c + d*x)*cot(a + b*x)*csc(a + b*x))/(2*b) + (im*d*PolyLog(2, -ℯ^(im*(a + b*x))))/(2*b^2) - (im*d*PolyLog(2, ℯ^(im*(a + b*x))))/(2*b^2)]
    @test_int [csc(a + b*x)^3/(c + d*x)^1, x, 0, Unintegrable(csc(a + b*x)^3/(c + d*x), x)]
    @test_int [csc(a + b*x)^3/(c + d*x)^2, x, 0, Unintegrable(csc(a + b*x)^3/(c + d*x)^2, x)]

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

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

    @test_int [sin(a + b*x)*(c + d*x)^(5/2), x, 8, (15*d^2*sqrt(c + d*x)*cos(a + b*x))/(4*b^3) - ((c + d*x)^(5/2)*cos(a + b*x))/b - (15*d^(5/2)*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(4*b^(7/2)) + (15*d^(5/2)*sqrt(pi/2)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(4*b^(7/2)) + (5*d*(c + d*x)^(3/2)*sin(a + b*x))/(2*b^2)]
    @test_int [sin(a + b*x)*(c + d*x)^(3/2), x, 7, -(((c + d*x)^(3/2)*cos(a + b*x))/b) - (3*d^(3/2)*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(2*b^(5/2)) - (3*d^(3/2)*sqrt(pi/2)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(2*b^(5/2)) + (3*d*sqrt(c + d*x)*sin(a + b*x))/(2*b^2)]
    @test_int [sin(a + b*x)*(c + d*x)^(1/2), x, 6, -((sqrt(c + d*x)*cos(a + b*x))/b) + (sqrt(d)*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/b^(3/2) - (sqrt(d)*sqrt(pi/2)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/b^(3/2)]
    @test_int [sin(a + b*x)/(c + d*x)^(1/2), x, 5, (sqrt(2*pi)*cos(a - (b*c)/d)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(sqrt(b)*sqrt(d)) + (sqrt(2*pi)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(sqrt(b)*sqrt(d))]
    @test_int [sin(a + b*x)/(c + d*x)^(3/2), x, 6, (2*sqrt(b)*sqrt(2*pi)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/d^(3/2) - (2*sqrt(b)*sqrt(2*pi)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/d^(3/2) - (2*sin(a + b*x))/(d*sqrt(c + d*x))]
    @test_int [sin(a + b*x)/(c + d*x)^(5/2), x, 7, -((4*b*cos(a + b*x))/(3*d^2*sqrt(c + d*x))) - (4*b^(3/2)*sqrt(2*pi)*cos(a - (b*c)/d)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(3*d^(5/2)) - (4*b^(3/2)*sqrt(2*pi)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(3*d^(5/2)) - (2*sin(a + b*x))/(3*d*(c + d*x)^(3/2))]
    @test_int [sin(a + b*x)/(c + d*x)^(7/2), x, 8, -((4*b*cos(a + b*x))/(15*d^2*(c + d*x)^(3/2))) - (8*b^(5/2)*sqrt(2*pi)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(15*d^(7/2)) + (8*b^(5/2)*sqrt(2*pi)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(15*d^(7/2)) - (2*sin(a + b*x))/(5*d*(c + d*x)^(5/2)) + (8*b^2*sin(a + b*x))/(15*d^3*sqrt(c + d*x))]

    @test_int [sin(a + b*x)^2*(c + d*x)^(5/2), x, 10, -((5*d*(c + d*x)^(3/2))/(16*b^2)) + (c + d*x)^(7/2)/(7*d) - (15*d^(5/2)*sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/(128*b^(7/2)) - (15*d^(5/2)*sqrt(pi)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/(128*b^(7/2)) - ((c + d*x)^(5/2)*cos(a + b*x)*sin(a + b*x))/(2*b) + (5*d*(c + d*x)^(3/2)*sin(a + b*x)^2)/(8*b^2) + (15*d^2*sqrt(c + d*x)*sin(2*a + 2*b*x))/(64*b^3)]
    @test_int [sin(a + b*x)^2*(c + d*x)^(3/2), x, 9, -((3*d*sqrt(c + d*x))/(16*b^2)) + (c + d*x)^(5/2)/(5*d) + (3*d^(3/2)*sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/(32*b^(5/2)) - (3*d^(3/2)*sqrt(pi)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/(32*b^(5/2)) - ((c + d*x)^(3/2)*cos(a + b*x)*sin(a + b*x))/(2*b) + (3*d*sqrt(c + d*x)*sin(a + b*x)^2)/(8*b^2)]
    @test_int [sin(a + b*x)^2*(c + d*x)^(1/2), x, 8, (c + d*x)^(3/2)/(3*d) + (sqrt(d)*sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/(8*b^(3/2)) + (sqrt(d)*sqrt(pi)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/(8*b^(3/2)) - (sqrt(c + d*x)*sin(2*a + 2*b*x))/(4*b)]
    @test_int [sin(a + b*x)^2/(c + d*x)^(1/2), x, 7, sqrt(c + d*x)/d - (sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/(2*sqrt(b)*sqrt(d)) + (sqrt(pi)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/(2*sqrt(b)*sqrt(d))]
    @test_int [sin(a + b*x)^2/(c + d*x)^(3/2), x, 7, (2*sqrt(b)*sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/d^(3/2) + (2*sqrt(b)*sqrt(pi)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/d^(3/2) - (2*sin(a + b*x)^2)/(d*sqrt(c + d*x))]
    @test_int [sin(a + b*x)^2/(c + d*x)^(5/2), x, 9, (8*b^(3/2)*sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/(3*d^(5/2)) - (8*b^(3/2)*sqrt(pi)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/(3*d^(5/2)) - (8*b*cos(a + b*x)*sin(a + b*x))/(3*d^2*sqrt(c + d*x)) - (2*sin(a + b*x)^2)/(3*d*(c + d*x)^(3/2))]
    @test_int [sin(a + b*x)^2/(c + d*x)^(7/2), x, 9, -((16*b^2)/(15*d^3*sqrt(c + d*x))) - (32*b^(5/2)*sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/(15*d^(7/2)) - (32*b^(5/2)*sqrt(pi)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/(15*d^(7/2)) - (8*b*cos(a + b*x)*sin(a + b*x))/(15*d^2*(c + d*x)^(3/2)) - (2*sin(a + b*x)^2)/(5*d*(c + d*x)^(5/2)) + (32*b^2*sin(a + b*x)^2)/(15*d^3*sqrt(c + d*x))]
    @test_int [sin(a + b*x)^2/(c + d*x)^(9/2), x, 11, -((16*b^2)/(105*d^3*(c + d*x)^(3/2))) - (128*b^(7/2)*sqrt(pi)*cos(2*a - (2*b*c)/d)*FresnelC((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi))))/(105*d^(9/2)) + (128*b^(7/2)*sqrt(pi)*FresnelS((2*sqrt(b)*sqrt(c + d*x))/(sqrt(d)*sqrt(pi)))*sin(2*a - (2*b*c)/d))/(105*d^(9/2)) - (8*b*cos(a + b*x)*sin(a + b*x))/(35*d^2*(c + d*x)^(5/2)) + (128*b^3*cos(a + b*x)*sin(a + b*x))/(105*d^4*sqrt(c + d*x)) - (2*sin(a + b*x)^2)/(7*d*(c + d*x)^(7/2)) + (32*b^2*sin(a + b*x)^2)/(105*d^3*(c + d*x)^(3/2))]

    @test_int [sin(a + b*x)^3*(c + d*x)^(5/2), x, 23, (45*d^2*sqrt(c + d*x)*cos(a + b*x))/(16*b^3) - (2*(c + d*x)^(5/2)*cos(a + b*x))/(3*b) - (5*d^2*sqrt(c + d*x)*cos(3*a + 3*b*x))/(144*b^3) - (45*d^(5/2)*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(16*b^(7/2)) + (5*d^(5/2)*sqrt(pi/6)*cos(3*a - (3*b*c)/d)*FresnelC((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d)))/(144*b^(7/2)) - (5*d^(5/2)*sqrt(pi/6)*FresnelS((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d))*sin(3*a - (3*b*c)/d))/(144*b^(7/2)) + (45*d^(5/2)*sqrt(pi/2)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(16*b^(7/2)) + (5*d*(c + d*x)^(3/2)*sin(a + b*x))/(3*b^2) - ((c + d*x)^(5/2)*cos(a + b*x)*sin(a + b*x)^2)/(3*b) + (5*d*(c + d*x)^(3/2)*sin(a + b*x)^3)/(18*b^2)]
    @test_int [sin(a + b*x)^3*(c + d*x)^(3/2), x, 20, -((2*(c + d*x)^(3/2)*cos(a + b*x))/(3*b)) - (9*d^(3/2)*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(8*b^(5/2)) + (d^(3/2)*sqrt(pi/6)*cos(3*a - (3*b*c)/d)*FresnelS((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d)))/(24*b^(5/2)) + (d^(3/2)*sqrt(pi/6)*FresnelC((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d))*sin(3*a - (3*b*c)/d))/(24*b^(5/2)) - (9*d^(3/2)*sqrt(pi/2)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(8*b^(5/2)) + (d*sqrt(c + d*x)*sin(a + b*x))/b^2 - ((c + d*x)^(3/2)*cos(a + b*x)*sin(a + b*x)^2)/(3*b) + (d*sqrt(c + d*x)*sin(a + b*x)^3)/(6*b^2)]
    @test_int [sin(a + b*x)^3*(c + d*x)^(1/2), x, 14, -((3*sqrt(c + d*x)*cos(a + b*x))/(4*b)) + (sqrt(c + d*x)*cos(3*a + 3*b*x))/(12*b) + (3*sqrt(d)*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(4*b^(3/2)) - (sqrt(d)*sqrt(pi/6)*cos(3*a - (3*b*c)/d)*FresnelC((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d)))/(12*b^(3/2)) + (sqrt(d)*sqrt(pi/6)*FresnelS((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d))*sin(3*a - (3*b*c)/d))/(12*b^(3/2)) - (3*sqrt(d)*sqrt(pi/2)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(4*b^(3/2))]
    @test_int [sin(a + b*x)^3/(c + d*x)^(1/2), x, 12, (3*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(2*sqrt(b)*sqrt(d)) - (sqrt(pi/6)*cos(3*a - (3*b*c)/d)*FresnelS((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d)))/(2*sqrt(b)*sqrt(d)) - (sqrt(pi/6)*FresnelC((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d))*sin(3*a - (3*b*c)/d))/(2*sqrt(b)*sqrt(d)) + (3*sqrt(pi/2)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(2*sqrt(b)*sqrt(d))]
    @test_int [sin(a + b*x)^3/(c + d*x)^(3/2), x, 12, (3*sqrt(b)*sqrt(pi/2)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/d^(3/2) - (sqrt(b)*sqrt((3*pi)/2)*cos(3*a - (3*b*c)/d)*FresnelC((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d)))/d^(3/2) + (sqrt(b)*sqrt((3*pi)/2)*FresnelS((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d))*sin(3*a - (3*b*c)/d))/d^(3/2) - (3*sqrt(b)*sqrt(pi/2)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/d^(3/2) - (2*sin(a + b*x)^3)/(d*sqrt(c + d*x))]
    @test_int [sin(a + b*x)^3/(c + d*x)^(5/2), x, 18, -((b^(3/2)*sqrt(2*pi)*cos(a - (b*c)/d)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/d^(5/2)) + (b^(3/2)*sqrt(6*pi)*cos(3*a - (3*b*c)/d)*FresnelS((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d)))/d^(5/2) + (b^(3/2)*sqrt(6*pi)*FresnelC((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d))*sin(3*a - (3*b*c)/d))/d^(5/2) - (b^(3/2)*sqrt(2*pi)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/d^(5/2) - (4*b*cos(a + b*x)*sin(a + b*x)^2)/(d^2*sqrt(c + d*x)) - (2*sin(a + b*x)^3)/(3*d*(c + d*x)^(3/2))]
    @test_int [sin(a + b*x)^3/(c + d*x)^(7/2), x, 19, -((2*b^(5/2)*sqrt(2*pi)*cos(a - (b*c)/d)*FresnelC((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d)))/(5*d^(7/2))) + (6*b^(5/2)*sqrt(6*pi)*cos(3*a - (3*b*c)/d)*FresnelC((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d)))/(5*d^(7/2)) - (6*b^(5/2)*sqrt(6*pi)*FresnelS((sqrt(b)*sqrt(6/pi)*sqrt(c + d*x))/sqrt(d))*sin(3*a - (3*b*c)/d))/(5*d^(7/2)) + (2*b^(5/2)*sqrt(2*pi)*FresnelS((sqrt(b)*sqrt(2/pi)*sqrt(c + d*x))/sqrt(d))*sin(a - (b*c)/d))/(5*d^(7/2)) - (16*b^2*sin(a + b*x))/(5*d^3*sqrt(c + d*x)) - (4*b*cos(a + b*x)*sin(a + b*x)^2)/(5*d^2*(c + d*x)^(3/2)) - (2*sin(a + b*x)^3)/(5*d*(c + d*x)^(5/2)) + (24*b^2*sin(a + b*x)^3)/(5*d^3*sqrt(c + d*x))]

    @test_int [sin(f*x)*(d*x)^(3/2), x, 4, -(((d*x)^(3/2)*cos(f*x))/f) - (3*d^(3/2)*sqrt(pi/2)*FresnelS((sqrt(f)*sqrt(2/pi)*sqrt(d*x))/sqrt(d)))/(2*f^(5/2)) + (3*d*sqrt(d*x)*sin(f*x))/(2*f^2)]
    @test_int [sin(f*x)*(d*x)^(1/2), x, 3, -((sqrt(d*x)*cos(f*x))/f) + (sqrt(d)*sqrt(pi/2)*FresnelC((sqrt(f)*sqrt(2/pi)*sqrt(d*x))/sqrt(d)))/f^(3/2)]
    @test_int [sin(f*x)/(d*x)^(1/2), x, 2, (sqrt(2*pi)*FresnelS((sqrt(f)*sqrt(2/pi)*sqrt(d*x))/sqrt(d)))/(sqrt(d)*sqrt(f))]
    @test_int [sin(f*x)/(d*x)^(3/2), x, 3, (2*sqrt(f)*sqrt(2*pi)*FresnelC((sqrt(f)*sqrt(2/pi)*sqrt(d*x))/sqrt(d)))/d^(3/2) - (2*sin(f*x))/(d*sqrt(d*x))]
    @test_int [sin(f*x)/(d*x)^(5/2), x, 4, -((4*f*cos(f*x))/(3*d^2*sqrt(d*x))) - (4*f^(3/2)*sqrt(2*pi)*FresnelS((sqrt(f)*sqrt(2/pi)*sqrt(d*x))/sqrt(d)))/(3*d^(5/2)) - (2*sin(f*x))/(3*d*(d*x)^(3/2))]

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

    @test_int [csc(a + b*x)*(c + d*x)^(1/2), x, 0, Unintegrable(sqrt(c + d*x)*csc(a + b*x), x)]
    @test_int [csc(a + b*x)/(c + d*x)^(1/2), x, 0, Unintegrable(csc(a + b*x)/sqrt(c + d*x), x)]

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

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

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

    @test_int [x^1/sin(e + f*x)^(3/2) + x*sqrt(sin(e + f*x)), x, 2, -((2*x*cos(e + f*x))/(f*sqrt(sin(e + f*x)))) + (4*sqrt(sin(e + f*x)))/f^2]
    @test_int [x^2/sin(e + f*x)^(3/2) + x^2*sqrt(sin(e + f*x)), x, 3, -((16*Elliptic.E((1/2)*(e - pi/2 + f*x), 2))/f^3) - (2*x^2*cos(e + f*x))/(f*sqrt(sin(e + f*x))) + (8*x*sqrt(sin(e + f*x)))/f^2]

    @test_int [x/sin(e + f*x)^(5/2) - x/(3*sqrt(sin(e + f*x))), x, 2, -((2*x*cos(e + f*x))/(3*f*sin(e + f*x)^(3/2))) - 4/(3*f^2*sqrt(sin(e + f*x)))]

    @test_int [x/sin(e + f*x)^(7/2) + (3/5)*x*sqrt(sin(e + f*x)), x, 3, -((2*x*cos(e + f*x))/(5*f*sin(e + f*x)^(5/2))) - 4/(15*f^2*sin(e + f*x)^(3/2)) - (6*x*cos(e + f*x))/(5*f*sqrt(sin(e + f*x))) + (12*sqrt(sin(e + f*x)))/(5*f^2)]

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

    @test_int [(c + d*x)^m*(b*sin(e + f*x))^n, x, 0, Unintegrable((c + d*x)^m*(b*sin(e + f*x))^n, x)]

    @test_int [sin(a + b*x)^3*(c + d*x)^m, x, 8, -((3*ℯ^(im*(a - (b*c)/d))*(c + d*x)^m*Gamma(1 + m, -((im*b*(c + d*x))/d)))/((-((im*b*(c + d*x))/d))^m*(8*b))) - (3*(c + d*x)^m*Gamma(1 + m, (im*b*(c + d*x))/d))/(ℯ^(im*(a - (b*c)/d))*((im*b*(c + d*x))/d)^m*(8*b)) + (3^(-1 - m)*ℯ^(3*im*(a - (b*c)/d))*(c + d*x)^m*Gamma(1 + m, -((3*im*b*(c + d*x))/d)))/((-((im*b*(c + d*x))/d))^m*(8*b)) + (3^(-1 - m)*(c + d*x)^m*Gamma(1 + m, (3*im*b*(c + d*x))/d))/(ℯ^(3*im*(a - (b*c)/d))*((im*b*(c + d*x))/d)^m*(8*b))]
    @test_int [sin(a + b*x)^2*(c + d*x)^m, x, 5, (c + d*x)^(1 + m)/(2*d*(1 + m)) + (im*2^(-3 - m)*ℯ^(2*im*(a - (b*c)/d))*(c + d*x)^m*Gamma(1 + m, -((2*im*b*(c + d*x))/d)))/((-((im*b*(c + d*x))/d))^m*b) - (im*2^(-3 - m)*(c + d*x)^m*Gamma(1 + m, (2*im*b*(c + d*x))/d))/(ℯ^(2*im*(a - (b*c)/d))*((im*b*(c + d*x))/d)^m*b)]
    @test_int [sin(a + b*x)^1*(c + d*x)^m, x, 3, -((ℯ^(im*(a - (b*c)/d))*(c + d*x)^m*Gamma(1 + m, -((im*b*(c + d*x))/d)))/((-((im*b*(c + d*x))/d))^m*(2*b))) - ((c + d*x)^m*Gamma(1 + m, (im*b*(c + d*x))/d))/(ℯ^(im*(a - (b*c)/d))*((im*b*(c + d*x))/d)^m*(2*b))]
    @test_int [csc(a + b*x)^1*(c + d*x)^m, x, 0, Unintegrable((c + d*x)^m*csc(a + b*x), x)]
    @test_int [csc(a + b*x)^2*(c + d*x)^m, x, 0, Unintegrable((c + d*x)^m*csc(a + b*x)^2, x)]

    @test_int [x^(m + 3)*sin(a + b*x), x, 3, (im*ℯ^(im*a)*x^m*Gamma(4 + m, (-im)*b*x))/(((-im)*b*x)^m*(2*b^4)) - (im*x^m*Gamma(4 + m, im*b*x))/(ℯ^(im*a)*(im*b*x)^m*(2*b^4))]
    @test_int [x^(m + 2)*sin(a + b*x), x, 3, (ℯ^(im*a)*x^m*Gamma(3 + m, (-im)*b*x))/(((-im)*b*x)^m*(2*b^3)) + (x^m*Gamma(3 + m, im*b*x))/(ℯ^(im*a)*(im*b*x)^m*(2*b^3))]
    @test_int [x^(m + 1)*sin(a + b*x), x, 3, -((im*ℯ^(im*a)*x^m*Gamma(2 + m, (-im)*b*x))/(((-im)*b*x)^m*(2*b^2))) + (im*x^m*Gamma(2 + m, im*b*x))/(ℯ^(im*a)*(im*b*x)^m*(2*b^2))]
    @test_int [x^(m + 0)*sin(a + b*x), x, 3, -((ℯ^(im*a)*x^m*Gamma(1 + m, (-im)*b*x))/(((-im)*b*x)^m*(2*b))) - (x^m*Gamma(1 + m, im*b*x))/(ℯ^(im*a)*(im*b*x)^m*(2*b))]
    @test_int [x^(m - 1)*sin(a + b*x), x, 3, ((1/2)*im*ℯ^(im*a)*x^m*Gamma(m, (-im)*b*x))/((-im)*b*x)^m - ((1/2)*im*x^m*Gamma(m, im*b*x))/(ℯ^(im*a)*(im*b*x)^m)]
    @test_int [x^(m - 2)*sin(a + b*x), x, 3, ((1/2)*b*ℯ^(im*a)*x^m*Gamma(-1 + m, (-im)*b*x))/((-im)*b*x)^m + ((1/2)*b*x^m*Gamma(-1 + m, im*b*x))/(ℯ^(im*a)*(im*b*x)^m)]
    @test_int [x^(m - 3)*sin(a + b*x), x, 3, ((-(1/2))*im*b^2*ℯ^(im*a)*x^m*Gamma(-2 + m, (-im)*b*x))/((-im)*b*x)^m + ((1/2)*im*b^2*x^m*Gamma(-2 + m, im*b*x))/(ℯ^(im*a)*(im*b*x)^m)]

    @test_int [x^(m + 3)*sin(a + b*x)^2, x, 5, x^(4 + m)/(2*(4 + m)) + (2^(-6 - m)*ℯ^(2*im*a)*x^m*Gamma(4 + m, -2*im*b*x))/(((-im)*b*x)^m*b^4) + (2^(-6 - m)*x^m*Gamma(4 + m, 2*im*b*x))/(ℯ^(2*im*a)*(im*b*x)^m*b^4)]
    @test_int [x^(m + 2)*sin(a + b*x)^2, x, 5, x^(3 + m)/(2*(3 + m)) - (im*2^(-5 - m)*ℯ^(2*im*a)*x^m*Gamma(3 + m, -2*im*b*x))/(((-im)*b*x)^m*b^3) + (im*2^(-5 - m)*x^m*Gamma(3 + m, 2*im*b*x))/(ℯ^(2*im*a)*(im*b*x)^m*b^3)]
    @test_int [x^(m + 1)*sin(a + b*x)^2, x, 5, x^(2 + m)/(2*(2 + m)) - (2^(-4 - m)*ℯ^(2*im*a)*x^m*Gamma(2 + m, -2*im*b*x))/(((-im)*b*x)^m*b^2) - (2^(-4 - m)*x^m*Gamma(2 + m, 2*im*b*x))/(ℯ^(2*im*a)*(im*b*x)^m*b^2)]
    @test_int [x^(m + 0)*sin(a + b*x)^2, x, 5, x^(1 + m)/(2*(1 + m)) + (im*2^(-3 - m)*ℯ^(2*im*a)*x^m*Gamma(1 + m, -2*im*b*x))/(((-im)*b*x)^m*b) - (im*2^(-3 - m)*x^m*Gamma(1 + m, 2*im*b*x))/(ℯ^(2*im*a)*(im*b*x)^m*b)]
    @test_int [x^(m - 1)*sin(a + b*x)^2, x, 5, x^m/(2*m) + (2^(-2 - m)*ℯ^(2*im*a)*x^m*Gamma(m, -2*im*b*x))/((-im)*b*x)^m + (2^(-2 - m)*x^m*Gamma(m, 2*im*b*x))/(ℯ^(2*im*a)*(im*b*x)^m)]
    @test_int [x^(m - 2)*sin(a + b*x)^2, x, 5, -(x^(-1 + m)/(2*(1 - m))) - (im*2^(-1 - m)*b*ℯ^(2*im*a)*x^m*Gamma(-1 + m, -2*im*b*x))/((-im)*b*x)^m + (im*2^(-1 - m)*b*x^m*Gamma(-1 + m, 2*im*b*x))/(ℯ^(2*im*a)*(im*b*x)^m)]
    @test_int [x^(m - 3)*sin(a + b*x)^2, x, 5, -(x^(-2 + m)/(2*(2 - m))) - (b^2*ℯ^(2*im*a)*x^m*Gamma(-2 + m, -2*im*b*x))/(2^m*((-im)*b*x)^m) - (b^2*x^m*Gamma(-2 + m, 2*im*b*x))/(2^m*ℯ^(2*im*a)*(im*b*x)^m)]

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

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

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

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

    @test_int [x^1/csc(e + f*x)^(3/2) - x*sqrt(csc(e + f*x))/3, x, 4, 4/(9*f^2*csc(e + f*x)^(3/2)) - (2*x*cos(e + f*x))/(3*f*sqrt(csc(e + f*x)))]
    @test_int [x^2/csc(e + f*x)^(3/2) - (1/3)*x^2*sqrt(csc(e + f*x)), x, 7, (8*x)/(9*f^2*csc(e + f*x)^(3/2)) + (16*cos(e + f*x))/(27*f^3*sqrt(csc(e + f*x))) - (2*x^2*cos(e + f*x))/(3*f*sqrt(csc(e + f*x))) - (16*sqrt(csc(e + f*x))*Elliptic.F((1/2)*(e - pi/2 + f*x), 2)*sqrt(sin(e + f*x)))/(27*f^3)]

    @test_int [x/csc(e + f*x)^(5/2) - 3*x/(5*sqrt(csc(e + f*x))), x, 4, 4/(25*f^2*csc(e + f*x)^(5/2)) - (2*x*cos(e + f*x))/(5*f*csc(e + f*x)^(3/2))]

    @test_int [x/csc(e + f*x)^(7/2) - (5/21)*x*sqrt(csc(e + f*x)), x, 5, 4/(49*f^2*csc(e + f*x)^(7/2)) - (2*x*cos(e + f*x))/(7*f*csc(e + f*x)^(5/2)) + 20/(63*f^2*csc(e + f*x)^(3/2)) - (10*x*cos(e + f*x))/(21*f*sqrt(csc(e + f*x)))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(c+d*x)^m*(a+b*sin(e+f*x))^n*with*a^2-b^2=0=#

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

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

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

    @test_int [(a + a*sin(e + f*x))^2*(c + d*x)^3, x, 10, -((3*a^2*c*d^2*x)/(4*f^2)) - (3*a^2*d^3*x^2)/(8*f^2) + (3*a^2*(c + d*x)^4)/(8*d) + (12*a^2*d^2*(c + d*x)*cos(e + f*x))/f^3 - (2*a^2*(c + d*x)^3*cos(e + f*x))/f - (12*a^2*d^3*sin(e + f*x))/f^4 + (6*a^2*d*(c + d*x)^2*sin(e + f*x))/f^2 + (3*a^2*d^2*(c + d*x)*cos(e + f*x)*sin(e + f*x))/(4*f^3) - (a^2*(c + d*x)^3*cos(e + f*x)*sin(e + f*x))/(2*f) - (3*a^2*d^3*sin(e + f*x)^2)/(8*f^4) + (3*a^2*d*(c + d*x)^2*sin(e + f*x)^2)/(4*f^2)]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*x)^2, x, 9, -((a^2*d^2*x)/(4*f^2)) + (a^2*(c + d*x)^3)/(2*d) + (4*a^2*d^2*cos(e + f*x))/f^3 - (2*a^2*(c + d*x)^2*cos(e + f*x))/f + (4*a^2*d*(c + d*x)*sin(e + f*x))/f^2 + (a^2*d^2*cos(e + f*x)*sin(e + f*x))/(4*f^3) - (a^2*(c + d*x)^2*cos(e + f*x)*sin(e + f*x))/(2*f) + (a^2*d*(c + d*x)*sin(e + f*x)^2)/(2*f^2)]
    @test_int [(a + a*sin(e + f*x))^2*(c + d*x)^1, x, 6, (1/2)*a^2*c*x + (1/4)*a^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) - (2*a^2*(c + d*x)*cos(e + f*x))/f + (2*a^2*d*sin(e + f*x))/f^2 - (a^2*(c + d*x)*cos(e + f*x)*sin(e + f*x))/(2*f) + (a^2*d*sin(e + f*x)^2)/(4*f^2)]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*x)^1, x, 9, -((a^2*cos(2*e - (2*c*f)/d)*CosIntegral((2*c*f)/d + 2*f*x))/(2*d)) + (3*a^2*log(c + d*x))/(2*d) + (2*a^2*CosIntegral((c*f)/d + f*x)*sin(e - (c*f)/d))/d + (2*a^2*cos(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d + (a^2*sin(2*e - (2*c*f)/d)*SinIntegral((2*c*f)/d + 2*f*x))/(2*d)]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*x)^2, x, 9, (2*a^2*f*cos(e - (c*f)/d)*CosIntegral((c*f)/d + f*x))/d^2 + (a^2*f*CosIntegral((2*c*f)/d + 2*f*x)*sin(2*e - (2*c*f)/d))/d^2 - (4*a^2*sin(e/2 + pi/4 + (f*x)/2)^4)/(d*(c + d*x)) - (2*a^2*f*sin(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d^2 + (a^2*f*cos(2*e - (2*c*f)/d)*SinIntegral((2*c*f)/d + 2*f*x))/d^2]
    @test_int [(a + a*sin(e + f*x))^2/(c + d*x)^3, x, 15, (a^2*f^2*cos(2*e - (2*c*f)/d)*CosIntegral((2*c*f)/d + 2*f*x))/d^3 - (a^2*f^2*CosIntegral((c*f)/d + f*x)*sin(e - (c*f)/d))/d^3 - (4*a^2*f*cos(e/2 + pi/4 + (f*x)/2)*sin(e/2 + pi/4 + (f*x)/2)^3)/(d^2*(c + d*x)) - (2*a^2*sin(e/2 + pi/4 + (f*x)/2)^4)/(d*(c + d*x)^2) - (a^2*f^2*cos(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d^3 - (a^2*f^2*sin(2*e - (2*c*f)/d)*SinIntegral((2*c*f)/d + 2*f*x))/d^3]

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

    @test_int [1/(a + a*sin(e + f*x))*(c + d*x)^3, x, 7, -((im*(c + d*x)^3)/(a*f)) - ((c + d*x)^3*cot(e/2 + pi/4 + (f*x)/2))/(a*f) + (6*d*(c + d*x)^2*log(1 - im*ℯ^(im*(e + f*x))))/(a*f^2) - (12*im*d^2*(c + d*x)*PolyLog(2, im*ℯ^(im*(e + f*x))))/(a*f^3) + (12*d^3*PolyLog(3, im*ℯ^(im*(e + f*x))))/(a*f^4)]
    @test_int [1/(a + a*sin(e + f*x))*(c + d*x)^2, x, 6, -((im*(c + d*x)^2)/(a*f)) - ((c + d*x)^2*cot(e/2 + pi/4 + (f*x)/2))/(a*f) + (4*d*(c + d*x)*log(1 - im*ℯ^(im*(e + f*x))))/(a*f^2) - (4*im*d^2*PolyLog(2, im*ℯ^(im*(e + f*x))))/(a*f^3)]
    @test_int [1/(a + a*sin(e + f*x))*(c + d*x)^1, x, 3, -(((c + d*x)*cot(e/2 + pi/4 + (f*x)/2))/(a*f)) + (2*d*log(sin(e/2 + pi/4 + (f*x)/2)))/(a*f^2)]
    @test_int [1/(a + a*sin(e + f*x))/(c + d*x)^1, x, 0, Unintegrable(1/((c + d*x)*(a + a*sin(e + f*x))), x)]
    @test_int [1/(a + a*sin(e + f*x))/(c + d*x)^2, x, 0, Unintegrable(1/((c + d*x)^2*(a + a*sin(e + f*x))), x)]

    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*x)^3, x, 10, -((im*(c + d*x)^3)/(3*a^2*f)) - (2*d^2*(c + d*x)*cot(e/2 + pi/4 + (f*x)/2))/(a^2*f^3) - ((c + d*x)^3*cot(e/2 + pi/4 + (f*x)/2))/(3*a^2*f) - (d*(c + d*x)^2*csc(e/2 + pi/4 + (f*x)/2)^2)/(2*a^2*f^2) - ((c + d*x)^3*cot(e/2 + pi/4 + (f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)^2)/(6*a^2*f) + (2*d*(c + d*x)^2*log(1 - im*ℯ^(im*(e + f*x))))/(a^2*f^2) + (4*d^3*log(sin(e/2 + pi/4 + (f*x)/2)))/(a^2*f^4) - (4*im*d^2*(c + d*x)*PolyLog(2, im*ℯ^(im*(e + f*x))))/(a^2*f^3) + (4*d^3*PolyLog(3, im*ℯ^(im*(e + f*x))))/(a^2*f^4)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*x)^2, x, 9, -((im*(c + d*x)^2)/(3*a^2*f)) - (2*d^2*cot(e/2 + pi/4 + (f*x)/2))/(3*a^2*f^3) - ((c + d*x)^2*cot(e/2 + pi/4 + (f*x)/2))/(3*a^2*f) - (d*(c + d*x)*csc(e/2 + pi/4 + (f*x)/2)^2)/(3*a^2*f^2) - ((c + d*x)^2*cot(e/2 + pi/4 + (f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)^2)/(6*a^2*f) + (4*d*(c + d*x)*log(1 - im*ℯ^(im*(e + f*x))))/(3*a^2*f^2) - (4*im*d^2*PolyLog(2, im*ℯ^(im*(e + f*x))))/(3*a^2*f^3)]
    @test_int [1/(a + a*sin(e + f*x))^2*(c + d*x)^1, x, 4, -(((c + d*x)*cot(e/2 + pi/4 + (f*x)/2))/(3*a^2*f)) - (d*csc(e/2 + pi/4 + (f*x)/2)^2)/(6*a^2*f^2) - ((c + d*x)*cot(e/2 + pi/4 + (f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)^2)/(6*a^2*f) + (2*d*log(sin(e/2 + pi/4 + (f*x)/2)))/(3*a^2*f^2)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c + d*x)^1, x, 0, Unintegrable(1/((c + d*x)*(a + a*sin(e + f*x))^2), x)]
    @test_int [1/(a + a*sin(e + f*x))^2/(c + d*x)^2, x, 0, Unintegrable(1/((c + d*x)^2*(a + a*sin(e + f*x))^2), x)]

    @test_int [1/(a - a*sin(e + f*x))*(c + d*x)^3, x, 7, -((im*(c + d*x)^3)/(a*f)) + (6*d*(c + d*x)^2*log(1 + im*ℯ^(im*(e + f*x))))/(a*f^2) - (12*im*d^2*(c + d*x)*PolyLog(2, (-im)*ℯ^(im*(e + f*x))))/(a*f^3) + (12*d^3*PolyLog(3, (-im)*ℯ^(im*(e + f*x))))/(a*f^4) + ((c + d*x)^3*tan(e/2 + pi/4 + (f*x)/2))/(a*f)]
    @test_int [1/(a - a*sin(e + f*x))*(c + d*x)^2, x, 6, -((im*(c + d*x)^2)/(a*f)) + (4*d*(c + d*x)*log(1 + im*ℯ^(im*(e + f*x))))/(a*f^2) - (4*im*d^2*PolyLog(2, (-im)*ℯ^(im*(e + f*x))))/(a*f^3) + ((c + d*x)^2*tan(e/2 + pi/4 + (f*x)/2))/(a*f)]
    @test_int [1/(a - a*sin(e + f*x))*(c + d*x)^1, x, 3, (2*d*log(cos(e/2 + pi/4 + (f*x)/2)))/(a*f^2) + ((c + d*x)*tan(e/2 + pi/4 + (f*x)/2))/(a*f)]
    @test_int [1/(a - a*sin(e + f*x))/(c + d*x)^1, x, 0, Unintegrable(1/((c + d*x)*(a - a*sin(e + f*x))), x)]
    @test_int [1/(a - a*sin(e + f*x))/(c + d*x)^2, x, 0, Unintegrable(1/((c + d*x)^2*(a - a*sin(e + f*x))), x)]

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

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

    @test_int [x^3*sqrt(a + a*sin(c + d*x)), x, 5, -((96*sqrt(a + a*sin(c + d*x)))/d^4) + (12*x^2*sqrt(a + a*sin(c + d*x)))/d^2 + (48*x*cot(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x)))/d^3 - (2*x^3*cot(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [x^2*sqrt(a + a*sin(c + d*x)), x, 4, (8*x*sqrt(a + a*sin(c + d*x)))/d^2 + (16*cot(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x)))/d^3 - (2*x^2*cot(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [x^1*sqrt(a + a*sin(c + d*x)), x, 3, (4*sqrt(a + a*sin(c + d*x)))/d^2 - (2*x*cot(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x)))/d]
    @test_int [sqrt(a + a*sin(c + d*x))/x^1, x, 4, CosIntegral((d*x)/2)*csc(c/2 + pi/4 + (d*x)/2)*sin((1/4)*(2*c + pi))*sqrt(a + a*sin(c + d*x)) + cos((1/4)*(2*c + pi))*csc(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x))*SinIntegral((d*x)/2)]
    @test_int [sqrt(a + a*sin(c + d*x))/x^2, x, 5, -(sqrt(a + a*sin(c + d*x))/x) - (1/2)*d*CosIntegral((d*x)/2)*csc(c/2 + pi/4 + (d*x)/2)*sin((1/4)*(2*c - pi))*sqrt(a + a*sin(c + d*x)) - (1/2)*d*csc(c/2 + pi/4 + (d*x)/2)*sin((1/4)*(2*c + pi))*sqrt(a + a*sin(c + d*x))*SinIntegral((d*x)/2)]
    @test_int [sqrt(a + a*sin(c + d*x))/x^3, x, 6, -(sqrt(a + a*sin(c + d*x))/(2*x^2)) - (d*cot(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x)))/(4*x) - (1/8)*d^2*CosIntegral((d*x)/2)*csc(c/2 + pi/4 + (d*x)/2)*sin((1/4)*(2*c + pi))*sqrt(a + a*sin(c + d*x)) - (1/8)*d^2*cos((1/4)*(2*c + pi))*csc(c/2 + pi/4 + (d*x)/2)*sqrt(a + a*sin(c + d*x))*SinIntegral((d*x)/2)]

    @test_int [x^3*(a + a*sin(e + f*x))^(3/2), x, 9, -((1280*a*sqrt(a + a*sin(e + f*x)))/(9*f^4)) + (16*a*x^2*sqrt(a + a*sin(e + f*x)))/f^2 + (640*a*x*cot(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(9*f^3) - (8*a*x^3*cot(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(3*f) + (32*a*x*cos(e/2 + pi/4 + (f*x)/2)*sin(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(9*f^3) - (4*a*x^3*cos(e/2 + pi/4 + (f*x)/2)*sin(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(3*f) - (64*a*sin(e/2 + pi/4 + (f*x)/2)^2*sqrt(a + a*sin(e + f*x)))/(27*f^4) + (8*a*x^2*sin(e/2 + pi/4 + (f*x)/2)^2*sqrt(a + a*sin(e + f*x)))/(3*f^2)]
    @test_int [x^2*(a + a*sin(e + f*x))^(3/2), x, 7, (32*a*x*sqrt(a + a*sin(e + f*x)))/(3*f^2) + (224*a*cot(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(9*f^3) - (8*a*x^2*cot(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(3*f) - (32*a*cos(e/2 + pi/4 + (f*x)/2)^2*cot(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(27*f^3) - (4*a*x^2*cos(e/2 + pi/4 + (f*x)/2)*sin(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(3*f) + (16*a*x*sin(e/2 + pi/4 + (f*x)/2)^2*sqrt(a + a*sin(e + f*x)))/(9*f^2)]
    @test_int [x^1*(a + a*sin(e + f*x))^(3/2), x, 4, (16*a*sqrt(a + a*sin(e + f*x)))/(3*f^2) - (8*a*x*cot(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(3*f) - (4*a*x*cos(e/2 + pi/4 + (f*x)/2)*sin(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(3*f) + (8*a*sin(e/2 + pi/4 + (f*x)/2)^2*sqrt(a + a*sin(e + f*x)))/(9*f^2)]
    @test_int [(a + a*sin(e + f*x))^(3/2)/x^1, x, 9, (1/2)*a*cos((3/4)*(2*e - pi))*CosIntegral((3*f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)) + (3/2)*a*CosIntegral((f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)*sin((1/4)*(2*e + pi))*sqrt(a + a*sin(e + f*x)) + (3/2)*a*cos((1/4)*(2*e + pi))*csc(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x))*SinIntegral((f*x)/2) - (1/2)*a*csc(e/2 + pi/4 + (f*x)/2)*sin((3/4)*(2*e - pi))*sqrt(a + a*sin(e + f*x))*SinIntegral((3*f*x)/2)]
    @test_int [(a + a*sin(e + f*x))^(3/2)/x^2, x, 9, (-(3/4))*a*f*CosIntegral((f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)*sin((1/4)*(2*e - pi))*sqrt(a + a*sin(e + f*x)) + (3/4)*a*f*CosIntegral((3*f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)*sin((1/4)*(6*e + pi))*sqrt(a + a*sin(e + f*x)) - (2*a*sin(e/2 + pi/4 + (f*x)/2)^2*sqrt(a + a*sin(e + f*x)))/x - (3/4)*a*f*csc(e/2 + pi/4 + (f*x)/2)*sin((1/4)*(2*e + pi))*sqrt(a + a*sin(e + f*x))*SinIntegral((f*x)/2) + (3/4)*a*f*cos((1/4)*(6*e + pi))*csc(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x))*SinIntegral((3*f*x)/2)]
    @test_int [(a + a*sin(e + f*x))^(3/2)/x^3, x, 13, (-(9/16))*a*f^2*cos((3/4)*(2*e - pi))*CosIntegral((3*f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)) - (3/16)*a*f^2*CosIntegral((f*x)/2)*csc(e/2 + pi/4 + (f*x)/2)*sin((1/4)*(2*e + pi))*sqrt(a + a*sin(e + f*x)) - (3*a*f*cos(e/2 + pi/4 + (f*x)/2)*sin(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x)))/(2*x) - (a*sin(e/2 + pi/4 + (f*x)/2)^2*sqrt(a + a*sin(e + f*x)))/x^2 - (3/16)*a*f^2*cos((1/4)*(2*e + pi))*csc(e/2 + pi/4 + (f*x)/2)*sqrt(a + a*sin(e + f*x))*SinIntegral((f*x)/2) + (9/16)*a*f^2*csc(e/2 + pi/4 + (f*x)/2)*sin((3/4)*(2*e - pi))*sqrt(a + a*sin(e + f*x))*SinIntegral((3*f*x)/2)]

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

    @test_int [x^3/sqrt(a + a*sin(c + d*x)), x, 10, -((4*x^3*atanh(ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d*sqrt(a + a*sin(c + d*x)))) + (12*im*x^2*PolyLog(2, -ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^2*sqrt(a + a*sin(c + d*x))) - (12*im*x^2*PolyLog(2, ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^2*sqrt(a + a*sin(c + d*x))) - (48*x*PolyLog(3, -ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^3*sqrt(a + a*sin(c + d*x))) + (48*x*PolyLog(3, ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^3*sqrt(a + a*sin(c + d*x))) - (96*im*PolyLog(4, -ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^4*sqrt(a + a*sin(c + d*x))) + (96*im*PolyLog(4, ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^4*sqrt(a + a*sin(c + d*x)))]
    @test_int [x^2/sqrt(a + a*sin(c + d*x)), x, 8, -((4*x^2*atanh(ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d*sqrt(a + a*sin(c + d*x)))) + (8*im*x*PolyLog(2, -ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^2*sqrt(a + a*sin(c + d*x))) - (8*im*x*PolyLog(2, ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^2*sqrt(a + a*sin(c + d*x))) - (16*PolyLog(3, -ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^3*sqrt(a + a*sin(c + d*x))) + (16*PolyLog(3, ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^3*sqrt(a + a*sin(c + d*x)))]
    @test_int [x^1/sqrt(a + a*sin(c + d*x)), x, 6, -((4*x*atanh(ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d*sqrt(a + a*sin(c + d*x)))) + (4*im*PolyLog(2, -ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^2*sqrt(a + a*sin(c + d*x))) - (4*im*PolyLog(2, ℯ^((1/4)*im*(2*c + pi + 2*d*x)))*sin(c/2 + pi/4 + (d*x)/2))/(d^2*sqrt(a + a*sin(c + d*x)))]
    @test_int [1/(x^1*sqrt(a + a*sin(c + d*x))), x, 0, Unintegrable(1/(x*sqrt(a + a*sin(c + d*x))), x)]
    @test_int [1/(x^2*sqrt(a + a*sin(c + d*x))), x, 0, Unintegrable(1/(x^2*sqrt(a + a*sin(c + d*x))), x)]

    @test_int [x^3/(a + a*sin(e + f*x))^(3/2), x, 16, -((3*x^2)/(a*f^2*sqrt(a + a*sin(e + f*x)))) - (x^3*cot(e/2 + pi/4 + (f*x)/2))/(2*a*f*sqrt(a + a*sin(e + f*x))) - (24*x*atanh(ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^3*sqrt(a + a*sin(e + f*x))) - (x^3*atanh(ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f*sqrt(a + a*sin(e + f*x))) + (24*im*PolyLog(2, -ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^4*sqrt(a + a*sin(e + f*x))) + (3*im*x^2*PolyLog(2, -ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^2*sqrt(a + a*sin(e + f*x))) - (24*im*PolyLog(2, ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^4*sqrt(a + a*sin(e + f*x))) - (3*im*x^2*PolyLog(2, ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^2*sqrt(a + a*sin(e + f*x))) - (12*x*PolyLog(3, -ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^3*sqrt(a + a*sin(e + f*x))) + (12*x*PolyLog(3, ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^3*sqrt(a + a*sin(e + f*x))) - (24*im*PolyLog(4, -ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^4*sqrt(a + a*sin(e + f*x))) + (24*im*PolyLog(4, ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^4*sqrt(a + a*sin(e + f*x)))]
    @test_int [x^2/(a + a*sin(e + f*x))^(3/2), x, 10, -((2*x)/(a*f^2*sqrt(a + a*sin(e + f*x)))) - (x^2*cot(e/2 + pi/4 + (f*x)/2))/(2*a*f*sqrt(a + a*sin(e + f*x))) - (x^2*atanh(ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f*sqrt(a + a*sin(e + f*x))) - (4*atanh(cos(e/2 + pi/4 + (f*x)/2))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^3*sqrt(a + a*sin(e + f*x))) + (2*im*x*PolyLog(2, -ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^2*sqrt(a + a*sin(e + f*x))) - (2*im*x*PolyLog(2, ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^2*sqrt(a + a*sin(e + f*x))) - (4*PolyLog(3, -ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^3*sqrt(a + a*sin(e + f*x))) + (4*PolyLog(3, ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^3*sqrt(a + a*sin(e + f*x)))]
    @test_int [x^1/(a + a*sin(e + f*x))^(3/2), x, 7, -(1/(a*f^2*sqrt(a + a*sin(e + f*x)))) - (x*cot(e/2 + pi/4 + (f*x)/2))/(2*a*f*sqrt(a + a*sin(e + f*x))) - (x*atanh(ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f*sqrt(a + a*sin(e + f*x))) + (im*PolyLog(2, -ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^2*sqrt(a + a*sin(e + f*x))) - (im*PolyLog(2, ℯ^((1/4)*im*(2*e + pi + 2*f*x)))*sin(e/2 + pi/4 + (f*x)/2))/(a*f^2*sqrt(a + a*sin(e + f*x)))]
    @test_int [1/(x^1*(a + a*sin(e + f*x))^(3/2)), x, 0, Unintegrable(1/(x*(a + a*sin(e + f*x))^(3/2)), x)]
    @test_int [1/(x^2*(a + a*sin(e + f*x))^(3/2)), x, 0, Unintegrable(1/(x^2*(a + a*sin(e + f*x))^(3/2)), x)]

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

    #= Used*to*hang*Rubi =#
    @test_int [(a + a*sin(c + d*x))^(1/3)/x, x, 0, Unintegrable((a + a*sin(c + d*x))^(1/3)/x, x)]

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

    @test_int [(c + d*x)^m*(a + a*sin(e + f*x))^n, x, 0, Unintegrable((c + d*x)^m*(a + a*sin(e + f*x))^n, x)]

    @test_int [(c + d*x)^m*(a + a*sin(e + f*x))^3, x, 12, (5*a^3*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (15*a^3*ℯ^(im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*(8*f)) - (15*a^3*(c + d*x)^m*Gamma(1 + m, (im*f*(c + d*x))/d))/(ℯ^(im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*(8*f)) + (3*im*2^(-3 - m)*a^3*ℯ^(2*im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((2*im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*f) - (3*im*2^(-3 - m)*a^3*(c + d*x)^m*Gamma(1 + m, (2*im*f*(c + d*x))/d))/(ℯ^(2*im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*f) + (3^(-1 - m)*a^3*ℯ^(3*im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((3*im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*(8*f)) + (3^(-1 - m)*a^3*(c + d*x)^m*Gamma(1 + m, (3*im*f*(c + d*x))/d))/(ℯ^(3*im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*(8*f))]
    @test_int [(c + d*x)^m*(a + a*sin(e + f*x))^2, x, 9, (3*a^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (a^2*ℯ^(im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*f) - (a^2*(c + d*x)^m*Gamma(1 + m, (im*f*(c + d*x))/d))/(ℯ^(im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*f) + (im*2^(-3 - m)*a^2*ℯ^(2*im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((2*im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*f) - (im*2^(-3 - m)*a^2*(c + d*x)^m*Gamma(1 + m, (2*im*f*(c + d*x))/d))/(ℯ^(2*im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*f)]
    @test_int [(c + d*x)^m*(a + a*sin(e + f*x))^1, x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) - (a*ℯ^(im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*(2*f)) - (a*(c + d*x)^m*Gamma(1 + m, (im*f*(c + d*x))/d))/(ℯ^(im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*(2*f))]
    @test_int [(c + d*x)^m/(a + a*sin(e + f*x))^1, x, 0, Unintegrable((c + d*x)^m/(a + a*sin(e + f*x)), x)]
    @test_int [(c + d*x)^m/(a + a*sin(e + f*x))^2, x, 0, Unintegrable((c + d*x)^m/(a + a*sin(e + f*x))^2, x)]

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

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

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

    @test_int [(a + b*sin(e + f*x))*(c + d*x)^3, x, 6, (a*(c + d*x)^4)/(4*d) + (6*b*d^2*(c + d*x)*cos(e + f*x))/f^3 - (b*(c + d*x)^3*cos(e + f*x))/f - (6*b*d^3*sin(e + f*x))/f^4 + (3*b*d*(c + d*x)^2*sin(e + f*x))/f^2]
    @test_int [(a + b*sin(e + f*x))*(c + d*x)^2, x, 5, (a*(c + d*x)^3)/(3*d) + (2*b*d^2*cos(e + f*x))/f^3 - (b*(c + d*x)^2*cos(e + f*x))/f + (2*b*d*(c + d*x)*sin(e + f*x))/f^2]
    @test_int [(a + b*sin(e + f*x))*(c + d*x)^1, x, 4, (a*(c + d*x)^2)/(2*d) - (b*(c + d*x)*cos(e + f*x))/f + (b*d*sin(e + f*x))/f^2]
    @test_int [(a + b*sin(e + f*x))/(c + d*x)^1, x, 5, (a*log(c + d*x))/d + (b*CosIntegral((c*f)/d + f*x)*sin(e - (c*f)/d))/d + (b*cos(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d]
    @test_int [(a + b*sin(e + f*x))/(c + d*x)^2, x, 6, -(a/(d*(c + d*x))) + (b*f*cos(e - (c*f)/d)*CosIntegral((c*f)/d + f*x))/d^2 - (b*sin(e + f*x))/(d*(c + d*x)) - (b*f*sin(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d^2]
    @test_int [(a + b*sin(e + f*x))/(c + d*x)^3, x, 7, -(a/(2*d*(c + d*x)^2)) - (b*f*cos(e + f*x))/(2*d^2*(c + d*x)) - (b*f^2*CosIntegral((c*f)/d + f*x)*sin(e - (c*f)/d))/(2*d^3) - (b*sin(e + f*x))/(2*d*(c + d*x)^2) - (b*f^2*cos(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/(2*d^3)]

    @test_int [(a + b*sin(e + f*x))^2*(c + d*x)^3, x, 10, -((3*b^2*c*d^2*x)/(4*f^2)) - (3*b^2*d^3*x^2)/(8*f^2) + (a^2*(c + d*x)^4)/(4*d) + (b^2*(c + d*x)^4)/(8*d) + (12*a*b*d^2*(c + d*x)*cos(e + f*x))/f^3 - (2*a*b*(c + d*x)^3*cos(e + f*x))/f - (12*a*b*d^3*sin(e + f*x))/f^4 + (6*a*b*d*(c + d*x)^2*sin(e + f*x))/f^2 + (3*b^2*d^2*(c + d*x)*cos(e + f*x)*sin(e + f*x))/(4*f^3) - (b^2*(c + d*x)^3*cos(e + f*x)*sin(e + f*x))/(2*f) - (3*b^2*d^3*sin(e + f*x)^2)/(8*f^4) + (3*b^2*d*(c + d*x)^2*sin(e + f*x)^2)/(4*f^2)]
    @test_int [(a + b*sin(e + f*x))^2*(c + d*x)^2, x, 9, -((b^2*d^2*x)/(4*f^2)) + (a^2*(c + d*x)^3)/(3*d) + (b^2*(c + d*x)^3)/(6*d) + (4*a*b*d^2*cos(e + f*x))/f^3 - (2*a*b*(c + d*x)^2*cos(e + f*x))/f + (4*a*b*d*(c + d*x)*sin(e + f*x))/f^2 + (b^2*d^2*cos(e + f*x)*sin(e + f*x))/(4*f^3) - (b^2*(c + d*x)^2*cos(e + f*x)*sin(e + f*x))/(2*f) + (b^2*d*(c + d*x)*sin(e + f*x)^2)/(2*f^2)]
    @test_int [(a + b*sin(e + f*x))^2*(c + d*x)^1, x, 6, (1/2)*b^2*c*x + (1/4)*b^2*d*x^2 + (a^2*(c + d*x)^2)/(2*d) - (2*a*b*(c + d*x)*cos(e + f*x))/f + (2*a*b*d*sin(e + f*x))/f^2 - (b^2*(c + d*x)*cos(e + f*x)*sin(e + f*x))/(2*f) + (b^2*d*sin(e + f*x)^2)/(4*f^2)]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*x)^1, x, 10, -((b^2*cos(2*e - (2*c*f)/d)*CosIntegral((2*c*f)/d + 2*f*x))/(2*d)) + (a^2*log(c + d*x))/d + (b^2*log(c + d*x))/(2*d) + (2*a*b*CosIntegral((c*f)/d + f*x)*sin(e - (c*f)/d))/d + (2*a*b*cos(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d + (b^2*sin(2*e - (2*c*f)/d)*SinIntegral((2*c*f)/d + 2*f*x))/(2*d)]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*x)^2, x, 11, -(a^2/(d*(c + d*x))) + (2*a*b*f*cos(e - (c*f)/d)*CosIntegral((c*f)/d + f*x))/d^2 + (b^2*f*CosIntegral((2*c*f)/d + 2*f*x)*sin(2*e - (2*c*f)/d))/d^2 - (2*a*b*sin(e + f*x))/(d*(c + d*x)) - (b^2*sin(e + f*x)^2)/(d*(c + d*x)) - (2*a*b*f*sin(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d^2 + (b^2*f*cos(2*e - (2*c*f)/d)*SinIntegral((2*c*f)/d + 2*f*x))/d^2]
    @test_int [(a + b*sin(e + f*x))^2/(c + d*x)^3, x, 14, -(a^2/(2*d*(c + d*x)^2)) - (a*b*f*cos(e + f*x))/(d^2*(c + d*x)) + (b^2*f^2*cos(2*e - (2*c*f)/d)*CosIntegral((2*c*f)/d + 2*f*x))/d^3 - (a*b*f^2*CosIntegral((c*f)/d + f*x)*sin(e - (c*f)/d))/d^3 - (a*b*sin(e + f*x))/(d*(c + d*x)^2) - (b^2*f*cos(e + f*x)*sin(e + f*x))/(d^2*(c + d*x)) - (b^2*sin(e + f*x)^2)/(2*d*(c + d*x)^2) - (a*b*f^2*cos(e - (c*f)/d)*SinIntegral((c*f)/d + f*x))/d^3 - (b^2*f^2*sin(2*e - (2*c*f)/d)*SinIntegral((2*c*f)/d + 2*f*x))/d^3]

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

    @test_int [(c + d*x)^3/(a + b*sin(e + f*x)), x, 12, -((im*(c + d*x)^3*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f)) + (im*(c + d*x)^3*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f) - (3*d*(c + d*x)^2*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^2) + (3*d*(c + d*x)^2*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^2) - (6*im*d^2*(c + d*x)*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^3) + (6*im*d^2*(c + d*x)*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^3) + (6*d^3*PolyLog(4, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^4) - (6*d^3*PolyLog(4, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^4)]
    @test_int [(c + d*x)^2/(a + b*sin(e + f*x)), x, 10, -((im*(c + d*x)^2*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f)) + (im*(c + d*x)^2*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f) - (2*d*(c + d*x)*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^2) + (2*d*(c + d*x)*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^2) - (2*im*d^2*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^3) + (2*im*d^2*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^3)]
    @test_int [(c + d*x)^1/(a + b*sin(e + f*x)), x, 8, -((im*(c + d*x)*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f)) + (im*(c + d*x)*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f) - (d*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^2) + (d*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/(sqrt(a^2 - b^2)*f^2)]
    @test_int [1/((c + d*x)^1*(a + b*sin(e + f*x))), x, 0, Unintegrable(1/((c + d*x)*(a + b*sin(e + f*x))), x)]
    @test_int [1/((c + d*x)^2*(a + b*sin(e + f*x))), x, 0, Unintegrable(1/((c + d*x)^2*(a + b*sin(e + f*x))), x)]

    @test_int [(c + d*x)^3/(a + b*sin(e + f*x))^2, x, 22, (im*(c + d*x)^3)/((a^2 - b^2)*f) - (3*d*(c + d*x)^2*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*f^2) - (im*a*(c + d*x)^3*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f) - (3*d*(c + d*x)^2*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*f^2) + (im*a*(c + d*x)^3*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f) + (6*im*d^2*(c + d*x)*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*f^3) - (3*a*d*(c + d*x)^2*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^2) + (6*im*d^2*(c + d*x)*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*f^3) + (3*a*d*(c + d*x)^2*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^2) - (6*d^3*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*f^4) - (6*im*a*d^2*(c + d*x)*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^3) - (6*d^3*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*f^4) + (6*im*a*d^2*(c + d*x)*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^3) + (6*a*d^3*PolyLog(4, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^4) - (6*a*d^3*PolyLog(4, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^4) + (b*(c + d*x)^3*cos(e + f*x))/((a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [(c + d*x)^2/(a + b*sin(e + f*x))^2, x, 18, (im*(c + d*x)^2)/((a^2 - b^2)*f) - (2*d*(c + d*x)*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*f^2) - (im*a*(c + d*x)^2*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f) - (2*d*(c + d*x)*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*f^2) + (im*a*(c + d*x)^2*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f) + (2*im*d^2*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*f^3) - (2*a*d*(c + d*x)*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^2) + (2*im*d^2*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*f^3) + (2*a*d*(c + d*x)*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^2) - (2*im*a*d^2*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^3) + (2*im*a*d^2*PolyLog(3, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^3) + (b*(c + d*x)^2*cos(e + f*x))/((a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [(c + d*x)^1/(a + b*sin(e + f*x))^2, x, 11, -((im*a*(c + d*x)*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f)) + (im*a*(c + d*x)*log(1 - (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f) - (d*log(a + b*sin(e + f*x)))/((a^2 - b^2)*f^2) - (a*d*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^2) + (a*d*PolyLog(2, (im*b*ℯ^(im*(e + f*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*f^2) + (b*(c + d*x)*cos(e + f*x))/((a^2 - b^2)*f*(a + b*sin(e + f*x)))]
    @test_int [1/((c + d*x)^1*(a + b*sin(e + f*x))^2), x, 0, Unintegrable(1/((c + d*x)*(a + b*sin(e + f*x))^2), x)]
    @test_int [1/((c + d*x)^2*(a + b*sin(e + f*x))^2), x, 0, Unintegrable(1/((c + d*x)^2*(a + b*sin(e + f*x))^2), x)]

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

    @test_int [(c + d*x)^m*(a + b*sin(e + f*x))^n, x, 0, Unintegrable((c + d*x)^m*(a + b*sin(e + f*x))^n, x)]

    @test_int [(c + d*x)^m*(a + b*sin(e + f*x))^3, x, 18, (a^3*(c + d*x)^(1 + m))/(d*(1 + m)) + (3*a*b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (3*a^2*b*ℯ^(im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*(2*f)) - (3*b^3*ℯ^(im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*(8*f)) - (3*a^2*b*(c + d*x)^m*Gamma(1 + m, (im*f*(c + d*x))/d))/(ℯ^(im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*(2*f)) - (3*b^3*(c + d*x)^m*Gamma(1 + m, (im*f*(c + d*x))/d))/(ℯ^(im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*(8*f)) + (3*im*2^(-3 - m)*a*b^2*ℯ^(2*im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((2*im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*f) - (3*im*2^(-3 - m)*a*b^2*(c + d*x)^m*Gamma(1 + m, (2*im*f*(c + d*x))/d))/(ℯ^(2*im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*f) + (3^(-1 - m)*b^3*ℯ^(3*im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((3*im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*(8*f)) + (3^(-1 - m)*b^3*(c + d*x)^m*Gamma(1 + m, (3*im*f*(c + d*x))/d))/(ℯ^(3*im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*(8*f))]
    @test_int [(c + d*x)^m*(a + b*sin(e + f*x))^2, x, 10, (a^2*(c + d*x)^(1 + m))/(d*(1 + m)) + (b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) - (a*b*ℯ^(im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*f) - (a*b*(c + d*x)^m*Gamma(1 + m, (im*f*(c + d*x))/d))/(ℯ^(im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*f) + (im*2^(-3 - m)*b^2*ℯ^(2*im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((2*im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*f) - (im*2^(-3 - m)*b^2*(c + d*x)^m*Gamma(1 + m, (2*im*f*(c + d*x))/d))/(ℯ^(2*im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*f)]
    @test_int [(c + d*x)^m*(a + b*sin(e + f*x))^1, x, 5, (a*(c + d*x)^(1 + m))/(d*(1 + m)) - (b*ℯ^(im*(e - (c*f)/d))*(c + d*x)^m*Gamma(1 + m, -((im*f*(c + d*x))/d)))/((-((im*f*(c + d*x))/d))^m*(2*f)) - (b*(c + d*x)^m*Gamma(1 + m, (im*f*(c + d*x))/d))/(ℯ^(im*(e - (c*f)/d))*((im*f*(c + d*x))/d)^m*(2*f))]
    @test_int [(c + d*x)^m/(a + b*sin(e + f*x))^1, x, 0, Unintegrable((c + d*x)^m/(a + b*sin(e + f*x)), x)]
    @test_int [(c + d*x)^m/(a + b*sin(e + f*x))^2, x, 0, Unintegrable((c + d*x)^m/(a + b*sin(e + f*x))^2, x)]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e+f*x)^m*sin(c+d*x)^n*(a+b*sin(c+d*x))^p*with*a^2-b^2=0=#

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

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

    @test_int [((e + f*x)^3*sin(c + d*x))/(a + a*sin(c + d*x)), x, 9, (im*(e + f*x)^3)/(a*d) + (e + f*x)^4/(4*a*f) + ((e + f*x)^3*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (6*f*(e + f*x)^2*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + ((12*im)*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - (12*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^4)]
    @test_int [((e + f*x)^2*sin(c + d*x))/(a + a*sin(c + d*x)), x, 8, (im*(e + f*x)^2)/(a*d) + (e + f*x)^3/(3*a*f) + ((e + f*x)^2*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (4*f*(e + f*x)*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + ((4*im)*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3)]
    @test_int [((e + f*x)*sin(c + d*x))/(a + a*sin(c + d*x)), x, 5, (e*x)/a + (f*x^2)/(2*a) + ((e + f*x)*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (2*f*log(sin(c/2 + pi/4 + (d*x)/2)))/(a*d^2)]
    @test_int [sin(c + d*x)/(a + a*sin(c + d*x)), x, 2, x/a + cos(c + d*x)/(d*(a + a*sin(c + d*x)))]
    @test_int [sin(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))), x, 0, Unintegrable(sin(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [sin(c + d*x)/((e + f*x)^2*(a + a*sin(c + d*x))), x, 0, Unintegrable(sin(c + d*x)/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

    @test_int [((e + f*x)^3*sin(c + d*x)^2)/(a + a*sin(c + d*x)), x, 14, ((-im)*(e + f*x)^3)/(a*d) - (e + f*x)^4/(4*a*f) + (6*f^2*(e + f*x)*cos(c + d*x))/(a*d^3) - ((e + f*x)^3*cos(c + d*x))/(a*d) - ((e + f*x)^3*cot(c/2 + pi/4 + (d*x)/2))/(a*d) + (6*f*(e + f*x)^2*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) - ((12*im)*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) + (12*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^4) - (6*f^3*sin(c + d*x))/(a*d^4) + (3*f*(e + f*x)^2*sin(c + d*x))/(a*d^2)]
    @test_int [((e + f*x)^2*sin(c + d*x)^2)/(a + a*sin(c + d*x)), x, 12, ((-im)*(e + f*x)^2)/(a*d) - (e + f*x)^3/(3*a*f) + (2*f^2*cos(c + d*x))/(a*d^3) - ((e + f*x)^2*cos(c + d*x))/(a*d) - ((e + f*x)^2*cot(c/2 + pi/4 + (d*x)/2))/(a*d) + (4*f*(e + f*x)*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) - ((4*im)*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) + (2*f*(e + f*x)*sin(c + d*x))/(a*d^2)]
    @test_int [((e + f*x)*sin(c + d*x)^2)/(a + a*sin(c + d*x)), x, 8, -((e*x)/a) - (f*x^2)/(2*a) - ((e + f*x)*cos(c + d*x))/(a*d) - ((e + f*x)*cot(c/2 + pi/4 + (d*x)/2))/(a*d) + (2*f*log(sin(c/2 + pi/4 + (d*x)/2)))/(a*d^2) + (f*sin(c + d*x))/(a*d^2)]
    @test_int [sin(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, -(x/a) - cos(c + d*x)/(a*d) - cos(c + d*x)/(a*d*(1 + sin(c + d*x)))]
    @test_int [sin(c + d*x)^2/((e + f*x)*(a + a*sin(c + d*x))), x, 0, Unintegrable(sin(c + d*x)^2/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [sin(c + d*x)^2/((e + f*x)^2*(a + a*sin(c + d*x))), x, 0, Unintegrable(sin(c + d*x)^2/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

    @test_int [((e + f*x)^3*sin(c + d*x)^3)/(a + a*sin(c + d*x)), x, 19, (-3*e*f^2*x)/(4*a*d^2) - (3*f^3*x^2)/(8*a*d^2) + (im*(e + f*x)^3)/(a*d) + (3*(e + f*x)^4)/(8*a*f) - (6*f^2*(e + f*x)*cos(c + d*x))/(a*d^3) + ((e + f*x)^3*cos(c + d*x))/(a*d) + ((e + f*x)^3*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (6*f*(e + f*x)^2*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + ((12*im)*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - (12*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^4) + (6*f^3*sin(c + d*x))/(a*d^4) - (3*f*(e + f*x)^2*sin(c + d*x))/(a*d^2) + (3*f^2*(e + f*x)*cos(c + d*x)*sin(c + d*x))/(4*a*d^3) - ((e + f*x)^3*cos(c + d*x)*sin(c + d*x))/(2*a*d) - (3*f^3*sin(c + d*x)^2)/(8*a*d^4) + (3*f*(e + f*x)^2*sin(c + d*x)^2)/(4*a*d^2)]
    @test_int [((e + f*x)^2*sin(c + d*x)^3)/(a + a*sin(c + d*x)), x, 17, -(f^2*x)/(4*a*d^2) + (im*(e + f*x)^2)/(a*d) + (e + f*x)^3/(2*a*f) - (2*f^2*cos(c + d*x))/(a*d^3) + ((e + f*x)^2*cos(c + d*x))/(a*d) + ((e + f*x)^2*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (4*f*(e + f*x)*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + ((4*im)*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - (2*f*(e + f*x)*sin(c + d*x))/(a*d^2) + (f^2*cos(c + d*x)*sin(c + d*x))/(4*a*d^3) - ((e + f*x)^2*cos(c + d*x)*sin(c + d*x))/(2*a*d) + (f*(e + f*x)*sin(c + d*x)^2)/(2*a*d^2)]
    @test_int [((e + f*x)*sin(c + d*x)^3)/(a + a*sin(c + d*x)), x, 11, (3*e*x)/(2*a) + (3*f*x^2)/(4*a) + ((e + f*x)*cos(c + d*x))/(a*d) + ((e + f*x)*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (2*f*log(sin(c/2 + pi/4 + (d*x)/2)))/(a*d^2) - (f*sin(c + d*x))/(a*d^2) - ((e + f*x)*cos(c + d*x)*sin(c + d*x))/(2*a*d) + (f*sin(c + d*x)^2)/(4*a*d^2)]
    @test_int [sin(c + d*x)^3/(a + a*sin(c + d*x)), x, 2, (3*x)/(2*a) + (2*cos(c + d*x))/(a*d) - (3*cos(c + d*x)*sin(c + d*x))/(2*a*d) + (cos(c + d*x)*sin(c + d*x)^2)/(d*(a + a*sin(c + d*x)))]
    @test_int [sin(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))), x, 0, Unintegrable(sin(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [sin(c + d*x)^3/((e + f*x)^2*(a + a*sin(c + d*x))), x, 0, Unintegrable(sin(c + d*x)^3/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

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

    @test_int [((e + f*x)^3*csc(c + d*x))/(a + a*sin(c + d*x)), x, 17, (im*(e + f*x)^3)/(a*d) - (2*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a*d) + ((e + f*x)^3*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (6*f*(e + f*x)^2*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + ((3*im)*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) + ((12*im)*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - ((3*im)*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) - (6*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) - (12*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^4) + (6*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) - ((6*im)*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a*d^4) + ((6*im)*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a*d^4)]
    @test_int [((e + f*x)^2*csc(c + d*x))/(a + a*sin(c + d*x)), x, 14, (im*(e + f*x)^2)/(a*d) - (2*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d) + ((e + f*x)^2*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (4*f*(e + f*x)*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + ((2*im)*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) + ((4*im)*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - ((2*im)*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) - (2*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (2*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3)]
    @test_int [((e + f*x)*csc(c + d*x))/(a + a*sin(c + d*x)), x, 9, (-2*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d) + ((e + f*x)*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - (2*f*log(sin(c/2 + pi/4 + (d*x)/2)))/(a*d^2) + (im*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - (im*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2)]
    @test_int [csc(c + d*x)/(a + a*sin(c + d*x)), x, 3, -(atanh(cos(c + d*x))/(a*d)) + cos(c + d*x)/(d*(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))), x, 0, Unintegrable(csc(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [csc(c + d*x)/((e + f*x)^2*(a + a*sin(c + d*x))), x, 0, Unintegrable(csc(c + d*x)/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

    @test_int [((e + f*x)^3*csc(c + d*x)^2)/(a + a*sin(c + d*x)), x, 24, ((-2*im)*(e + f*x)^3)/(a*d) + (2*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a*d) - ((e + f*x)^3*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - ((e + f*x)^3*cot(c + d*x))/(a*d) + (6*f*(e + f*x)^2*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + (3*f*(e + f*x)^2*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) - ((3*im)*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((12*im)*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) + ((3*im)*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) - ((3*im)*f^2*(e + f*x)*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) + (6*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (12*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^4) - (6*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) + (3*f^3*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a*d^4) + ((6*im)*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a*d^4) - ((6*im)*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a*d^4)]
    @test_int [((e + f*x)^2*csc(c + d*x)^2)/(a + a*sin(c + d*x)), x, 20, ((-2*im)*(e + f*x)^2)/(a*d) + (2*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d) - ((e + f*x)^2*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - ((e + f*x)^2*cot(c + d*x))/(a*d) + (4*f*(e + f*x)*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) + (2*f*(e + f*x)*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) - ((2*im)*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((4*im)*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) + ((2*im)*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) - (im*f^2*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) + (2*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) - (2*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3)]
    @test_int [((e + f*x)*csc(c + d*x)^2)/(a + a*sin(c + d*x)), x, 12, (2*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d) - ((e + f*x)*cot(c/2 + pi/4 + (d*x)/2))/(a*d) - ((e + f*x)*cot(c + d*x))/(a*d) + (2*f*log(sin(c/2 + pi/4 + (d*x)/2)))/(a*d^2) + (f*log(sin(c + d*x)))/(a*d^2) - (im*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) + (im*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2)]
    @test_int [csc(c + d*x)^2/(a + a*sin(c + d*x)), x, 5, atanh(cos(c + d*x))/(a*d) - (2*cot(c + d*x))/(a*d) + cot(c + d*x)/(d*(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^2/((e + f*x)*(a + a*sin(c + d*x))), x, 0, Unintegrable(csc(c + d*x)^2/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [csc(c + d*x)^2/((e + f*x)^2*(a + a*sin(c + d*x))), x, 0, Unintegrable(csc(c + d*x)^2/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

    @test_int [((e + f*x)^3*csc(c + d*x)^3)/(a + a*sin(c + d*x)), x, 40, ((2*im)*(e + f*x)^3)/(a*d) - (6*f^2*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d^3) - (3*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a*d) + ((e + f*x)^3*cot(c/2 + pi/4 + (d*x)/2))/(a*d) + ((e + f*x)^3*cot(c + d*x))/(a*d) - (3*f*(e + f*x)^2*csc(c + d*x))/(2*a*d^2) - ((e + f*x)^3*cot(c + d*x)*csc(c + d*x))/(2*a*d) - (6*f*(e + f*x)^2*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) - (3*f*(e + f*x)^2*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) + ((3*im)*f^3*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^4) + (((9*im)/2)*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) + ((12*im)*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - ((3*im)*f^3*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^4) - (((9*im)/2)*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + ((3*im)*f^2*(e + f*x)*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) - (9*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) - (12*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^4) + (9*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) - (3*f^3*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a*d^4) - ((9*im)*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a*d^4) + ((9*im)*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a*d^4)]
    @test_int [((e + f*x)^2*csc(c + d*x)^3)/(a + a*sin(c + d*x)), x, 30, ((2*im)*(e + f*x)^2)/(a*d) - (3*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d) - (f^2*atanh(cos(c + d*x)))/(a*d^3) + ((e + f*x)^2*cot(c/2 + pi/4 + (d*x)/2))/(a*d) + ((e + f*x)^2*cot(c + d*x))/(a*d) - (f*(e + f*x)*csc(c + d*x))/(a*d^2) - ((e + f*x)^2*cot(c + d*x)*csc(c + d*x))/(2*a*d) - (4*f*(e + f*x)*log(1 - im*ℯ^(im*(c + d*x))))/(a*d^2) - (2*f*(e + f*x)*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) + ((3*im)*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) + ((4*im)*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - ((3*im)*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + (im*f^2*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) - (3*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (3*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3)]
    @test_int [((e + f*x)*csc(c + d*x)^3)/(a + a*sin(c + d*x)), x, 19, (-3*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d) + ((e + f*x)*cot(c/2 + pi/4 + (d*x)/2))/(a*d) + ((e + f*x)*cot(c + d*x))/(a*d) - (f*csc(c + d*x))/(2*a*d^2) - ((e + f*x)*cot(c + d*x)*csc(c + d*x))/(2*a*d) - (2*f*log(sin(c/2 + pi/4 + (d*x)/2)))/(a*d^2) - (f*log(sin(c + d*x)))/(a*d^2) + (((3*im)/2)*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - (((3*im)/2)*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2)]
    @test_int [csc(c + d*x)^3/(a + a*sin(c + d*x)), x, 6, (-3*atanh(cos(c + d*x)))/(2*a*d) + (2*cot(c + d*x))/(a*d) - (3*cot(c + d*x)*csc(c + d*x))/(2*a*d) + (cot(c + d*x)*csc(c + d*x))/(d*(a + a*sin(c + d*x)))]
    @test_int [csc(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))), x, 0, Unintegrable(csc(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [csc(c + d*x)^3/((e + f*x)^2*(a + a*sin(c + d*x))), x, 0, Unintegrable(csc(c + d*x)^3/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

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

    @test_int [((e + f*x)^m*sin(c + d*x)^2)/(a + a*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sin(c + d*x)^2)/(a + a*sin(c + d*x)), x)]
    @test_int [((e + f*x)^m*sin(c + d*x))/(a + a*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sin(c + d*x))/(a + a*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m/(a + a*sin(c + d*x)), x, 0, Unintegrable((e + f*x)^m/(a + a*sin(c + d*x)), x)]
    @test_int [((e + f*x)^m*csc(c + d*x))/(a + a*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*csc(c + d*x))/(a + a*sin(c + d*x)), x)]
    @test_int [((e + f*x)^m*csc(c + d*x)^2)/(a + a*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*csc(c + d*x)^2)/(a + a*sin(c + d*x)), x)]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x)^m*sin(c+d*x)^n / (a+b*sin(c+d*x))^1=#

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

    @test_int [((e + f*x)^3*sin(c + d*x))/(a + b*sin(c + d*x)), x, 14, (e + f*x)^4/(4*b*f) + (im*a*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) - (im*a*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) + (3*a*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - (3*a*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) + ((6*im)*a*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - ((6*im)*a*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - (6*a*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^4) + (6*a*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^4)]
    @test_int [((e + f*x)^2*sin(c + d*x))/(a + b*sin(c + d*x)), x, 12, (e + f*x)^3/(3*b*f) + (im*a*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) - (im*a*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) + (2*a*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - (2*a*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) + ((2*im)*a*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - ((2*im)*a*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3)]
    @test_int [((e + f*x)*sin(c + d*x))/(a + b*sin(c + d*x)), x, 10, (e*x)/b + (f*x^2)/(2*b) + (im*a*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) - (im*a*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) + (a*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - (a*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2)]
    @test_int [sin(c + d*x)/(a + b*sin(c + d*x)), x, 4, x/b - (2*a*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2)*d)]

    @test_int [((e + f*x)^3*sin(c + d*x)^2)/(a + b*sin(c + d*x)), x, 19, -(a*(e + f*x)^4)/(4*b^2*f) + (6*f^2*(e + f*x)*cos(c + d*x))/(b*d^3) - ((e + f*x)^3*cos(c + d*x))/(b*d) - (im*a^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d) + (im*a^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d) - (3*a^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^2) + (3*a^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^2) - ((6*im)*a^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^3) + ((6*im)*a^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^3) + (6*a^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^4) - (6*a^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^4) - (6*f^3*sin(c + d*x))/(b*d^4) + (3*f*(e + f*x)^2*sin(c + d*x))/(b*d^2)]
    @test_int [((e + f*x)^2*sin(c + d*x)^2)/(a + b*sin(c + d*x)), x, 16, -(a*(e + f*x)^3)/(3*b^2*f) + (2*f^2*cos(c + d*x))/(b*d^3) - ((e + f*x)^2*cos(c + d*x))/(b*d) - (im*a^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d) + (im*a^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d) - (2*a^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^2) + (2*a^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^2) - ((2*im)*a^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^3) + ((2*im)*a^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^3) + (2*f*(e + f*x)*sin(c + d*x))/(b*d^2)]
    @test_int [((e + f*x)*sin(c + d*x)^2)/(a + b*sin(c + d*x)), x, 13, -((a*e*x)/b^2) - (a*f*x^2)/(2*b^2) - ((e + f*x)*cos(c + d*x))/(b*d) - (im*a^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d) + (im*a^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d) - (a^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^2) + (a^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*sqrt(a^2 - b^2)*d^2) + (f*sin(c + d*x))/(b*d^2)]
    @test_int [sin(c + d*x)^2/(a + b*sin(c + d*x)), x, 6, -((a*x)/b^2) + (2*a^2*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^2*sqrt(a^2 - b^2)*d) - cos(c + d*x)/(b*d)]

    @test_int [((e + f*x)^3*sin(c + d*x)^3)/(a + b*sin(c + d*x)), x, 24, (-3*e*f^2*x)/(4*b*d^2) - (3*f^3*x^2)/(8*b*d^2) + (a^2*(e + f*x)^4)/(4*b^3*f) + (e + f*x)^4/(8*b*f) - (6*a*f^2*(e + f*x)*cos(c + d*x))/(b^2*d^3) + (a*(e + f*x)^3*cos(c + d*x))/(b^2*d) + (im*a^3*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d) - (im*a^3*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d) + (3*a^3*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^2) + ((6*im)*a^3*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^3) - ((6*im)*a^3*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^3) - (6*a^3*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^4) + (6*a^3*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^4) + (6*a*f^3*sin(c + d*x))/(b^2*d^4) - (3*a*f*(e + f*x)^2*sin(c + d*x))/(b^2*d^2) + (3*f^2*(e + f*x)*cos(c + d*x)*sin(c + d*x))/(4*b*d^3) - ((e + f*x)^3*cos(c + d*x)*sin(c + d*x))/(2*b*d) - (3*f^3*sin(c + d*x)^2)/(8*b*d^4) + (3*f*(e + f*x)^2*sin(c + d*x)^2)/(4*b*d^2)]
    @test_int [((e + f*x)^2*sin(c + d*x)^3)/(a + b*sin(c + d*x)), x, 21, -(f^2*x)/(4*b*d^2) + (a^2*(e + f*x)^3)/(3*b^3*f) + (e + f*x)^3/(6*b*f) - (2*a*f^2*cos(c + d*x))/(b^2*d^3) + (a*(e + f*x)^2*cos(c + d*x))/(b^2*d) + (im*a^3*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d) - (im*a^3*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d) + (2*a^3*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^2) - (2*a^3*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^2) + ((2*im)*a^3*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^3) - ((2*im)*a^3*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^3) - (2*a*f*(e + f*x)*sin(c + d*x))/(b^2*d^2) + (f^2*cos(c + d*x)*sin(c + d*x))/(4*b*d^3) - ((e + f*x)^2*cos(c + d*x)*sin(c + d*x))/(2*b*d) + (f*(e + f*x)*sin(c + d*x)^2)/(2*b*d^2)]
    @test_int [((e + f*x)*sin(c + d*x)^3)/(a + b*sin(c + d*x)), x, 16, (a^2*e*x)/b^3 + (e*x)/(2*b) + (a^2*f*x^2)/(2*b^3) + (f*x^2)/(4*b) + (a*(e + f*x)*cos(c + d*x))/(b^2*d) + (im*a^3*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d) - (im*a^3*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d) + (a^3*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^2) - (a^3*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*sqrt(a^2 - b^2)*d^2) - (a*f*sin(c + d*x))/(b^2*d^2) - ((e + f*x)*cos(c + d*x)*sin(c + d*x))/(2*b*d) + (f*sin(c + d*x)^2)/(4*b*d^2)]
    @test_int [sin(c + d*x)^3/(a + b*sin(c + d*x)), x, 6, ((2*a^2 + b^2)*x)/(2*b^3) - (2*a^3*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^3*sqrt(a^2 - b^2)*d) + (a*cos(c + d*x))/(b^2*d) - (cos(c + d*x)*sin(c + d*x))/(2*b*d)]

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

    @test_int [((e + f*x)^3*csc(c + d*x))/(a + b*sin(c + d*x)), x, 22, (-2*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a*d) + (im*b*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d) - (im*b*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d) + ((3*im)*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((3*im)*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + (3*b*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^2) - (3*b*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^2) - (6*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (6*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) + ((6*im)*b*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^3) - ((6*im)*b*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^3) - ((6*im)*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a*d^4) + ((6*im)*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a*d^4) - (6*b*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^4) + (6*b*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^4)]
    @test_int [((e + f*x)^2*csc(c + d*x))/(a + b*sin(c + d*x)), x, 18, (-2*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d) + (im*b*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d) - (im*b*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d) + ((2*im)*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((2*im)*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + (2*b*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^2) - (2*b*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^2) - (2*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (2*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) + ((2*im)*b*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^3) - ((2*im)*b*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^3)]
    @test_int [((e + f*x)*csc(c + d*x))/(a + b*sin(c + d*x)), x, 14, (-2*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d) + (im*b*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d) - (im*b*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d) + (im*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - (im*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + (b*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^2) - (b*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*sqrt(a^2 - b^2)*d^2)]
    @test_int [csc(c + d*x)/(a + b*sin(c + d*x)), x, 5, (-2*b*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a*sqrt(a^2 - b^2)*d) - atanh(cos(c + d*x))/(a*d)]

    @test_int [((e + f*x)^3*csc(c + d*x)^2)/(a + b*sin(c + d*x)), x, 29, ((-im)*(e + f*x)^3)/(a*d) + (2*b*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a^2*d) - ((e + f*x)^3*cot(c + d*x))/(a*d) - (im*b^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d) + (im*b^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d) + (3*f*(e + f*x)^2*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) - ((3*im)*b*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a^2*d^2) + ((3*im)*b*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a^2*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^2) + (3*b^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^2) - ((3*im)*f^2*(e + f*x)*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) + (6*b*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a^2*d^3) - ((6*im)*b^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^3) + ((6*im)*b^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^3) + (3*f^3*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a*d^4) + ((6*im)*b*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a^2*d^4) - ((6*im)*b*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a^2*d^4) + (6*b^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^4) - (6*b^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^4)]
    @test_int [((e + f*x)^2*csc(c + d*x)^2)/(a + b*sin(c + d*x)), x, 24, ((-im)*(e + f*x)^2)/(a*d) + (2*b*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a^2*d) - ((e + f*x)^2*cot(c + d*x))/(a*d) - (im*b^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d) + (im*b^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d) + (2*f*(e + f*x)*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) - ((2*im)*b*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a^2*d^2) + ((2*im)*b*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a^2*d^2) - (2*b^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^2) + (2*b^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^2) - (im*f^2*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) + (2*b*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a^2*d^3) - (2*b*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a^2*d^3) - ((2*im)*b^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^3) + ((2*im)*b^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^3)]
    @test_int [((e + f*x)*csc(c + d*x)^2)/(a + b*sin(c + d*x)), x, 17, (2*b*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a^2*d) - ((e + f*x)*cot(c + d*x))/(a*d) - (im*b^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d) + (im*b^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d) + (f*log(sin(c + d*x)))/(a*d^2) - (im*b*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a^2*d^2) + (im*b*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a^2*d^2) - (b^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^2) + (b^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*sqrt(a^2 - b^2)*d^2)]
    @test_int [csc(c + d*x)^2/(a + b*sin(c + d*x)), x, 7, (2*b^2*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^2*sqrt(a^2 - b^2)*d) + (b*atanh(cos(c + d*x)))/(a^2*d) - cot(c + d*x)/(a*d)]

    #= ::Subsubsection::Closed:: =#
    #=m*symbolic=#

    @test_int [((e + f*x)^m*sin(c + d*x)^2)/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sin(c + d*x)^2)/(a + b*sin(c + d*x)), x)]
    @test_int [((e + f*x)^m*sin(c + d*x))/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sin(c + d*x))/(a + b*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m/(a + b*sin(c + d*x)), x, 0, Unintegrable((e + f*x)^m/(a + b*sin(c + d*x)), x)]
    @test_int [((e + f*x)^m*csc(c + d*x))/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*csc(c + d*x))/(a + b*sin(c + d*x)), x)]
    @test_int [((e + f*x)^m*csc(c + d*x)^2)/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*csc(c + d*x)^2)/(a + b*sin(c + d*x)), x)]

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

    @test_int [((e + f*x)*sin(c + d*x))/(a + b*sin(c + d*x))^2, x, 21, (im*a^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) - (im*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) - (im*a^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) + (im*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) + (a*f*log(a + b*sin(c + d*x)))/(b*(a^2 - b^2)*d^2) + (a^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) - (f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - (a^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) + (f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - (a*(e + f*x)*cos(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [((e + f*x)^2*sin(c + d*x))/(a + b*sin(c + d*x))^2, x, 30, ((-im)*a*(e + f*x)^2)/(b*(a^2 - b^2)*d) + (2*a*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) + (im*a^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) - (im*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) + (2*a*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) - (im*a^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) + (im*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) - ((2*im)*a*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) + (2*a^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) - (2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - ((2*im)*a*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) - (2*a^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) + (2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) + ((2*im)*a^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) - ((2*im)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - ((2*im)*a^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) + ((2*im)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - (a*(e + f*x)^2*cos(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [((e + f*x)^3*sin(c + d*x))/(a + b*sin(c + d*x))^2, x, 36, ((-im)*a*(e + f*x)^3)/(b*(a^2 - b^2)*d) + (3*a*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) + (im*a^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) - (im*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) + (3*a*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) - (im*a^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) + (im*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d) - ((6*im)*a*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) + (3*a^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) - (3*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - ((6*im)*a*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) - (3*a^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) + (3*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) + (6*a*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^4) + ((6*im)*a^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) - ((6*im)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) + (6*a*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^4) - ((6*im)*a^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) + ((6*im)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - (6*a^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) + (6*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^4) + (6*a^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) - (6*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^4) - (a*(e + f*x)^3*cos(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]

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

    @test_int [((e + f*x)*sin(c + d*x))/(a + b*sin(c + d*x))^3, x, 48, (((3*im)/2)*a^3*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d) - (((3*im)/2)*a*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) - (((3*im)/2)*a^3*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d) + (((3*im)/2)*a*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*log(a + b*sin(c + d*x)))/(2*b*(a^2 - b^2)^2*d^2) - (f*log(a + b*sin(c + d*x)))/(b*(a^2 - b^2)*d^2) + (3*a^3*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(3/2)*d^2) - (3*a^3*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(5/2)*d^2) + (3*a*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(3/2)*d^2) - (a*(e + f*x)*cos(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) - (a*f)/(2*b*(a^2 - b^2)*d^2*(a + b*sin(c + d*x))) - (3*a^2*(e + f*x)*cos(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) + ((e + f*x)*cos(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [((e + f*x)^2*sin(c + d*x))/(a + b*sin(c + d*x))^3, x, 73, (((-3*im)/2)*a^2*(e + f*x)^2)/(b*(a^2 - b^2)^2*d) + (im*(e + f*x)^2)/(b*(a^2 - b^2)*d) + (2*a*f^2*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b*(a^2 - b^2)^(3/2)*d^3) + (3*a^2*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^2) - (2*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) + (((3*im)/2)*a^3*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d) - (((3*im)/2)*a*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) + (3*a^2*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^2) - (2*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) - (((3*im)/2)*a^3*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d) + (((3*im)/2)*a*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) - ((3*im)*a^2*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^3) + ((2*im)*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) + (3*a^3*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^2) - (3*a*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) - ((3*im)*a^2*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^3) + ((2*im)*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) - (3*a^3*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^2) + (3*a*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) + ((3*im)*a^3*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^3) - ((3*im)*a*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) - ((3*im)*a^3*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^3) + ((3*im)*a*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) - (a*(e + f*x)^2*cos(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) - (a*f*(e + f*x))/(b*(a^2 - b^2)*d^2*(a + b*sin(c + d*x))) - (3*a^2*(e + f*x)^2*cos(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) + ((e + f*x)^2*cos(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]
    @test_int [((e + f*x)^3*sin(c + d*x))/(a + b*sin(c + d*x))^3, x, 92, (((-3*im)/2)*a^2*(e + f*x)^3)/(b*(a^2 - b^2)^2*d) + (im*(e + f*x)^3)/(b*(a^2 - b^2)*d) - ((3*im)*a*f^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) + (9*a^2*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^2*d^2) - (3*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) + (((3*im)/2)*a^3*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d) - (((3*im)/2)*a*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) + ((3*im)*a*f^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) + (9*a^2*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^2*d^2) - (3*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^2) - (((3*im)/2)*a^3*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d) + (((3*im)/2)*a*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d) - (3*a*f^3*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) - ((9*im)*a^2*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^3) + ((6*im)*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) + (9*a^3*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(5/2)*d^2) - (9*a*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(3/2)*d^2) + (3*a*f^3*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) - ((9*im)*a^2*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^3) + ((6*im)*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) - (9*a^3*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(5/2)*d^2) + (9*a*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(3/2)*d^2) + (9*a^2*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^4) - (6*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^4) + ((9*im)*a^3*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^3) - ((9*im)*a*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) + (9*a^2*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^2*d^4) - (6*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^4) - ((9*im)*a^3*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^3) + ((9*im)*a*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) - (9*a^3*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^4) + (9*a*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) + (9*a^3*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(5/2)*d^4) - (9*a*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) - (a*(e + f*x)^3*cos(c + d*x))/(2*(a^2 - b^2)*d*(a + b*sin(c + d*x))^2) - (3*a*f*(e + f*x)^2)/(2*b*(a^2 - b^2)*d^2*(a + b*sin(c + d*x))) - (3*a^2*(e + f*x)^3*cos(c + d*x))/(2*(a^2 - b^2)^2*d*(a + b*sin(c + d*x))) + ((e + f*x)^3*cos(c + d*x))/((a^2 - b^2)*d*(a + b*sin(c + d*x)))]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e+f*x)^m*cos(c+d*x)^n*(a+b*sin(c+d*x))^p*with*a^2-b^2=0=#

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

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

    @test_int [(e + f*x)^3*cos(c + d*x)/(a + a*sin(c + d*x)), x, 6, -((im*(e + f*x)^4)/(4*a*f)) + (2*(e + f*x)^3*log(1 - im*ℯ^(im*(c + d*x))))/(a*d) - (6*im*f*(e + f*x)^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^2) + (12*f^2*(e + f*x)*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^3) + (12*im*f^3*PolyLog(4, im*ℯ^(im*(c + d*x))))/(a*d^4)]
    @test_int [(e + f*x)^2*cos(c + d*x)/(a + a*sin(c + d*x)), x, 5, -((im*(e + f*x)^3)/(3*a*f)) + (2*(e + f*x)^2*log(1 - im*ℯ^(im*(c + d*x))))/(a*d) - (4*im*f*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^2) + (4*f^2*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^3)]
    @test_int [(e + f*x)^1*cos(c + d*x)/(a + a*sin(c + d*x)), x, 4, -((im*(e + f*x)^2)/(2*a*f)) + (2*(e + f*x)*log(1 - im*ℯ^(im*(c + d*x))))/(a*d) - (2*im*f*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^2)]
    @test_int [(e + f*x)^0*cos(c + d*x)/(a + a*sin(c + d*x)), x, 2, log(1 + sin(c + d*x))/(a*d)]
    @test_int [1/(e + f*x)^1*cos(c + d*x)/(a + a*sin(c + d*x)), x, 0, Unintegrable(cos(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [1/(e + f*x)^2*cos(c + d*x)/(a + a*sin(c + d*x)), x, 0, Unintegrable(cos(c + d*x)/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

    @test_int [(e + f*x)^3*cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 6, (e + f*x)^4/(4*a*f) - (6*f^2*(e + f*x)*cos(c + d*x))/(a*d^3) + ((e + f*x)^3*cos(c + d*x))/(a*d) + (6*f^3*sin(c + d*x))/(a*d^4) - (3*f*(e + f*x)^2*sin(c + d*x))/(a*d^2)]
    @test_int [(e + f*x)^2*cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 5, (e + f*x)^3/(3*a*f) - (2*f^2*cos(c + d*x))/(a*d^3) + ((e + f*x)^2*cos(c + d*x))/(a*d) - (2*f*(e + f*x)*sin(c + d*x))/(a*d^2)]
    @test_int [(e + f*x)^1*cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 4, (e*x)/a + (f*x^2)/(2*a) + ((e + f*x)*cos(c + d*x))/(a*d) - (f*sin(c + d*x))/(a*d^2)]
    @test_int [(e + f*x)^0*cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 2, x/a + cos(c + d*x)/(a*d)]
    @test_int [1/(e + f*x)^1*cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 5, log(e + f*x)/(a*f) - (CosIntegral((d*e)/f + d*x)*sin(c - (d*e)/f))/(a*f) - (cos(c - (d*e)/f)*SinIntegral((d*e)/f + d*x))/(a*f)]
    @test_int [1/(e + f*x)^2*cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 6, -(1/(a*f*(e + f*x))) - (d*cos(c - (d*e)/f)*CosIntegral((d*e)/f + d*x))/(a*f^2) + sin(c + d*x)/(a*f*(e + f*x)) + (d*sin(c - (d*e)/f)*SinIntegral((d*e)/f + d*x))/(a*f^2)]

    @test_int [(e + f*x)^3*cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 10, -((3*f^3*x)/(8*a*d^3)) + (e + f*x)^3/(4*a*d) - (6*f^3*cos(c + d*x))/(a*d^4) + (3*f*(e + f*x)^2*cos(c + d*x))/(a*d^2) - (6*f^2*(e + f*x)*sin(c + d*x))/(a*d^3) + ((e + f*x)^3*sin(c + d*x))/(a*d) + (3*f^3*cos(c + d*x)*sin(c + d*x))/(8*a*d^4) - (3*f*(e + f*x)^2*cos(c + d*x)*sin(c + d*x))/(4*a*d^2) + (3*f^2*(e + f*x)*sin(c + d*x)^2)/(4*a*d^3) - ((e + f*x)^3*sin(c + d*x)^2)/(2*a*d)]
    @test_int [(e + f*x)^2*cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 7, (e*f*x)/(2*a*d) + (f^2*x^2)/(4*a*d) + (2*f*(e + f*x)*cos(c + d*x))/(a*d^2) - (2*f^2*sin(c + d*x))/(a*d^3) + ((e + f*x)^2*sin(c + d*x))/(a*d) - (f*(e + f*x)*cos(c + d*x)*sin(c + d*x))/(2*a*d^2) + (f^2*sin(c + d*x)^2)/(4*a*d^3) - ((e + f*x)^2*sin(c + d*x)^2)/(2*a*d)]
    @test_int [(e + f*x)^1*cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 6, (f*x)/(4*a*d) + (f*cos(c + d*x))/(a*d^2) + ((e + f*x)*sin(c + d*x))/(a*d) - (f*cos(c + d*x)*sin(c + d*x))/(4*a*d^2) - ((e + f*x)*sin(c + d*x)^2)/(2*a*d)]
    @test_int [(e + f*x)^0*cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 2, sin(c + d*x)/(a*d) - sin(c + d*x)^2/(2*a*d)]
    @test_int [1/(e + f*x)^1*cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 9, (cos(c - (d*e)/f)*CosIntegral((d*e)/f + d*x))/(a*f) - (CosIntegral((2*d*e)/f + 2*d*x)*sin(2*c - (2*d*e)/f))/(2*a*f) - (sin(c - (d*e)/f)*SinIntegral((d*e)/f + d*x))/(a*f) - (cos(2*c - (2*d*e)/f)*SinIntegral((2*d*e)/f + 2*d*x))/(2*a*f)]
    @test_int [1/(e + f*x)^2*cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 11, -(cos(c + d*x)/(a*f*(e + f*x))) - (d*cos(2*c - (2*d*e)/f)*CosIntegral((2*d*e)/f + 2*d*x))/(a*f^2) - (d*CosIntegral((d*e)/f + d*x)*sin(c - (d*e)/f))/(a*f^2) + sin(2*c + 2*d*x)/(2*a*f*(e + f*x)) - (d*cos(c - (d*e)/f)*SinIntegral((d*e)/f + d*x))/(a*f^2) + (d*sin(2*c - (2*d*e)/f)*SinIntegral((2*d*e)/f + 2*d*x))/(a*f^2)]

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

    @test_int [(e + f*x)^3*sec(c + d*x)/(a + a*sin(c + d*x)), x, 22, -((3*im*f*(e + f*x)^2)/(2*a*d^2)) - (6*im*f^2*(e + f*x)*atan(ℯ^(im*(c + d*x))))/(a*d^3) - (im*(e + f*x)^3*atan(ℯ^(im*(c + d*x))))/(a*d) + (3*f^2*(e + f*x)*log(1 + ℯ^(2*im*(c + d*x))))/(a*d^3) + (3*im*f^3*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(a*d^4) + (3*im*f*(e + f*x)^2*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(2*a*d^2) - (3*im*f^3*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^4) - (3*im*f*(e + f*x)^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(2*a*d^2) - (3*im*f^3*PolyLog(2, -ℯ^(2*im*(c + d*x))))/(2*a*d^4) - (3*f^2*(e + f*x)*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/(a*d^3) + (3*f^2*(e + f*x)*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^3) - (3*im*f^3*PolyLog(4, (-im)*ℯ^(im*(c + d*x))))/(a*d^4) + (3*im*f^3*PolyLog(4, im*ℯ^(im*(c + d*x))))/(a*d^4) - (3*f*(e + f*x)^2*sec(c + d*x))/(2*a*d^2) - ((e + f*x)^3*sec(c + d*x)^2)/(2*a*d) + (3*f*(e + f*x)^2*tan(c + d*x))/(2*a*d^2) + ((e + f*x)^3*sec(c + d*x)*tan(c + d*x))/(2*a*d)]
    @test_int [(e + f*x)^2*sec(c + d*x)/(a + a*sin(c + d*x)), x, 13, -((im*(e + f*x)^2*atan(ℯ^(im*(c + d*x))))/(a*d)) + (f^2*atanh(sin(c + d*x)))/(a*d^3) + (f^2*log(cos(c + d*x)))/(a*d^3) + (im*f*(e + f*x)*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(a*d^2) - (im*f*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^2) - (f^2*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/(a*d^3) + (f^2*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^3) - (f*(e + f*x)*sec(c + d*x))/(a*d^2) - ((e + f*x)^2*sec(c + d*x)^2)/(2*a*d) + (f*(e + f*x)*tan(c + d*x))/(a*d^2) + ((e + f*x)^2*sec(c + d*x)*tan(c + d*x))/(2*a*d)]
    @test_int [(e + f*x)^1*sec(c + d*x)/(a + a*sin(c + d*x)), x, 10, -((im*(e + f*x)*atan(ℯ^(im*(c + d*x))))/(a*d)) + (im*f*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(2*a*d^2) - (im*f*PolyLog(2, im*ℯ^(im*(c + d*x))))/(2*a*d^2) - (f*sec(c + d*x))/(2*a*d^2) - ((e + f*x)*sec(c + d*x)^2)/(2*a*d) + (f*tan(c + d*x))/(2*a*d^2) + ((e + f*x)*sec(c + d*x)*tan(c + d*x))/(2*a*d)]
    @test_int [(e + f*x)^0*sec(c + d*x)/(a + a*sin(c + d*x)), x, 4, atanh(sin(c + d*x))/(2*a*d) - 1/(2*d*(a + a*sin(c + d*x)))]
    @test_int [1/(e + f*x)^1*sec(c + d*x)/(a + a*sin(c + d*x)), x, 0, Unintegrable(sec(c + d*x)/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [1/(e + f*x)^2*sec(c + d*x)/(a + a*sin(c + d*x)), x, 0, Unintegrable(sec(c + d*x)/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

    @test_int [(e + f*x)^3*sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 20, -((2*im*(e + f*x)^3)/(3*a*d)) - (im*f*(e + f*x)^2*atan(ℯ^(im*(c + d*x))))/(a*d^2) + (f^3*atanh(sin(c + d*x)))/(a*d^4) + (2*f*(e + f*x)^2*log(1 + ℯ^(2*im*(c + d*x))))/(a*d^2) + (f^3*log(cos(c + d*x)))/(a*d^4) + (im*f^2*(e + f*x)*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(a*d^3) - (im*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(a*d^3) - (2*im*f^2*(e + f*x)*PolyLog(2, -ℯ^(2*im*(c + d*x))))/(a*d^3) - (f^3*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/(a*d^4) + (f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/(a*d^4) + (f^3*PolyLog(3, -ℯ^(2*im*(c + d*x))))/(a*d^4) - (f^2*(e + f*x)*sec(c + d*x))/(a*d^3) - (f*(e + f*x)^2*sec(c + d*x)^2)/(2*a*d^2) - ((e + f*x)^3*sec(c + d*x)^3)/(3*a*d) + (f^2*(e + f*x)*tan(c + d*x))/(a*d^3) + (2*(e + f*x)^3*tan(c + d*x))/(3*a*d) + (f*(e + f*x)^2*sec(c + d*x)*tan(c + d*x))/(2*a*d^2) + ((e + f*x)^3*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)]
    @test_int [(e + f*x)^2*sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 16, -((2*im*(e + f*x)^2)/(3*a*d)) - (2*im*f*(e + f*x)*atan(ℯ^(im*(c + d*x))))/(3*a*d^2) + (4*f*(e + f*x)*log(1 + ℯ^(2*im*(c + d*x))))/(3*a*d^2) + (im*f^2*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(3*a*d^3) - (im*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(3*a*d^3) - (2*im*f^2*PolyLog(2, -ℯ^(2*im*(c + d*x))))/(3*a*d^3) - (f^2*sec(c + d*x))/(3*a*d^3) - (f*(e + f*x)*sec(c + d*x)^2)/(3*a*d^2) - ((e + f*x)^2*sec(c + d*x)^3)/(3*a*d) + (f^2*tan(c + d*x))/(3*a*d^3) + (2*(e + f*x)^2*tan(c + d*x))/(3*a*d) + (f*(e + f*x)*sec(c + d*x)*tan(c + d*x))/(3*a*d^2) + ((e + f*x)^2*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)]
    @test_int [(e + f*x)^1*sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 7, (f*atanh(sin(c + d*x)))/(6*a*d^2) + (2*f*log(cos(c + d*x)))/(3*a*d^2) - (f*sec(c + d*x)^2)/(6*a*d^2) - ((e + f*x)*sec(c + d*x)^3)/(3*a*d) + (2*(e + f*x)*tan(c + d*x))/(3*a*d) + (f*sec(c + d*x)*tan(c + d*x))/(6*a*d^2) + ((e + f*x)*sec(c + d*x)^2*tan(c + d*x))/(3*a*d)]
    @test_int [(e + f*x)^0*sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 3, -(sec(c + d*x)/(3*d*(a + a*sin(c + d*x)))) + (2*tan(c + d*x))/(3*a*d)]
    @test_int [1/(e + f*x)^1*sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^2/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [1/(e + f*x)^2*sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^2/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

    @test_int [(e + f*x)^3*sec(c + d*x)^3/(a + a*sin(c + d*x)), x, 32, -((im*f*(e + f*x)^2)/(2*a*d^2)) - (5*im*f^2*(e + f*x)*atan(ℯ^(im*(c + d*x))))/(a*d^3) - (3*im*(e + f*x)^3*atan(ℯ^(im*(c + d*x))))/(4*a*d) + (f^2*(e + f*x)*log(1 + ℯ^(2*im*(c + d*x))))/(a*d^3) + (5*im*f^3*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(2*a*d^4) + (9*im*f*(e + f*x)^2*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(8*a*d^2) - (5*im*f^3*PolyLog(2, im*ℯ^(im*(c + d*x))))/(2*a*d^4) - (9*im*f*(e + f*x)^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/(8*a*d^2) - (im*f^3*PolyLog(2, -ℯ^(2*im*(c + d*x))))/(2*a*d^4) - (9*f^2*(e + f*x)*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/(4*a*d^3) + (9*f^2*(e + f*x)*PolyLog(3, im*ℯ^(im*(c + d*x))))/(4*a*d^3) - (9*im*f^3*PolyLog(4, (-im)*ℯ^(im*(c + d*x))))/(4*a*d^4) + (9*im*f^3*PolyLog(4, im*ℯ^(im*(c + d*x))))/(4*a*d^4) - (f^3*sec(c + d*x))/(4*a*d^4) - (9*f*(e + f*x)^2*sec(c + d*x))/(8*a*d^2) - (f^2*(e + f*x)*sec(c + d*x)^2)/(4*a*d^3) - (f*(e + f*x)^2*sec(c + d*x)^3)/(4*a*d^2) - ((e + f*x)^3*sec(c + d*x)^4)/(4*a*d) + (f^3*tan(c + d*x))/(4*a*d^4) + (f*(e + f*x)^2*tan(c + d*x))/(2*a*d^2) + (f^2*(e + f*x)*sec(c + d*x)*tan(c + d*x))/(4*a*d^3) + (3*(e + f*x)^3*sec(c + d*x)*tan(c + d*x))/(8*a*d) + (f*(e + f*x)^2*sec(c + d*x)^2*tan(c + d*x))/(4*a*d^2) + ((e + f*x)^3*sec(c + d*x)^3*tan(c + d*x))/(4*a*d)]
    @test_int [(e + f*x)^2*sec(c + d*x)^3/(a + a*sin(c + d*x)), x, 17, -((3*im*(e + f*x)^2*atan(ℯ^(im*(c + d*x))))/(4*a*d)) + (5*f^2*atanh(sin(c + d*x)))/(6*a*d^3) + (f^2*log(cos(c + d*x)))/(3*a*d^3) + (3*im*f*(e + f*x)*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(4*a*d^2) - (3*im*f*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/(4*a*d^2) - (3*f^2*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/(4*a*d^3) + (3*f^2*PolyLog(3, im*ℯ^(im*(c + d*x))))/(4*a*d^3) - (3*f*(e + f*x)*sec(c + d*x))/(4*a*d^2) - (f^2*sec(c + d*x)^2)/(12*a*d^3) - (f*(e + f*x)*sec(c + d*x)^3)/(6*a*d^2) - ((e + f*x)^2*sec(c + d*x)^4)/(4*a*d) + (f*(e + f*x)*tan(c + d*x))/(3*a*d^2) + (f^2*sec(c + d*x)*tan(c + d*x))/(12*a*d^3) + (3*(e + f*x)^2*sec(c + d*x)*tan(c + d*x))/(8*a*d) + (f*(e + f*x)*sec(c + d*x)^2*tan(c + d*x))/(6*a*d^2) + ((e + f*x)^2*sec(c + d*x)^3*tan(c + d*x))/(4*a*d)]
    @test_int [(e + f*x)^1*sec(c + d*x)^3/(a + a*sin(c + d*x)), x, 11, -((3*im*(e + f*x)*atan(ℯ^(im*(c + d*x))))/(4*a*d)) + (3*im*f*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/(8*a*d^2) - (3*im*f*PolyLog(2, im*ℯ^(im*(c + d*x))))/(8*a*d^2) - (3*f*sec(c + d*x))/(8*a*d^2) - (f*sec(c + d*x)^3)/(12*a*d^2) - ((e + f*x)*sec(c + d*x)^4)/(4*a*d) + (f*tan(c + d*x))/(4*a*d^2) + (3*(e + f*x)*sec(c + d*x)*tan(c + d*x))/(8*a*d) + ((e + f*x)*sec(c + d*x)^3*tan(c + d*x))/(4*a*d) + (f*tan(c + d*x)^3)/(12*a*d^2)]
    @test_int [(e + f*x)^0*sec(c + d*x)^3/(a + a*sin(c + d*x)), x, 4, (3*atanh(sin(c + d*x)))/(8*a*d) + 1/(8*d*(a - a*sin(c + d*x))) - a/(8*d*(a + a*sin(c + d*x))^2) - 1/(4*d*(a + a*sin(c + d*x)))]
    @test_int [1/(e + f*x)^1*sec(c + d*x)^3/(a + a*sin(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))), x)]
    @test_int [1/(e + f*x)^2*sec(c + d*x)^3/(a + a*sin(c + d*x)), x, 0, Unintegrable(sec(c + d*x)^3/((e + f*x)^2*(a + a*sin(c + d*x))), x)]

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

    @test_int [(e + f*x)^m*cos(c + d*x)^4/(a + a*sin(c + d*x)), x, 14, (e + f*x)^(1 + m)/(2*a*f*(1 + m)) + (ℯ^(im*(c - (d*e)/f))*(e + f*x)^m*Gamma(1 + m, -((im*d*(e + f*x))/f)))/((-((im*d*(e + f*x))/f))^m*(8*a*d)) + ((e + f*x)^m*Gamma(1 + m, (im*d*(e + f*x))/f))/(ℯ^(im*(c - (d*e)/f))*((im*d*(e + f*x))/f)^m*(8*a*d)) - (im*2^(-3 - m)*ℯ^(2*im*(c - (d*e)/f))*(e + f*x)^m*Gamma(1 + m, -((2*im*d*(e + f*x))/f)))/((-((im*d*(e + f*x))/f))^m*(a*d)) + (im*2^(-3 - m)*(e + f*x)^m*Gamma(1 + m, (2*im*d*(e + f*x))/f))/(ℯ^(2*im*(c - (d*e)/f))*((im*d*(e + f*x))/f)^m*(a*d)) + (3^(-1 - m)*ℯ^(3*im*(c - (d*e)/f))*(e + f*x)^m*Gamma(1 + m, -((3*im*d*(e + f*x))/f)))/((-((im*d*(e + f*x))/f))^m*(8*a*d)) + (3^(-1 - m)*(e + f*x)^m*Gamma(1 + m, (3*im*d*(e + f*x))/f))/(ℯ^(3*im*(c - (d*e)/f))*((im*d*(e + f*x))/f)^m*(8*a*d))]
    @test_int [(e + f*x)^m*cos(c + d*x)^3/(a + a*sin(c + d*x)), x, 9, -((im*ℯ^(im*(c - (d*e)/f))*(e + f*x)^m*Gamma(1 + m, -((im*d*(e + f*x))/f)))/((-((im*d*(e + f*x))/f))^m*(2*a*d))) + (im*(e + f*x)^m*Gamma(1 + m, (im*d*(e + f*x))/f))/(ℯ^(im*(c - (d*e)/f))*((im*d*(e + f*x))/f)^m*(2*a*d)) + (2^(-3 - m)*ℯ^(2*im*(c - (d*e)/f))*(e + f*x)^m*Gamma(1 + m, -((2*im*d*(e + f*x))/f)))/((-((im*d*(e + f*x))/f))^m*(a*d)) + (2^(-3 - m)*(e + f*x)^m*Gamma(1 + m, (2*im*d*(e + f*x))/f))/(ℯ^(2*im*(c - (d*e)/f))*((im*d*(e + f*x))/f)^m*(a*d))]
    @test_int [(e + f*x)^m*cos(c + d*x)^2/(a + a*sin(c + d*x)), x, 5, (e + f*x)^(1 + m)/(a*f*(1 + m)) + (ℯ^(im*(c - (d*e)/f))*(e + f*x)^m*Gamma(1 + m, -((im*d*(e + f*x))/f)))/((-((im*d*(e + f*x))/f))^m*(2*a*d)) + ((e + f*x)^m*Gamma(1 + m, (im*d*(e + f*x))/f))/(ℯ^(im*(c - (d*e)/f))*((im*d*(e + f*x))/f)^m*(2*a*d))]
    @test_int [(e + f*x)^m*cos(c + d*x)^1/(a + a*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*cos(c + d*x))/(a + a*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m*cos(c + d*x)^0/(a + a*sin(c + d*x)), x, 0, Unintegrable((e + f*x)^m/(a + a*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m*sec(c + d*x)^1/(a + a*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sec(c + d*x))/(a + a*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m*sec(c + d*x)^2/(a + a*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sec(c + d*x)^2)/(a + a*sin(c + d*x)), x)]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x)^m*cos(c+d*x)^n / (a+b*sin(c+d*x))^1=#

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

    @test_int [((e + f*x)^3*cos(c + d*x))/(a + b*sin(c + d*x)), x, 11, ((-im/4)*(e + f*x)^4)/(b*f) + ((e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d) + ((e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d) - ((3*im)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d^2) - ((3*im)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d^2) + (6*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d^3) + (6*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d^3) + ((6*im)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d^4) + ((6*im)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d^4)]
    @test_int [((e + f*x)^2*cos(c + d*x))/(a + b*sin(c + d*x)), x, 9, ((-im/3)*(e + f*x)^3)/(b*f) + ((e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d) + ((e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d) - ((2*im)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d^2) - ((2*im)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d^2) + (2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d^3) + (2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d^3)]
    @test_int [((e + f*x)^1*cos(c + d*x))/(a + b*sin(c + d*x)), x, 7, ((-im/2)*(e + f*x)^2)/(b*f) + ((e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d) + ((e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d) - (im*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*d^2) - (im*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*d^2)]
    @test_int [cos(c + d*x)/(a + b*sin(c + d*x)), x, 2, log(a + b*sin(c + d*x))/(b*d)]

    @test_int [((e + f*x)^3*cos(c + d*x)^2)/(a + b*sin(c + d*x)), x, 18, (a*(e + f*x)^4)/(4*b^2*f) - (6*f^2*(e + f*x)*cos(c + d*x))/(b*d^3) + ((e + f*x)^3*cos(c + d*x))/(b*d) + (im*sqrt(a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d) - (im*sqrt(a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d) + (3*sqrt(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d^2) - (3*sqrt(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d^2) + ((6*im)*sqrt(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d^3) - ((6*im)*sqrt(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d^3) - (6*sqrt(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d^4) + (6*sqrt(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d^4) + (6*f^3*sin(c + d*x))/(b*d^4) - (3*f*(e + f*x)^2*sin(c + d*x))/(b*d^2)]
    @test_int [((e + f*x)^2*cos(c + d*x)^2)/(a + b*sin(c + d*x)), x, 15, (a*(e + f*x)^3)/(3*b^2*f) - (2*f^2*cos(c + d*x))/(b*d^3) + ((e + f*x)^2*cos(c + d*x))/(b*d) + (im*sqrt(a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d) - (im*sqrt(a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d) + (2*sqrt(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d^2) - (2*sqrt(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d^2) + ((2*im)*sqrt(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d^3) - ((2*im)*sqrt(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d^3) - (2*f*(e + f*x)*sin(c + d*x))/(b*d^2)]
    @test_int [((e + f*x)^1*cos(c + d*x)^2)/(a + b*sin(c + d*x)), x, 12, (a*e*x)/b^2 + (a*f*x^2)/(2*b^2) + ((e + f*x)*cos(c + d*x))/(b*d) + (im*sqrt(a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d) - (im*sqrt(a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d) + (sqrt(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^2*d^2) - (sqrt(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^2*d^2) - (f*sin(c + d*x))/(b*d^2)]
    @test_int [cos(c + d*x)^2/(a + b*sin(c + d*x)), x, 5, (a*x)/b^2 - (2*sqrt(a^2 - b^2)*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(b^2*d) + cos(c + d*x)/(b*d)]

    @test_int [((e + f*x)^3*cos(c + d*x)^3)/(a + b*sin(c + d*x)), x, 21, (-3*f^3*x)/(8*b*d^3) + (e + f*x)^3/(4*b*d) + ((im/4)*(a^2 - b^2)*(e + f*x)^4)/(b^3*f) - (6*a*f^3*cos(c + d*x))/(b^2*d^4) + (3*a*f*(e + f*x)^2*cos(c + d*x))/(b^2*d^2) - ((a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d) - ((a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d) + ((3*im)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d^2) + ((3*im)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d^2) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d^3) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d^3) - ((6*im)*(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d^4) - ((6*im)*(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d^4) - (6*a*f^2*(e + f*x)*sin(c + d*x))/(b^2*d^3) + (a*(e + f*x)^3*sin(c + d*x))/(b^2*d) + (3*f^3*cos(c + d*x)*sin(c + d*x))/(8*b*d^4) - (3*f*(e + f*x)^2*cos(c + d*x)*sin(c + d*x))/(4*b*d^2) + (3*f^2*(e + f*x)*sin(c + d*x)^2)/(4*b*d^3) - ((e + f*x)^3*sin(c + d*x)^2)/(2*b*d)]
    @test_int [((e + f*x)^2*cos(c + d*x)^3)/(a + b*sin(c + d*x)), x, 16, (e*f*x)/(2*b*d) + (f^2*x^2)/(4*b*d) + ((im/3)*(a^2 - b^2)*(e + f*x)^3)/(b^3*f) + (2*a*f*(e + f*x)*cos(c + d*x))/(b^2*d^2) - ((a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d) - ((a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d) + ((2*im)*(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d^2) + ((2*im)*(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d^2) - (2*(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d^3) - (2*(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d^3) - (2*a*f^2*sin(c + d*x))/(b^2*d^3) + (a*(e + f*x)^2*sin(c + d*x))/(b^2*d) - (f*(e + f*x)*cos(c + d*x)*sin(c + d*x))/(2*b*d^2) + (f^2*sin(c + d*x)^2)/(4*b*d^3) - ((e + f*x)^2*sin(c + d*x)^2)/(2*b*d)]
    @test_int [((e + f*x)^1*cos(c + d*x)^3)/(a + b*sin(c + d*x)), x, 13, (f*x)/(4*b*d) + ((im/2)*(a^2 - b^2)*(e + f*x)^2)/(b^3*f) + (a*f*cos(c + d*x))/(b^2*d^2) - ((a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d) - ((a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d) + (im*(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b^3*d^2) + (im*(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b^3*d^2) + (a*(e + f*x)*sin(c + d*x))/(b^2*d) - (f*cos(c + d*x)*sin(c + d*x))/(4*b*d^2) - ((e + f*x)*sin(c + d*x)^2)/(2*b*d)]
    @test_int [cos(c + d*x)^3/(a + b*sin(c + d*x)), x, 3, -(((a^2 - b^2)*log(a + b*sin(c + d*x)))/(b^3*d)) + (a*sin(c + d*x))/(b^2*d) - sin(c + d*x)^2/(2*b*d)]

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

    @test_int [((e + f*x)^3*sec(c + d*x))/(a + b*sin(c + d*x)), x, 29, -((2*im*a*(e + f*x)^3*atan(ℯ^(im*(c + d*x))))/((a^2 - b^2)*d)) - (b*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d) - (b*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d) + (b*(e + f*x)^3*log(1 + ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d) + (3*im*a*f*(e + f*x)^2*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) - (3*im*a*f*(e + f*x)^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) + (3*im*b*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d^2) + (3*im*b*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d^2) - (3*im*b*f*(e + f*x)^2*PolyLog(2, -ℯ^(2*im*(c + d*x))))/(2*(a^2 - b^2)*d^2) - (6*a*f^2*(e + f*x)*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) + (6*a*f^2*(e + f*x)*PolyLog(3, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d^3) - (6*b*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d^3) + (3*b*f^2*(e + f*x)*PolyLog(3, -ℯ^(2*im*(c + d*x))))/(2*(a^2 - b^2)*d^3) - (6*im*a*f^3*PolyLog(4, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^4) + (6*im*a*f^3*PolyLog(4, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^4) - (6*im*b*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d^4) - (6*im*b*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d^4) + (3*im*b*f^3*PolyLog(4, -ℯ^(2*im*(c + d*x))))/(4*(a^2 - b^2)*d^4)]
    @test_int [((e + f*x)^2*sec(c + d*x))/(a + b*sin(c + d*x)), x, 24, -((2*im*a*(e + f*x)^2*atan(ℯ^(im*(c + d*x))))/((a^2 - b^2)*d)) - (b*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d) - (b*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d) + (b*(e + f*x)^2*log(1 + ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d) + (2*im*a*f*(e + f*x)*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) - (2*im*a*f*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) + (2*im*b*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d^2) + (2*im*b*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d^2) - (im*b*f*(e + f*x)*PolyLog(2, -ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^2) - (2*a*f^2*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) + (2*a*f^2*PolyLog(3, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) - (2*b*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d^3) - (2*b*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d^3) + (b*f^2*PolyLog(3, -ℯ^(2*im*(c + d*x))))/(2*(a^2 - b^2)*d^3)]
    @test_int [((e + f*x)^1*sec(c + d*x))/(a + b*sin(c + d*x)), x, 19, -((2*im*a*(e + f*x)*atan(ℯ^(im*(c + d*x))))/((a^2 - b^2)*d)) - (b*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d) - (b*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d) + (b*(e + f*x)*log(1 + ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d) + (im*a*f*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) - (im*a*f*PolyLog(2, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) + (im*b*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)*d^2) + (im*b*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)*d^2) - (im*b*f*PolyLog(2, -ℯ^(2*im*(c + d*x))))/(2*(a^2 - b^2)*d^2)]
    @test_int [sec(c + d*x)/(a + b*sin(c + d*x)), x, 6, -(log(1 - sin(c + d*x))/(2*(a + b)*d)) + log(1 + sin(c + d*x))/(2*(a - b)*d) - (b*log(a + b*sin(c + d*x)))/((a^2 - b^2)*d)]

    @test_int [((e + f*x)^3*sec(c + d*x)^2)/(a + b*sin(c + d*x)), x, 29, If(13>=8, -((im*a*(e + f*x)^3)/((a^2 - b^2)*d)) - (6*im*b*f*(e + f*x)^2*atan(ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) + (im*b^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) - (im*b^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) + (3*a*f*(e + f*x)^2*log(1 + ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^2) + (6*im*b*f^2*(e + f*x)*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) - (6*im*b*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) + (3*b^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (3*im*a*f^2*(e + f*x)*PolyLog(2, -ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^3) - (6*b*f^3*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^4) + (6*b*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^4) + (6*im*b^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) - (6*im*b^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) + (3*a*f^3*PolyLog(3, -ℯ^(2*im*(c + d*x))))/(2*(a^2 - b^2)*d^4) - (6*b^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^4) + (6*b^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^4) - (b*(e + f*x)^3*sec(c + d*x))/((a^2 - b^2)*d) + (a*(e + f*x)^3*tan(c + d*x))/((a^2 - b^2)*d), -((im*a*(e + f*x)^3)/((a^2 - b^2)*d)) - (6*im*b*f*(e + f*x)^2*atan(ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) + (im*b^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) - (im*b^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) + (3*a*f*(e + f*x)^2*log(1 + ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^2) + (6*im*b*f^2*(e + f*x)*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) - (6*im*b*f^2*(e + f*x)*PolyLog(2, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) + (3*b^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (3*b^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (3*im*a*f^2*(e + f*x)*PolyLog(2, -ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^3) - (6*b*f^3*PolyLog(3, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^4) + (6*b*f^3*PolyLog(3, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^4) + (6*im*b^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) - (6*im*b^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) + (3*a*f^3*PolyLog(3, -ℯ^(2*im*(c + d*x))))/(2*(a^2 - b^2)*d^4) - (6*b^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^4) + (6*b^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^4) - (b*(e + f*x)^3*sec(c + d*x))/((a^2 - b^2)*d) + (a*(e + f*x)^3*tan(c + d*x))/((a^2 - b^2)*d))]
    @test_int [((e + f*x)^2*sec(c + d*x)^2)/(a + b*sin(c + d*x)), x, 24, If(13>=8, -((im*a*(e + f*x)^2)/((a^2 - b^2)*d)) - (4*im*b*f*(e + f*x)*atan(ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) + (im*b^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) - (im*b^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) + (2*a*f*(e + f*x)*log(1 + ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^2) + (2*im*b*f^2*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) - (2*im*b*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) + (2*b^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (im*a*f^2*PolyLog(2, -ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^3) + (2*im*b^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) - (2*im*b^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) - (b*(e + f*x)^2*sec(c + d*x))/((a^2 - b^2)*d) + (a*(e + f*x)^2*tan(c + d*x))/((a^2 - b^2)*d), -((im*a*(e + f*x)^2)/((a^2 - b^2)*d)) - (4*im*b*f*(e + f*x)*atan(ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^2) + (im*b^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) - (im*b^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) + (2*a*f*(e + f*x)*log(1 + ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^2) + (2*im*b*f^2*PolyLog(2, (-im)*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) - (2*im*b*f^2*PolyLog(2, im*ℯ^(im*(c + d*x))))/((a^2 - b^2)*d^3) + (2*b^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (2*b^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (im*a*f^2*PolyLog(2, -ℯ^(2*im*(c + d*x))))/((a^2 - b^2)*d^3) + (2*im*b^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) - (2*im*b^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^3) - (b*(e + f*x)^2*sec(c + d*x))/((a^2 - b^2)*d) + (a*(e + f*x)^2*tan(c + d*x))/((a^2 - b^2)*d))]
    @test_int [((e + f*x)^1*sec(c + d*x)^2)/(a + b*sin(c + d*x)), x, 15, If(13>=8, (b*f*atanh(sin(c + d*x)))/((a^2 - b^2)*d^2) + (im*b^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) - (im*b^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) + (a*f*log(cos(c + d*x)))/((a^2 - b^2)*d^2) + (b^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (b^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (b*(e + f*x)*sec(c + d*x))/((a^2 - b^2)*d) + (a*(e + f*x)*tan(c + d*x))/((a^2 - b^2)*d), (b*f*atanh(sin(c + d*x)))/((a^2 - b^2)*d^2) + (im*b^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) - (im*b^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d) + (a*f*log(cos(c + d*x)))/((a^2 - b^2)*d^2) + (b^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (b^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/((a^2 - b^2)^(3/2)*d^2) - (b*(e + f*x)*sec(c + d*x))/((a^2 - b^2)*d) + (a*(e + f*x)*tan(c + d*x))/((a^2 - b^2)*d))]
    @test_int [sec(c + d*x)^2/(a + b*sin(c + d*x)), x, 5, -((2*b^2*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/((a^2 - b^2)^(3/2)*d)) - (sec(c + d*x)*(b - a*sin(c + d*x)))/((a^2 - b^2)*d)]

    #= ::Subsubsection::Closed:: =#
    #=m*symbolic=#

    @test_int [(e + f*x)^m*cos(c + d*x)^2/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*cos(c + d*x)^2)/(a + b*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m*cos(c + d*x)^1/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*cos(c + d*x))/(a + b*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m*cos(c + d*x)^0/(a + b*sin(c + d*x)), x, 0, Unintegrable((e + f*x)^m/(a + b*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m*sec(c + d*x)^1/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sec(c + d*x))/(a + b*sin(c + d*x)), x)]
    @test_int [(e + f*x)^m*sec(c + d*x)^2/(a + b*sin(c + d*x)), x, 0, Unintegrable(((e + f*x)^m*sec(c + d*x)^2)/(a + b*sin(c + d*x)), x)]

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

    @test_int [(e + f*x)^1*cos(c + d*x)/(a + b*sin(c + d*x))^2, x, 4, (2*f*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*sqrt(a^2 - b^2)*d^2) - (e + f*x)/(b*d*(a + b*sin(c + d*x)))]
    @test_int [(e + f*x)^2*cos(c + d*x)/(a + b*sin(c + d*x))^2, x, 9, -((2*im*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2)) + (2*im*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - (2*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) + (2*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - (e + f*x)^2/(b*d*(a + b*sin(c + d*x)))]
    @test_int [(e + f*x)^3*cos(c + d*x)/(a + b*sin(c + d*x))^2, x, 11, -((3*im*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2)) + (3*im*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^2) - (6*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) + (6*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^3) - (6*im*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^4) + (6*im*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*sqrt(a^2 - b^2)*d^4) - (e + f*x)^3/(b*d*(a + b*sin(c + d*x)))]

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

    @test_int [(e + f*x)^1*cos(c + d*x)/(a + b*sin(c + d*x))^3, x, 6, (a*f*atan((b + a*tan((1/2)*(c + d*x)))/sqrt(a^2 - b^2)))/(b*(a^2 - b^2)^(3/2)*d^2) - (e + f*x)/(2*b*d*(a + b*sin(c + d*x))^2) + (f*cos(c + d*x))/(2*(a^2 - b^2)*d^2*(a + b*sin(c + d*x)))]
    @test_int [(e + f*x)^2*cos(c + d*x)/(a + b*sin(c + d*x))^3, x, 12, -((im*a*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2)) + (im*a*f*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^2) - (f^2*log(a + b*sin(c + d*x)))/(b*(a^2 - b^2)*d^3) - (a*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) + (a*f^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) - (e + f*x)^2/(2*b*d*(a + b*sin(c + d*x))^2) + (f*(e + f*x)*cos(c + d*x))/((a^2 - b^2)*d^2*(a + b*sin(c + d*x)))]
    @test_int [(e + f*x)^3*cos(c + d*x)/(a + b*sin(c + d*x))^3, x, 19, (3*im*f*(e + f*x)^2)/(2*b*(a^2 - b^2)*d^2) - (3*f^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) - (3*im*a*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(3/2)*d^2) - (3*f^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^3) + (3*im*a*f*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(2*b*(a^2 - b^2)^(3/2)*d^2) + (3*im*f^3*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^4) - (3*a*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) + (3*im*f^3*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)*d^4) + (3*a*f^2*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^3) - (3*im*a*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) + (3*im*a*f^3*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(b*(a^2 - b^2)^(3/2)*d^4) - (e + f*x)^3/(2*b*d*(a + b*sin(c + d*x))^2) + (3*f*(e + f*x)^2*cos(c + d*x))/(2*(a^2 - b^2)*d^2*(a + b*sin(c + d*x)))]

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

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*(e+f*x)^m*cos(c+d*x)^n*cot(c+d*x)^p / (a+b*sin(c+d*x))^1=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*(e+f*x)^m*cos(c+d*x)^n*cot(c+d*x)^1 / (a+b*sin(c+d*x))=#

    @test_int [((e + f*x)^3*cos(c + d*x)*cot(c + d*x))/(a + b*sin(c + d*x)), x, 33, -(e + f*x)^4/(4*b*f) - (2*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a*d) - (im*sqrt(a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d) + (im*sqrt(a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d) + ((3*im)*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((3*im)*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) - (3*sqrt(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d^2) + (3*sqrt(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d^2) - (6*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (6*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) - ((6*im)*sqrt(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d^3) + ((6*im)*sqrt(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d^3) - ((6*im)*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a*d^4) + ((6*im)*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a*d^4) + (6*sqrt(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d^4) - (6*sqrt(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d^4)]
    @test_int [((e + f*x)^2*cos(c + d*x)*cot(c + d*x))/(a + b*sin(c + d*x)), x, 27, -(e + f*x)^3/(3*b*f) - (2*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d) - (im*sqrt(a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d) + (im*sqrt(a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d) + ((2*im)*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((2*im)*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) - (2*sqrt(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d^2) + (2*sqrt(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d^2) - (2*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (2*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) - ((2*im)*sqrt(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d^3) + ((2*im)*sqrt(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d^3)]
    @test_int [((e + f*x)^1*cos(c + d*x)*cot(c + d*x))/(a + b*sin(c + d*x)), x, 21, -((e*x)/b) - (f*x^2)/(2*b) - (2*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d) - (im*sqrt(a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d) + (im*sqrt(a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d) + (im*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - (im*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) - (sqrt(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b*d^2) + (sqrt(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b*d^2)]
    @test_int [(cos(c + d*x)*cot(c + d*x))/(a + b*sin(c + d*x)), x, 6, -(x/b) + (2*sqrt(a^2 - b^2)*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a*b*d) - atanh(cos(c + d*x))/(a*d)]

    @test_int [((e + f*x)^3*cos(c + d*x)^2*cot(c + d*x))/(a + b*sin(c + d*x)), x, 34, ((-im/4)*(e + f*x)^4)/(a*f) - ((im/4)*(a^2 - b^2)*(e + f*x)^4)/(a*b^2*f) + (6*f^3*cos(c + d*x))/(b*d^4) - (3*f*(e + f*x)^2*cos(c + d*x))/(b*d^2) + ((a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d) + ((e + f*x)^3*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d) - ((3*im)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d^2) - ((3*im)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d^2) - (((3*im)/2)*f*(e + f*x)^2*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^2) + (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d^3) + (3*f^2*(e + f*x)*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a*d^3) + ((6*im)*(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d^4) + ((6*im)*(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d^4) + (((3*im)/4)*f^3*PolyLog(4, ℯ^((2*im)*(c + d*x))))/(a*d^4) + (6*f^2*(e + f*x)*sin(c + d*x))/(b*d^3) - ((e + f*x)^3*sin(c + d*x))/(b*d)]
    @test_int [((e + f*x)^2*cos(c + d*x)^2*cot(c + d*x))/(a + b*sin(c + d*x)), x, 26, ((-im/3)*(e + f*x)^3)/(a*f) - ((im/3)*(a^2 - b^2)*(e + f*x)^3)/(a*b^2*f) - (2*f*(e + f*x)*cos(c + d*x))/(b*d^2) + ((a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d) + ((e + f*x)^2*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d) - ((2*im)*(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d^2) - ((2*im)*(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d^2) - (im*f*(e + f*x)*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^2) + (2*(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d^3) + (2*(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d^3) + (f^2*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a*d^3) + (2*f^2*sin(c + d*x))/(b*d^3) - ((e + f*x)^2*sin(c + d*x))/(b*d)]
    @test_int [((e + f*x)^1*cos(c + d*x)^2*cot(c + d*x))/(a + b*sin(c + d*x)), x, 22, ((-im/2)*(e + f*x)^2)/(a*f) - ((im/2)*(a^2 - b^2)*(e + f*x)^2)/(a*b^2*f) - (f*cos(c + d*x))/(b*d^2) + ((a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d) + ((a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d) + ((e + f*x)*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d) - (im*(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^2*d^2) - (im*(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^2*d^2) - ((im/2)*f*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^2) - ((e + f*x)*sin(c + d*x))/(b*d)]
    @test_int [(cos(c + d*x)^2*cot(c + d*x))/(a + b*sin(c + d*x)), x, 4, log(sin(c + d*x))/(a*d) + ((a^2 - b^2)*log(a + b*sin(c + d*x)))/(a*b^2*d) - sin(c + d*x)/(b*d)]

    @test_int [((e + f*x)^3*cos(c + d*x)^3*cot(c + d*x))/(a + b*sin(c + d*x)), x, 53, (3*e*f^2*x)/(4*b*d^2) + (3*f^3*x^2)/(8*b*d^2) - (e + f*x)^4/(8*b*f) + ((a^2 - b^2)*(e + f*x)^4)/(4*b^3*f) - (2*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a*d) - (6*f^2*(e + f*x)*cos(c + d*x))/(a*d^3) - (6*(a^2 - b^2)*f^2*(e + f*x)*cos(c + d*x))/(a*b^2*d^3) + ((e + f*x)^3*cos(c + d*x))/(a*d) + ((a^2 - b^2)*(e + f*x)^3*cos(c + d*x))/(a*b^2*d) + (3*f^3*cos(c + d*x)^2)/(8*b*d^4) - (3*f*(e + f*x)^2*cos(c + d*x)^2)/(4*b*d^2) + (im*(a^2 - b^2)^(3/2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d) - (im*(a^2 - b^2)^(3/2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d) + ((3*im)*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((3*im)*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d^2) - (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d^2) - (6*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (6*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) + ((6*im)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d^3) - ((6*im)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d^3) - ((6*im)*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a*d^4) + ((6*im)*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a*d^4) - (6*(a^2 - b^2)^(3/2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d^4) + (6*(a^2 - b^2)^(3/2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d^4) + (6*f^3*sin(c + d*x))/(a*d^4) + (6*(a^2 - b^2)*f^3*sin(c + d*x))/(a*b^2*d^4) - (3*f*(e + f*x)^2*sin(c + d*x))/(a*d^2) - (3*(a^2 - b^2)*f*(e + f*x)^2*sin(c + d*x))/(a*b^2*d^2) + (3*f^2*(e + f*x)*cos(c + d*x)*sin(c + d*x))/(4*b*d^3) - ((e + f*x)^3*cos(c + d*x)*sin(c + d*x))/(2*b*d)]
    @test_int [((e + f*x)^2*cos(c + d*x)^3*cot(c + d*x))/(a + b*sin(c + d*x)), x, 41, (f^2*x)/(4*b*d^2) - (e + f*x)^3/(6*b*f) + ((a^2 - b^2)*(e + f*x)^3)/(3*b^3*f) - (2*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d) - (2*f^2*cos(c + d*x))/(a*d^3) - (2*(a^2 - b^2)*f^2*cos(c + d*x))/(a*b^2*d^3) + ((e + f*x)^2*cos(c + d*x))/(a*d) + ((a^2 - b^2)*(e + f*x)^2*cos(c + d*x))/(a*b^2*d) - (f*(e + f*x)*cos(c + d*x)^2)/(2*b*d^2) + (im*(a^2 - b^2)^(3/2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d) - (im*(a^2 - b^2)^(3/2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d) + ((2*im)*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - ((2*im)*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d^2) - (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d^2) - (2*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^3) + (2*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^3) + ((2*im)*(a^2 - b^2)^(3/2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d^3) - ((2*im)*(a^2 - b^2)^(3/2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d^3) - (2*f*(e + f*x)*sin(c + d*x))/(a*d^2) - (2*(a^2 - b^2)*f*(e + f*x)*sin(c + d*x))/(a*b^2*d^2) + (f^2*cos(c + d*x)*sin(c + d*x))/(4*b*d^3) - ((e + f*x)^2*cos(c + d*x)*sin(c + d*x))/(2*b*d)]
    @test_int [((e + f*x)^1*cos(c + d*x)^3*cot(c + d*x))/(a + b*sin(c + d*x)), x, 31, -(e*x)/(2*b) + ((a^2 - b^2)*e*x)/b^3 - (f*x^2)/(4*b) + ((a^2 - b^2)*f*x^2)/(2*b^3) - (2*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d) + ((e + f*x)*cos(c + d*x))/(a*d) + ((a^2 - b^2)*(e + f*x)*cos(c + d*x))/(a*b^2*d) - (f*cos(c + d*x)^2)/(4*b*d^2) + (im*(a^2 - b^2)^(3/2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d) - (im*(a^2 - b^2)^(3/2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d) + (im*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^2) - (im*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^2) + ((a^2 - b^2)^(3/2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a*b^3*d^2) - ((a^2 - b^2)^(3/2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a*b^3*d^2) - (f*sin(c + d*x))/(a*d^2) - ((a^2 - b^2)*f*sin(c + d*x))/(a*b^2*d^2) - ((e + f*x)*cos(c + d*x)*sin(c + d*x))/(2*b*d)]
    @test_int [(cos(c + d*x)^3*cot(c + d*x))/(a + b*sin(c + d*x)), x, 6, ((2*a^2 - 3*b^2)*x)/(2*b^3) - (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a*b^3*d) - atanh(cos(c + d*x))/(a*d) + (a*cos(c + d*x))/(b^2*d) - (cos(c + d*x)*sin(c + d*x))/(2*b*d)]

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

    @test_int [((e + f*x)^3*cos(c + d*x)*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 48, ((im/4)*b*(e + f*x)^4)/(a^2*f) + ((im/4)*(a^2 - b^2)*(e + f*x)^4)/(a^2*b*f) - (6*f*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d^2) - ((e + f*x)^3*csc(c + d*x))/(a*d) - ((a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d) - ((a^2 - b^2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d) - (b*(e + f*x)^3*log(1 - ℯ^((2*im)*(c + d*x))))/(a^2*d) + ((6*im)*f^2*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^3) - ((6*im)*f^2*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^3) + ((3*im)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d^2) + ((3*im)*(a^2 - b^2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d^2) + (((3*im)/2)*b*f*(e + f*x)^2*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a^2*d^2) - (6*f^3*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^4) + (6*f^3*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^4) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d^3) - (6*(a^2 - b^2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d^3) - (3*b*f^2*(e + f*x)*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a^2*d^3) - ((6*im)*(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d^4) - ((6*im)*(a^2 - b^2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d^4) - (((3*im)/4)*b*f^3*PolyLog(4, ℯ^((2*im)*(c + d*x))))/(a^2*d^4)]
    @test_int [((e + f*x)^2*cos(c + d*x)*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 37, ((im/3)*b*(e + f*x)^3)/(a^2*f) + ((im/3)*(a^2 - b^2)*(e + f*x)^3)/(a^2*b*f) - (4*f*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d^2) - ((e + f*x)^2*csc(c + d*x))/(a*d) - ((a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d) - ((a^2 - b^2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d) - (b*(e + f*x)^2*log(1 - ℯ^((2*im)*(c + d*x))))/(a^2*d) + ((2*im)*f^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^3) - ((2*im)*f^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^3) + ((2*im)*(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d^2) + ((2*im)*(a^2 - b^2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d^2) + (im*b*f*(e + f*x)*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a^2*d^2) - (2*(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d^3) - (2*(a^2 - b^2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d^3) - (b*f^2*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a^2*d^3)]
    @test_int [((e + f*x)^1*cos(c + d*x)*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 28, ((im/2)*b*(e + f*x)^2)/(a^2*f) + ((im/2)*(a^2 - b^2)*(e + f*x)^2)/(a^2*b*f) - (f*atanh(cos(c + d*x)))/(a*d^2) - ((e + f*x)*csc(c + d*x))/(a*d) - ((a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d) - ((a^2 - b^2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d) - (b*(e + f*x)*log(1 - ℯ^((2*im)*(c + d*x))))/(a^2*d) + (im*(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b*d^2) + (im*(a^2 - b^2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b*d^2) + ((im/2)*b*f*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a^2*d^2)]
    @test_int [(cos(c + d*x)*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - (b*log(sin(c + d*x)))/(a^2*d) - ((1 - b^2/a^2)*log(a + b*sin(c + d*x)))/(b*d)]

    @test_int [((e + f*x)^3*cos(c + d*x)^2*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 66, ((-im)*(e + f*x)^3)/(a*d) - (e + f*x)^4/(4*a*f) - ((a^2 - b^2)*(e + f*x)^4)/(4*a*b^2*f) + (2*b*(e + f*x)^3*atanh(ℯ^(im*(c + d*x))))/(a^2*d) + (6*b*f^2*(e + f*x)*cos(c + d*x))/(a^2*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*cos(c + d*x))/(a^2*b*d^3) - (b*(e + f*x)^3*cos(c + d*x))/(a^2*d) - ((a^2 - b^2)*(e + f*x)^3*cos(c + d*x))/(a^2*b*d) - ((e + f*x)^3*cot(c + d*x))/(a*d) - (im*(a^2 - b^2)^(3/2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d) + (im*(a^2 - b^2)^(3/2)*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d) + (3*f*(e + f*x)^2*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) - ((3*im)*b*f*(e + f*x)^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a^2*d^2) + ((3*im)*b*f*(e + f*x)^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a^2*d^2) - (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d^2) + (3*(a^2 - b^2)^(3/2)*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d^2) - ((3*im)*f^2*(e + f*x)*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) + (6*b*f^2*(e + f*x)*PolyLog(3, -ℯ^(im*(c + d*x))))/(a^2*d^3) - (6*b*f^2*(e + f*x)*PolyLog(3, ℯ^(im*(c + d*x))))/(a^2*d^3) - ((6*im)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d^3) + ((6*im)*(a^2 - b^2)^(3/2)*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d^3) + (3*f^3*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a*d^4) + ((6*im)*b*f^3*PolyLog(4, -ℯ^(im*(c + d*x))))/(a^2*d^4) - ((6*im)*b*f^3*PolyLog(4, ℯ^(im*(c + d*x))))/(a^2*d^4) + (6*(a^2 - b^2)^(3/2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d^4) - (6*(a^2 - b^2)^(3/2)*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d^4) - (6*b*f^3*sin(c + d*x))/(a^2*d^4) - (6*(a^2 - b^2)*f^3*sin(c + d*x))/(a^2*b*d^4) + (3*b*f*(e + f*x)^2*sin(c + d*x))/(a^2*d^2) + (3*(a^2 - b^2)*f*(e + f*x)^2*sin(c + d*x))/(a^2*b*d^2)]
    @test_int [((e + f*x)^2*cos(c + d*x)^2*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 53, ((-im)*(e + f*x)^2)/(a*d) - (e + f*x)^3/(3*a*f) - ((a^2 - b^2)*(e + f*x)^3)/(3*a*b^2*f) + (2*b*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a^2*d) + (2*b*f^2*cos(c + d*x))/(a^2*d^3) + (2*(a^2 - b^2)*f^2*cos(c + d*x))/(a^2*b*d^3) - (b*(e + f*x)^2*cos(c + d*x))/(a^2*d) - ((a^2 - b^2)*(e + f*x)^2*cos(c + d*x))/(a^2*b*d) - ((e + f*x)^2*cot(c + d*x))/(a*d) - (im*(a^2 - b^2)^(3/2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d) + (im*(a^2 - b^2)^(3/2)*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d) + (2*f*(e + f*x)*log(1 - ℯ^((2*im)*(c + d*x))))/(a*d^2) - ((2*im)*b*f*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a^2*d^2) + ((2*im)*b*f*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a^2*d^2) - (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d^2) + (2*(a^2 - b^2)^(3/2)*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d^2) - (im*f^2*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a*d^3) + (2*b*f^2*PolyLog(3, -ℯ^(im*(c + d*x))))/(a^2*d^3) - (2*b*f^2*PolyLog(3, ℯ^(im*(c + d*x))))/(a^2*d^3) - ((2*im)*(a^2 - b^2)^(3/2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d^3) + ((2*im)*(a^2 - b^2)^(3/2)*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d^3) + (2*b*f*(e + f*x)*sin(c + d*x))/(a^2*d^2) + (2*(a^2 - b^2)*f*(e + f*x)*sin(c + d*x))/(a^2*b*d^2)]
    @test_int [((e + f*x)^1*cos(c + d*x)^2*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 38, -((e*x)/a) + ((1 - a^2/b^2)*e*x)/a - (f*x^2)/(2*a) + ((1 - a^2/b^2)*f*x^2)/(2*a) + (2*b*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a^2*d) - (b*(e + f*x)*cos(c + d*x))/(a^2*d) - ((a^2 - b^2)*(e + f*x)*cos(c + d*x))/(a^2*b*d) - ((e + f*x)*cot(c + d*x))/(a*d) - (im*(a^2 - b^2)^(3/2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d) + (im*(a^2 - b^2)^(3/2)*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d) + (f*log(sin(c + d*x)))/(a*d^2) - (im*b*f*PolyLog(2, -ℯ^(im*(c + d*x))))/(a^2*d^2) + (im*b*f*PolyLog(2, ℯ^(im*(c + d*x))))/(a^2*d^2) - ((a^2 - b^2)^(3/2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^2*d^2) + ((a^2 - b^2)^(3/2)*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^2*d^2) + (b*f*sin(c + d*x))/(a^2*d^2) + ((a^2 - b^2)*f*sin(c + d*x))/(a^2*b*d^2)]
    @test_int [(cos(c + d*x)^2*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 6, -((a*x)/b^2) + (2*(a^2 - b^2)^(3/2)*atan((b + a*tan((c + d*x)/2))/sqrt(a^2 - b^2)))/(a^2*b^2*d) + (b*atanh(cos(c + d*x)))/(a^2*d) - cos(c + d*x)/(b*d) - cot(c + d*x)/(a*d)]

    @test_int [((e + f*x)^3*cos(c + d*x)^3*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 85, (3*b*f^3*x)/(8*a^2*d^3) + (3*(a^2 - b^2)*f^3*x)/(8*a^2*b*d^3) - (b*(e + f*x)^3)/(4*a^2*d) - ((a^2 - b^2)*(e + f*x)^3)/(4*a^2*b*d) + ((im/4)*b*(e + f*x)^4)/(a^2*f) - ((im/4)*(a^2 - b^2)^2*(e + f*x)^4)/(a^2*b^3*f) - (6*f*(e + f*x)^2*atanh(ℯ^(im*(c + d*x))))/(a*d^2) + (6*f^3*cos(c + d*x))/(a*d^4) + (6*(a^2 - b^2)*f^3*cos(c + d*x))/(a*b^2*d^4) - (3*f*(e + f*x)^2*cos(c + d*x))/(a*d^2) - (3*(a^2 - b^2)*f*(e + f*x)^2*cos(c + d*x))/(a*b^2*d^2) - ((e + f*x)^3*csc(c + d*x))/(a*d) + ((a^2 - b^2)^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d) + ((a^2 - b^2)^2*(e + f*x)^3*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d) - (b*(e + f*x)^3*log(1 - ℯ^((2*im)*(c + d*x))))/(a^2*d) + ((6*im)*f^2*(e + f*x)*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^3) - ((6*im)*f^2*(e + f*x)*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^3) - ((3*im)*(a^2 - b^2)^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d^2) - ((3*im)*(a^2 - b^2)^2*f*(e + f*x)^2*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d^2) + (((3*im)/2)*b*f*(e + f*x)^2*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a^2*d^2) - (6*f^3*PolyLog(3, -ℯ^(im*(c + d*x))))/(a*d^4) + (6*f^3*PolyLog(3, ℯ^(im*(c + d*x))))/(a*d^4) + (6*(a^2 - b^2)^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d^3) + (6*(a^2 - b^2)^2*f^2*(e + f*x)*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d^3) - (3*b*f^2*(e + f*x)*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a^2*d^3) + ((6*im)*(a^2 - b^2)^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d^4) + ((6*im)*(a^2 - b^2)^2*f^3*PolyLog(4, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d^4) - (((3*im)/4)*b*f^3*PolyLog(4, ℯ^((2*im)*(c + d*x))))/(a^2*d^4) + (6*f^2*(e + f*x)*sin(c + d*x))/(a*d^3) + (6*(a^2 - b^2)*f^2*(e + f*x)*sin(c + d*x))/(a*b^2*d^3) - ((e + f*x)^3*sin(c + d*x))/(a*d) - ((a^2 - b^2)*(e + f*x)^3*sin(c + d*x))/(a*b^2*d) - (3*b*f^3*cos(c + d*x)*sin(c + d*x))/(8*a^2*d^4) - (3*(a^2 - b^2)*f^3*cos(c + d*x)*sin(c + d*x))/(8*a^2*b*d^4) + (3*b*f*(e + f*x)^2*cos(c + d*x)*sin(c + d*x))/(4*a^2*d^2) + (3*(a^2 - b^2)*f*(e + f*x)^2*cos(c + d*x)*sin(c + d*x))/(4*a^2*b*d^2) - (3*b*f^2*(e + f*x)*sin(c + d*x)^2)/(4*a^2*d^3) - (3*(a^2 - b^2)*f^2*(e + f*x)*sin(c + d*x)^2)/(4*a^2*b*d^3) + (b*(e + f*x)^3*sin(c + d*x)^2)/(2*a^2*d) + ((a^2 - b^2)*(e + f*x)^3*sin(c + d*x)^2)/(2*a^2*b*d)]
    @test_int [((e + f*x)^2*cos(c + d*x)^3*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 60, -(b*e*f*x)/(2*a^2*d) - ((a^2 - b^2)*e*f*x)/(2*a^2*b*d) - (b*f^2*x^2)/(4*a^2*d) - ((a^2 - b^2)*f^2*x^2)/(4*a^2*b*d) + ((im/3)*b*(e + f*x)^3)/(a^2*f) - ((im/3)*(a^2 - b^2)^2*(e + f*x)^3)/(a^2*b^3*f) - (4*f*(e + f*x)*atanh(ℯ^(im*(c + d*x))))/(a*d^2) - (2*f*(e + f*x)*cos(c + d*x))/(a*d^2) - (2*(a^2 - b^2)*f*(e + f*x)*cos(c + d*x))/(a*b^2*d^2) - ((e + f*x)^2*csc(c + d*x))/(a*d) + ((a^2 - b^2)^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d) + ((a^2 - b^2)^2*(e + f*x)^2*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d) - (b*(e + f*x)^2*log(1 - ℯ^((2*im)*(c + d*x))))/(a^2*d) + ((2*im)*f^2*PolyLog(2, -ℯ^(im*(c + d*x))))/(a*d^3) - ((2*im)*f^2*PolyLog(2, ℯ^(im*(c + d*x))))/(a*d^3) - ((2*im)*(a^2 - b^2)^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d^2) - ((2*im)*(a^2 - b^2)^2*f*(e + f*x)*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d^2) + (im*b*f*(e + f*x)*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a^2*d^2) + (2*(a^2 - b^2)^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d^3) + (2*(a^2 - b^2)^2*f^2*PolyLog(3, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d^3) - (b*f^2*PolyLog(3, ℯ^((2*im)*(c + d*x))))/(2*a^2*d^3) + (2*f^2*sin(c + d*x))/(a*d^3) + (2*(a^2 - b^2)*f^2*sin(c + d*x))/(a*b^2*d^3) - ((e + f*x)^2*sin(c + d*x))/(a*d) - ((a^2 - b^2)*(e + f*x)^2*sin(c + d*x))/(a*b^2*d) + (b*f*(e + f*x)*cos(c + d*x)*sin(c + d*x))/(2*a^2*d^2) + ((a^2 - b^2)*f*(e + f*x)*cos(c + d*x)*sin(c + d*x))/(2*a^2*b*d^2) - (b*f^2*sin(c + d*x)^2)/(4*a^2*d^3) - ((a^2 - b^2)*f^2*sin(c + d*x)^2)/(4*a^2*b*d^3) + (b*(e + f*x)^2*sin(c + d*x)^2)/(2*a^2*d) + ((a^2 - b^2)*(e + f*x)^2*sin(c + d*x)^2)/(2*a^2*b*d)]
    @test_int [((e + f*x)^1*cos(c + d*x)^3*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 45, -(b*f*x)/(4*a^2*d) - ((a^2 - b^2)*f*x)/(4*a^2*b*d) + ((im/2)*b*(e + f*x)^2)/(a^2*f) - ((im/2)*(a^2 - b^2)^2*(e + f*x)^2)/(a^2*b^3*f) - (f*atanh(cos(c + d*x)))/(a*d^2) - (f*cos(c + d*x))/(a*d^2) - ((a^2 - b^2)*f*cos(c + d*x))/(a*b^2*d^2) - ((e + f*x)*csc(c + d*x))/(a*d) + ((a^2 - b^2)^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d) + ((a^2 - b^2)^2*(e + f*x)*log(1 - (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d) - (b*(e + f*x)*log(1 - ℯ^((2*im)*(c + d*x))))/(a^2*d) - (im*(a^2 - b^2)^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a - sqrt(a^2 - b^2))))/(a^2*b^3*d^2) - (im*(a^2 - b^2)^2*f*PolyLog(2, (im*b*ℯ^(im*(c + d*x)))/(a + sqrt(a^2 - b^2))))/(a^2*b^3*d^2) + ((im/2)*b*f*PolyLog(2, ℯ^((2*im)*(c + d*x))))/(a^2*d^2) - ((e + f*x)*sin(c + d*x))/(a*d) - ((a^2 - b^2)*(e + f*x)*sin(c + d*x))/(a*b^2*d) + (b*f*cos(c + d*x)*sin(c + d*x))/(4*a^2*d^2) + ((a^2 - b^2)*f*cos(c + d*x)*sin(c + d*x))/(4*a^2*b*d^2) + (b*(e + f*x)*sin(c + d*x)^2)/(2*a^2*d) + ((a^2 - b^2)*(e + f*x)*sin(c + d*x)^2)/(2*a^2*b*d)]
    @test_int [(cos(c + d*x)^3*cot(c + d*x)^2)/(a + b*sin(c + d*x)), x, 4, -(csc(c + d*x)/(a*d)) - (b*log(sin(c + d*x)))/(a^2*d) + ((a^2 - b^2)^2*log(a + b*sin(c + d*x)))/(a^2*b^3*d) - (a*sin(c + d*x))/(b^2*d) + sin(c + d*x)^2/(2*b*d)]
    end

max_stars_repo_path: test/4 Trig functions/4.1 Sine/4.1.10 (c+d x)^m (a+b sin)^n.jl