@testset "4.5.2.1 (a+b sec)^m (c+d sec)^n" begin
    (a, b, c, d, e, f, m, n, p, x, ) = @variables a b c d e f m n p x

    #= ::Package:: =#

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

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

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

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

    @test_int [(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^5, x, 15, a^2*c^5*x - (19*a^2*c^5*atanh(sin(e + f*x)))/(16*f) - (a^2*c^5*tan(e + f*x))/f + (17*a^2*c^5*sec(e + f*x)*tan(e + f*x))/(16*f) + (a^2*c^5*sec(e + f*x)^3*tan(e + f*x))/(8*f) + (a^2*c^5*tan(e + f*x)^3)/(3*f) - (3*a^2*c^5*sec(e + f*x)*tan(e + f*x)^3)/(4*f) - (a^2*c^5*sec(e + f*x)^3*tan(e + f*x)^3)/(6*f) + (3*a^2*c^5*tan(e + f*x)^5)/(5*f)]
    @test_int [(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^4, x, 11, a^2*c^4*x - (3*a^2*c^4*atanh(sin(e + f*x)))/(4*f) - (a^2*c^4*tan(e + f*x))/f + (3*a^2*c^4*sec(e + f*x)*tan(e + f*x))/(4*f) + (a^2*c^4*tan(e + f*x)^3)/(3*f) - (a^2*c^4*sec(e + f*x)*tan(e + f*x)^3)/(2*f) + (a^2*c^4*tan(e + f*x)^5)/(5*f)]
    @test_int [(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^3, x, 5, a^2*c^3*x - (3*a^2*c^3*atanh(sin(e + f*x)))/(8*f) - (a^2*(8*c^3 - 3*c^3*sec(e + f*x))*tan(e + f*x))/(8*f) + (a^2*(4*c^3 - 3*c^3*sec(e + f*x))*tan(e + f*x)^3)/(12*f)]
    @test_int [(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^2, x, 4, a^2*c^2*x - (a^2*c^2*tan(e + f*x))/f + (a^2*c^2*tan(e + f*x)^3)/(3*f)]
    @test_int [(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^1, x, 4, a^2*c*x + (a^2*c*atanh(sin(e + f*x)))/(2*f) - (c*(2*a^2 + a^2*sec(e + f*x))*tan(e + f*x))/(2*f)]
    @test_int [(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^1, x, 8, (a^2*x)/c - (a^2*atanh(sin(e + f*x)))/(c*f) - (4*a^2*tan(e + f*x))/(c*f*(1 - sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^2, x, 9, (a^2*x)/c^2 - (4*a^2*tan(e + f*x))/(3*c^2*f*(1 - sec(e + f*x))^2) - (4*a^2*tan(e + f*x))/(3*c^2*f*(1 - sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^3, x, 12, (a^2*x)/c^3 - (4*a^2*tan(e + f*x))/(5*c^3*f*(1 - sec(e + f*x))^3) - (8*a^2*tan(e + f*x))/(15*c^3*f*(1 - sec(e + f*x))^2) - (23*a^2*tan(e + f*x))/(15*c^3*f*(1 - sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^4, x, 15, (a^2*x)/c^4 - (4*a^2*tan(e + f*x))/(7*c^4*f*(1 - sec(e + f*x))^4) - (12*a^2*tan(e + f*x))/(35*c^4*f*(1 - sec(e + f*x))^3) - (59*a^2*tan(e + f*x))/(105*c^4*f*(1 - sec(e + f*x))^2) - (164*a^2*tan(e + f*x))/(105*c^4*f*(1 - sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^5, x, 18, (a^2*x)/c^5 - (4*a^2*tan(e + f*x))/(9*c^5*f*(1 - sec(e + f*x))^5) - (16*a^2*tan(e + f*x))/(63*c^5*f*(1 - sec(e + f*x))^4) - (37*a^2*tan(e + f*x))/(105*c^5*f*(1 - sec(e + f*x))^3) - (179*a^2*tan(e + f*x))/(315*c^5*f*(1 - sec(e + f*x))^2) - (494*a^2*tan(e + f*x))/(315*c^5*f*(1 - sec(e + f*x)))]

    @test_int [(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^5, x, 13, a^3*c^5*x - (5*a^3*c^5*atanh(sin(e + f*x)))/(8*f) - (a^3*c^5*tan(e + f*x))/f + (5*a^3*c^5*sec(e + f*x)*tan(e + f*x))/(8*f) + (a^3*c^5*tan(e + f*x)^3)/(3*f) - (5*a^3*c^5*sec(e + f*x)*tan(e + f*x)^3)/(12*f) - (a^3*c^5*tan(e + f*x)^5)/(5*f) + (a^3*c^5*sec(e + f*x)*tan(e + f*x)^5)/(3*f) - (a^3*c^5*tan(e + f*x)^7)/(7*f)]
    @test_int [(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^4, x, 6, a^3*c^4*x - (5*a^3*c^4*atanh(sin(e + f*x)))/(16*f) - (a^3*(16*c^4 - 5*c^4*sec(e + f*x))*tan(e + f*x))/(16*f) + (a^3*(8*c^4 - 5*c^4*sec(e + f*x))*tan(e + f*x)^3)/(24*f) - (a^3*(6*c^4 - 5*c^4*sec(e + f*x))*tan(e + f*x)^5)/(30*f)]
    @test_int [(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^3, x, 5, a^3*c^3*x - (a^3*c^3*tan(e + f*x))/f + (a^3*c^3*tan(e + f*x)^3)/(3*f) - (a^3*c^3*tan(e + f*x)^5)/(5*f)]
    @test_int [(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^2, x, 5, a^3*c^2*x + (3*a^3*c^2*atanh(sin(e + f*x)))/(8*f) - (c^2*(8*a^3 + 3*a^3*sec(e + f*x))*tan(e + f*x))/(8*f) + (c^2*(4*a^3 + 3*a^3*sec(e + f*x))*tan(e + f*x)^3)/(12*f)]
    @test_int [(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^1, x, 9, a^3*c*x + (a^3*c*atanh(sin(e + f*x)))/f - (a^3*c*tan(e + f*x))/f - (a^3*c*sec(e + f*x)*tan(e + f*x))/f - (a^3*c*tan(e + f*x)^3)/(3*f)]
    @test_int [(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^1, x, 15, (a^3*x)/c - (4*a^3*atanh(sin(e + f*x)))/(c*f) + (8*a^3*cot(e + f*x))/(c*f) + (8*a^3*csc(e + f*x))/(c*f) - (a^3*tan(e + f*x))/(c*f)]
    @test_int [(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^2, x, 13, (a^3*x)/c^2 + (a^3*atanh(sin(e + f*x)))/(c^2*f) - (8*a^3*tan(e + f*x))/(3*c^2*f*(1 - sec(e + f*x))^2) + (4*a^3*tan(e + f*x))/(3*c^2*f*(1 - sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^3, x, 15, (a^3*x)/c^3 - (8*a^3*tan(e + f*x))/(5*c^3*f*(1 - sec(e + f*x))^3) + (4*a^3*tan(e + f*x))/(15*c^3*f*(1 - sec(e + f*x))^2) - (26*a^3*tan(e + f*x))/(15*c^3*f*(1 - sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^4, x, 19, (a^3*x)/c^4 - (8*a^3*tan(e + f*x))/(7*c^4*f*(1 - sec(e + f*x))^4) + (4*a^3*tan(e + f*x))/(35*c^4*f*(1 - sec(e + f*x))^3) - (62*a^3*tan(e + f*x))/(105*c^4*f*(1 - sec(e + f*x))^2) - (167*a^3*tan(e + f*x))/(105*c^4*f*(1 - sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^5, x, 23, (a^3*x)/c^5 - (8*a^3*tan(e + f*x))/(9*c^5*f*(1 - sec(e + f*x))^5) + (4*a^3*tan(e + f*x))/(63*c^5*f*(1 - sec(e + f*x))^4) - (38*a^3*tan(e + f*x))/(105*c^5*f*(1 - sec(e + f*x))^3) - (181*a^3*tan(e + f*x))/(315*c^5*f*(1 - sec(e + f*x))^2) - (496*a^3*tan(e + f*x))/(315*c^5*f*(1 - sec(e + f*x)))]

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

    @test_int [1/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^5, x, 26, (c^5*x)/a^2 - (47*c^5*atanh(sin(e + f*x)))/(2*a^2*f) + (13*c^5*tan(e + f*x))/(2*a^2*f) + (112*c^5*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x))) - (32*c^5*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2) + ((c^5 - c^5*sec(e + f*x))*tan(e + f*x))/(2*a^2*f), (c^5*x)/a^2 - (47*c^5*atanh(sin(e + f*x)))/(2*a^2*f) - (48*c^5*cot(e + f*x))/(a^2*f) - (64*c^5*cot(e + f*x)^3)/(3*a^2*f) + (33*c^5*csc(e + f*x))/(2*a^2*f) + (131*c^5*csc(e + f*x)^3)/(6*a^2*f) - (c^5*csc(e + f*x)^3*sec(e + f*x)^2)/(2*a^2*f) + (7*c^5*tan(e + f*x))/(a^2*f)]
    @test_int [1/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^4, x, 21, (c^4*x)/a^2 - (6*c^4*atanh(sin(e + f*x)))/(a^2*f) - (16*c^4*cot(e + f*x))/(a^2*f) - (32*c^4*cot(e + f*x)^3)/(3*a^2*f) + (32*c^4*csc(e + f*x)^3)/(3*a^2*f) + (c^4*tan(e + f*x))/(a^2*f)]
    @test_int [1/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^3, x, 13, (c^3*x)/a^2 - (c^3*atanh(sin(e + f*x)))/(a^2*f) - (8*c^3*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x))^2) + (4*c^3*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^2, x, 9, (c^2*x)/a^2 - (4*c^2*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x))^2) - (4*c^2*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^1, x, 7, (c*x)/a^2 - (2*c*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x))^2) - (5*c*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^1, x, 4, x/(a^2*c) + (cot(e + f*x)*(3 - 2*sec(e + f*x)))/(3*a^2*c*f) - (cot(e + f*x)^3*(1 - sec(e + f*x)))/(3*a^2*c*f)]
    @test_int [1/(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^2, x, 4, x/(a^2*c^2) + cot(e + f*x)/(a^2*c^2*f) - cot(e + f*x)^3/(3*a^2*c^2*f)]
    @test_int [1/(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^3, x, 5, x/(a^2*c^3) + (cot(e + f*x)^5*(1 + sec(e + f*x)))/(5*a^2*c^3*f) - (cot(e + f*x)^3*(5 + 4*sec(e + f*x)))/(15*a^2*c^3*f) + (cot(e + f*x)*(15 + 8*sec(e + f*x)))/(15*a^2*c^3*f)]
    @test_int [1/(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^4, x, 13, x/(a^2*c^4) + cot(e + f*x)/(a^2*c^4*f) - cot(e + f*x)^3/(3*a^2*c^4*f) + cot(e + f*x)^5/(5*a^2*c^4*f) - (2*cot(e + f*x)^7)/(7*a^2*c^4*f) + (2*csc(e + f*x))/(a^2*c^4*f) - (2*csc(e + f*x)^3)/(a^2*c^4*f) + (6*csc(e + f*x)^5)/(5*a^2*c^4*f) - (2*csc(e + f*x)^7)/(7*a^2*c^4*f)]
    @test_int [1/(a + a*sec(e + f*x))^2/(c - c*sec(e + f*x))^5, x, 17, x/(a^2*c^5) + cot(e + f*x)/(a^2*c^5*f) - cot(e + f*x)^3/(3*a^2*c^5*f) + cot(e + f*x)^5/(5*a^2*c^5*f) - cot(e + f*x)^7/(7*a^2*c^5*f) + (4*cot(e + f*x)^9)/(9*a^2*c^5*f) + (3*csc(e + f*x))/(a^2*c^5*f) - (13*csc(e + f*x)^3)/(3*a^2*c^5*f) + (21*csc(e + f*x)^5)/(5*a^2*c^5*f) - (15*csc(e + f*x)^7)/(7*a^2*c^5*f) + (4*csc(e + f*x)^9)/(9*a^2*c^5*f)]

    @test_int [1/(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^5, x, 29, (c^5*x)/a^3 + (8*c^5*atanh(sin(e + f*x)))/(a^3*f) + (32*c^5*cot(e + f*x))/(a^3*f) + (128*c^5*cot(e + f*x)^3)/(3*a^3*f) + (128*c^5*cot(e + f*x)^5)/(5*a^3*f) - (16*c^5*csc(e + f*x))/(a^3*f) + (64*c^5*csc(e + f*x)^3)/(3*a^3*f) - (128*c^5*csc(e + f*x)^5)/(5*a^3*f) - (c^5*tan(e + f*x))/(a^3*f)]
    @test_int [1/(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^4, x, 20, (c^4*x)/a^3 + (c^4*atanh(sin(e + f*x)))/(a^3*f) - (3*c^4*tan(e + f*x))/(a^3*f*(1 + sec(e + f*x))^3) - (c^4*sec(e + f*x)^2*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x))^3) + (14*c^4*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x))^2) - (23*c^4*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^3, x, 15, (c^3*x)/a^3 - (8*c^3*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x))^3) + (4*c^3*tan(e + f*x))/(15*a^3*f*(1 + sec(e + f*x))^2) - (26*c^3*tan(e + f*x))/(15*a^3*f*(1 + sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^2, x, 12, (c^2*x)/a^3 - (4*c^2*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x))^3) - (8*c^2*tan(e + f*x))/(15*a^3*f*(1 + sec(e + f*x))^2) - (23*c^2*tan(e + f*x))/(15*a^3*f*(1 + sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^1, x, 9, (c*x)/a^3 - (2*c*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x))^3) - (3*c*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x))^2) - (8*c*tan(e + f*x))/(5*a^3*f*(1 + sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^1, x, 12, x/(a^3*c) + cot(e + f*x)/(a^3*c*f) - cot(e + f*x)^3/(3*a^3*c*f) + (2*cot(e + f*x)^5)/(5*a^3*c*f) - (2*csc(e + f*x))/(a^3*c*f) + (4*csc(e + f*x)^3)/(3*a^3*c*f) - (2*csc(e + f*x)^5)/(5*a^3*c*f)]
    @test_int [1/(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^2, x, 5, x/(a^3*c^2) + (cot(e + f*x)*(15 - 8*sec(e + f*x)))/(15*a^3*c^2*f) - (cot(e + f*x)^3*(5 - 4*sec(e + f*x)))/(15*a^3*c^2*f) + (cot(e + f*x)^5*(1 - sec(e + f*x)))/(5*a^3*c^2*f)]
    @test_int [1/(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^3, x, 5, x/(a^3*c^3) + cot(e + f*x)/(a^3*c^3*f) - cot(e + f*x)^3/(3*a^3*c^3*f) + cot(e + f*x)^5/(5*a^3*c^3*f)]
    @test_int [1/(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^4, x, 6, x/(a^3*c^4) - (cot(e + f*x)^7*(1 + sec(e + f*x)))/(7*a^3*c^4*f) + (cot(e + f*x)^5*(7 + 6*sec(e + f*x)))/(35*a^3*c^4*f) + (cot(e + f*x)*(35 + 16*sec(e + f*x)))/(35*a^3*c^4*f) - (cot(e + f*x)^3*(35 + 24*sec(e + f*x)))/(105*a^3*c^4*f)]
    @test_int [1/(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^5, x, 14, x/(a^3*c^5) + cot(e + f*x)/(a^3*c^5*f) - cot(e + f*x)^3/(3*a^3*c^5*f) + cot(e + f*x)^5/(5*a^3*c^5*f) - cot(e + f*x)^7/(7*a^3*c^5*f) + (2*cot(e + f*x)^9)/(9*a^3*c^5*f) + (2*csc(e + f*x))/(a^3*c^5*f) - (8*csc(e + f*x)^3)/(3*a^3*c^5*f) + (12*csc(e + f*x)^5)/(5*a^3*c^5*f) - (8*csc(e + f*x)^7)/(7*a^3*c^5*f) + (2*csc(e + f*x)^9)/(9*a^3*c^5*f)]
    @test_int [1/(a + a*sec(e + f*x))^3/(c - c*sec(e + f*x))^6, x, 18, x/(a^3*c^6) + cot(e + f*x)/(a^3*c^6*f) - cot(e + f*x)^3/(3*a^3*c^6*f) + cot(e + f*x)^5/(5*a^3*c^6*f) - cot(e + f*x)^7/(7*a^3*c^6*f) + cot(e + f*x)^9/(9*a^3*c^6*f) - (4*cot(e + f*x)^11)/(11*a^3*c^6*f) + (3*csc(e + f*x))/(a^3*c^6*f) - (16*csc(e + f*x)^3)/(3*a^3*c^6*f) + (34*csc(e + f*x)^5)/(5*a^3*c^6*f) - (36*csc(e + f*x)^7)/(7*a^3*c^6*f) + (19*csc(e + f*x)^9)/(9*a^3*c^6*f) - (4*csc(e + f*x)^11)/(11*a^3*c^6*f)]

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

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

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

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

    @test_int [(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^3, x, 5, (2*a^(5/2)*c^3*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/f - (2*a^3*c^3*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) + (2*a^4*c^3*tan(e + f*x)^3)/(3*f*(a + a*sec(e + f*x))^(3/2)) - (2*a^5*c^3*tan(e + f*x)^5)/(5*f*(a + a*sec(e + f*x))^(5/2)) - (6*a^6*c^3*tan(e + f*x)^7)/(7*f*(a + a*sec(e + f*x))^(7/2)) - (2*a^7*c^3*tan(e + f*x)^9)/(9*f*(a + a*sec(e + f*x))^(9/2))]
    @test_int [(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^2, x, 5, (2*a^(5/2)*c^2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/f - (2*a^3*c^2*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) + (2*a^4*c^2*tan(e + f*x)^3)/(3*f*(a + a*sec(e + f*x))^(3/2)) + (6*a^5*c^2*tan(e + f*x)^5)/(5*f*(a + a*sec(e + f*x))^(5/2)) + (2*a^6*c^2*tan(e + f*x)^7)/(7*f*(a + a*sec(e + f*x))^(7/2))]
    @test_int [(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^1, x, 5, (2*a^(5/2)*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/f - (2*a^3*c*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) - (2*a^4*c*tan(e + f*x)^3)/(f*(a + a*sec(e + f*x))^(3/2)) - (2*a^5*c*tan(e + f*x)^5)/(5*f*(a + a*sec(e + f*x))^(5/2))]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c - c*sec(e + f*x))^1, x, 5, (2*a^(5/2)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c*f) + (8*a^2*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c*f) - (2*a^3*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c - c*sec(e + f*x))^2, x, 5, (2*a^(5/2)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c^2*f) - (8*a*cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(3*c^2*f)]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c - c*sec(e + f*x))^3, x, 5, (2*a^(5/2)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c^3*f) + (2*a^2*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c^3*f) + (8*cot(e + f*x)^5*(a + a*sec(e + f*x))^(5/2))/(5*c^3*f)]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c - c*sec(e + f*x))^4, x, 5, (2*a^(5/2)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c^4*f) + (2*a^2*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c^4*f) - (2*a*cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(3*c^4*f) - (8*cot(e + f*x)^7*(a + a*sec(e + f*x))^(7/2))/(7*a*c^4*f)]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c - c*sec(e + f*x))^5, x, 5, (2*a^(5/2)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c^5*f) + (2*a^2*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(c^5*f) - (2*a*cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(3*c^5*f) + (2*cot(e + f*x)^5*(a + a*sec(e + f*x))^(5/2))/(5*c^5*f) + (8*cot(e + f*x)^9*(a + a*sec(e + f*x))^(9/2))/(9*a^2*c^5*f)]

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

    @test_int [1/sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^4, x, 8, (2*c^4*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*f) - (16*sqrt(2)*c^4*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*f) + (14*c^4*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) - (2*a*c^4*tan(e + f*x)^3)/(f*(a + a*sec(e + f*x))^(3/2)) + (2*a^2*c^4*tan(e + f*x)^5)/(5*f*(a + a*sec(e + f*x))^(5/2))]
    @test_int [1/sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^3, x, 7, (2*c^3*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*f) - (8*sqrt(2)*c^3*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*f) + (6*c^3*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) - (2*a*c^3*tan(e + f*x)^3)/(3*f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [1/sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^2, x, 6, (2*c^2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*f) - (4*sqrt(2)*c^2*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*f) + (2*c^2*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^1, x, 5, (2*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*f) - (2*sqrt(2)*c*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(a)*f)]
    @test_int [1/sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^1, x, 6, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*c*f) - atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))))/(sqrt(2)*sqrt(a)*c*f) + (cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(a*c*f)]
    @test_int [1/sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^2, x, 7, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*c^2*f) - atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))))/(2*sqrt(2)*sqrt(a)*c^2*f) + (3*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(2*a*c^2*f) - (cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(3*a^2*c^2*f)]
    @test_int [1/sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^3, x, 8, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(sqrt(a)*c^3*f) - atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x))))/(4*sqrt(2)*sqrt(a)*c^3*f) + (7*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(4*a*c^3*f) - (cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(2*a^2*c^3*f) + (cot(e + f*x)^5*(a + a*sec(e + f*x))^(5/2))/(5*a^3*c^3*f)]

    @test_int [1/(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^4, x, 8, (2*c^4*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*f) + (12*sqrt(2)*c^4*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(a^(3/2)*f) - (14*c^4*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))) + (8*c^4*tan(e + f*x)^3)/(3*f*(a + a*sec(e + f*x))^(3/2)) - (a*c^4*sec((1/2)*(e + f*x))^2*sin(e + f*x)*tan(e + f*x)^4)/(f*(a + a*sec(e + f*x))^(5/2))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^3, x, 7, (2*c^3*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*f) + (2*sqrt(2)*c^3*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(a^(3/2)*f) - (4*c^3*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))) + (c^3*sec((1/2)*(e + f*x))^2*sin(e + f*x)*tan(e + f*x)^2)/(f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^2, x, 7, (2*c^2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*f) - (sqrt(2)*c^2*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(a^(3/2)*f) - (2*c^2*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(3/2)), (2*c^2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*f) - (sqrt(2)*c^2*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(a^(3/2)*f) - (c^2*sec((1/2)*(e + f*x))^2*sin(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^1, x, 6, (2*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*f) - (3*c*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(2)*a^(3/2)*f) - (c*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(3/2)), (2*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*f) - (3*c*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(2)*a^(3/2)*f) - (c*sec((1/2)*(e + f*x))^2*sin(e + f*x))/(2*a*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)/(c - c*sec(e + f*x))^1, x, 7, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*c*f) - (7*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(4*sqrt(2)*a^(3/2)*c*f) + (cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(4*a^2*c*f) + (cos(e + f*x)*cot(e + f*x)*sec((1/2)*(e + f*x))^2*sqrt(a + a*sec(e + f*x)))/(4*a^2*c*f)]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)/(c - c*sec(e + f*x))^2, x, 8, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*c^2*f) - (9*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(8*sqrt(2)*a^(3/2)*c^2*f) + (7*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(8*a^2*c^2*f) + (cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(12*a^3*c^2*f) - (cos(e + f*x)*cot(e + f*x)^3*sec((1/2)*(e + f*x))^2*(a + a*sec(e + f*x))^(3/2))/(4*a^3*c^2*f)]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)/(c - c*sec(e + f*x))^3, x, 9, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*c^3*f) - (11*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(16*sqrt(2)*a^(3/2)*c^3*f) + (21*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(16*a^2*c^3*f) - (5*cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(24*a^3*c^3*f) - (3*cot(e + f*x)^5*(a + a*sec(e + f*x))^(5/2))/(20*a^4*c^3*f) + (cos(e + f*x)*cot(e + f*x)^5*sec((1/2)*(e + f*x))^2*(a + a*sec(e + f*x))^(5/2))/(4*a^4*c^3*f)]

    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^5, x, 9, (2*c^5*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*f) - (23*sqrt(2)*c^5*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(a^(5/2)*f) + (21*c^5*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))) - (19*c^5*tan(e + f*x)^3)/(6*a*f*(a + a*sec(e + f*x))^(3/2)) + (3*c^5*sec((1/2)*(e + f*x))^2*sin(e + f*x)*tan(e + f*x)^4)/(4*f*(a + a*sec(e + f*x))^(5/2)) + (a*c^5*sec((1/2)*(e + f*x))^4*sin(e + f*x)^2*tan(e + f*x)^5)/(4*f*(a + a*sec(e + f*x))^(7/2))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^4, x, 8, (2*c^4*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*f) - (11*c^4*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(sqrt(2)*a^(5/2)*f) + (7*c^4*tan(e + f*x))/(2*a^2*f*sqrt(a + a*sec(e + f*x))) - (c^4*sec((1/2)*(e + f*x))^2*sin(e + f*x)*tan(e + f*x)^2)/(4*a*f*(a + a*sec(e + f*x))^(3/2)) - (c^4*sec((1/2)*(e + f*x))^4*sin(e + f*x)^2*tan(e + f*x)^3)/(4*f*(a + a*sec(e + f*x))^(5/2))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^3, x, 7, (2*c^3*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*f) - (7*c^3*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(2*sqrt(2)*a^(5/2)*f) - (c^3*sec((1/2)*(e + f*x))^2*sin(e + f*x))/(4*a^2*f*sqrt(a + a*sec(e + f*x))) + (c^3*sec((1/2)*(e + f*x))^4*sin(e + f*x)^2*tan(e + f*x))/(4*a*f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^2, x, 7, (2*c^2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*f) - (11*c^2*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(4*sqrt(2)*a^(5/2)*f) - (3*c^2*sec((1/2)*(e + f*x))^2*sin(e + f*x))/(8*a^2*f*sqrt(a + a*sec(e + f*x))) - (c^2*cos(e + f*x)*sec((1/2)*(e + f*x))^4*sin(e + f*x))/(4*a^2*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^1, x, 7, (2*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*f) - (23*c*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(8*sqrt(2)*a^(5/2)*f) - (c*tan(e + f*x))/(2*f*(a + a*sec(e + f*x))^(5/2)) - (7*c*tan(e + f*x))/(8*a*f*(a + a*sec(e + f*x))^(3/2)), (2*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*f) - (23*c*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(8*sqrt(2)*a^(5/2)*f) - (7*c*sec((1/2)*(e + f*x))^2*sin(e + f*x))/(16*a^2*f*sqrt(a + a*sec(e + f*x))) - (c*cos(e + f*x)*sec((1/2)*(e + f*x))^4*sin(e + f*x))/(8*a^2*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)/(c - c*sec(e + f*x))^1, x, 8, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*c*f) - (71*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(32*sqrt(2)*a^(5/2)*c*f) - (7*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(32*a^3*c*f) + (13*cos(e + f*x)*cot(e + f*x)*sec((1/2)*(e + f*x))^2*sqrt(a + a*sec(e + f*x)))/(32*a^3*c*f) + (cos(e + f*x)^2*cot(e + f*x)*sec((1/2)*(e + f*x))^4*sqrt(a + a*sec(e + f*x)))/(16*a^3*c*f)]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)/(c - c*sec(e + f*x))^2, x, 9, (2*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*c^2*f) - (107*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(64*sqrt(2)*a^(5/2)*c^2*f) + (21*cot(e + f*x)*sqrt(a + a*sec(e + f*x)))/(64*a^3*c^2*f) + (43*cot(e + f*x)^3*(a + a*sec(e + f*x))^(3/2))/(96*a^4*c^2*f) - (15*cos(e + f*x)*cot(e + f*x)^3*sec((1/2)*(e + f*x))^2*(a + a*sec(e + f*x))^(3/2))/(32*a^4*c^2*f) - (cos(e + f*x)^2*cot(e + f*x)^3*sec((1/2)*(e + f*x))^4*(a + a*sec(e + f*x))^(3/2))/(16*a^4*c^2*f)]

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

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

    @test_int [sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(7/2), x, 5, (a*c^4*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (a*c^3*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) - (a*c^2*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x))) - (a*c*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2), x, 4, (a*c^3*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (a*c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) - (a*c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2), x, 3, (a*c^2*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (a*c*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(1/2), x, 2, (a*c*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^(1/2), x, 2, (a*log(1 - cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^(3/2), x, 3, -((a*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2))) + (a*log(1 - cos(e + f*x))*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^(5/2), x, 4, -((a*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2))) - (a*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)) + (a*log(1 - cos(e + f*x))*tan(e + f*x))/(c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))/(c - c*sec(e + f*x))^(7/2), x, 5, -((a*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(7/2))) - (a*tan(e + f*x))/(2*c*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)) - (a*tan(e + f*x))/(c^2*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)) + (a*log(1 - cos(e + f*x))*tan(e + f*x))/(c^3*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    @test_int [(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(5/2), x, 5, (a^2*c^3*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (a^2*c^2*sqrt(c - c*sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) - (a^2*c*(c - c*sec(e + f*x))^(3/2)*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x))) + (a^2*(c - c*sec(e + f*x))^(5/2)*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(3/2), x, 3, (a^2*c^2*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (a^2*c^2*tan(e + f*x)^3)/(2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(1/2), x, 3, (a^2*c*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (a*c*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(f*sqrt(c - c*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)/(c - c*sec(e + f*x))^(1/2), x, 3, (a^2*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (2*a^2*log(1 - sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)/(c - c*sec(e + f*x))^(3/2), x, 3, -((2*a^2*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2))) + (a^2*log(1 - cos(e + f*x))*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)/(c - c*sec(e + f*x))^(5/2), x, 4, -((a^2*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2))) - (a^2*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)) + (a^2*log(1 - cos(e + f*x))*tan(e + f*x))/(c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)/(c - c*sec(e + f*x))^(7/2), x, 5, -((2*a^2*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(7/2))) - (a^2*tan(e + f*x))/(2*c*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)) - (a^2*tan(e + f*x))/(c^2*f*sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)) + (a^2*log(1 - cos(e + f*x))*tan(e + f*x))/(c^3*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

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

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

    @test_int [(c - c*sec(e + f*x))^(7/2)/sqrt(a + a*sec(e + f*x)), x, 3, (c^4*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (8*c^4*log(1 + sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (4*c^4*sec(e + f*x)*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (c^4*sec(e + f*x)^2*tan(e + f*x))/(2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(5/2)/sqrt(a + a*sec(e + f*x)), x, 3, (c^3*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (4*c^3*log(1 + sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (c^3*sec(e + f*x)*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(3/2)/sqrt(a + a*sec(e + f*x)), x, 3, (c^2*log(cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (2*c^2*log(1 + sec(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(1/2)/sqrt(a + a*sec(e + f*x)), x, 2, (c*log(1 + cos(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(1/2)), x, 2, (log(sin(e + f*x))*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(3/2)), x, 3, tan(e + f*x)/(2*c*f*(1 - cos(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (3*log(1 - cos(e + f*x))*tan(e + f*x))/(4*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (log(1 + cos(e + f*x))*tan(e + f*x))/(4*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))), (log(cos(e + f*x))*tan(e + f*x))/(c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (3*log(1 - sec(e + f*x))*tan(e + f*x))/(4*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (log(1 + sec(e + f*x))*tan(e + f*x))/(4*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(2*c*f*(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/(sqrt(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(5/2)), x, 3, (log(cos(e + f*x))*tan(e + f*x))/(c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (7*log(1 - sec(e + f*x))*tan(e + f*x))/(8*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (log(1 + sec(e + f*x))*tan(e + f*x))/(8*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(4*c^2*f*(1 - sec(e + f*x))^2*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (3*tan(e + f*x))/(4*c^2*f*(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    @test_int [(c - c*sec(e + f*x))^(7/2)/(a + a*sec(e + f*x))^(3/2), x, 3, (c^4*log(cos(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (4*c^4*log(1 + sec(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (c^4*sec(e + f*x)*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (8*c^4*tan(e + f*x))/(a*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(5/2)/(a + a*sec(e + f*x))^(3/2), x, 3, -((4*c^3*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x)))) + (c^3*log(cos(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(3/2)/(a + a*sec(e + f*x))^(3/2), x, 3, -((2*c^2*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x)))) + (c^2*log(1 + cos(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(1/2)/(a + a*sec(e + f*x))^(3/2), x, 3, -((c*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x)))) + (c*log(1 + cos(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/((a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(1/2)), x, 3, (log(cos(e + f*x))*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (log(1 - sec(e + f*x))*tan(e + f*x))/(4*a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (3*log(1 + sec(e + f*x))*tan(e + f*x))/(4*a*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(2*a*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/((a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(3/2)), x, 3, cot(e + f*x)/(2*a*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (log(sin(e + f*x))*tan(e + f*x))/(a*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/((a + a*sec(e + f*x))^(3/2)*(c - c*sec(e + f*x))^(5/2)), x, 3, (log(cos(e + f*x))*tan(e + f*x))/(a*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (11*log(1 - sec(e + f*x))*tan(e + f*x))/(16*a*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (5*log(1 + sec(e + f*x))*tan(e + f*x))/(16*a*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(8*a*c^2*f*(1 - sec(e + f*x))^2*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(2*a*c^2*f*(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(8*a*c^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

    @test_int [(c - c*sec(e + f*x))^(7/2)/(a + a*sec(e + f*x))^(5/2), x, 3, (c^4*log(cos(e + f*x))*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (2*c^4*log(1 + sec(e + f*x))*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (4*c^4*tan(e + f*x))/(a^2*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (4*c^4*tan(e + f*x))/(a^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(5/2)/(a + a*sec(e + f*x))^(5/2), x, 3, -((2*c^3*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x)))) + (c^3*log(1 + cos(e + f*x))*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(3/2)/(a + a*sec(e + f*x))^(5/2), x, 4, -((c^2*tan(e + f*x))/(f*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x)))) - (c^2*tan(e + f*x))/(a*f*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x))) + (c^2*log(1 + cos(e + f*x))*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [(c - c*sec(e + f*x))^(1/2)/(a + a*sec(e + f*x))^(5/2), x, 4, -((c*tan(e + f*x))/(2*f*(a + a*sec(e + f*x))^(5/2)*sqrt(c - c*sec(e + f*x)))) - (c*tan(e + f*x))/(a*f*(a + a*sec(e + f*x))^(3/2)*sqrt(c - c*sec(e + f*x))) + (c*log(1 + cos(e + f*x))*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/((a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(1/2)), x, 3, (log(cos(e + f*x))*tan(e + f*x))/(a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (log(1 - sec(e + f*x))*tan(e + f*x))/(8*a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (7*log(1 + sec(e + f*x))*tan(e + f*x))/(8*a^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(4*a^2*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - (3*tan(e + f*x))/(4*a^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/((a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(3/2)), x, 3, (log(cos(e + f*x))*tan(e + f*x))/(a^2*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (5*log(1 - sec(e + f*x))*tan(e + f*x))/(16*a^2*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (11*log(1 + sec(e + f*x))*tan(e + f*x))/(16*a^2*c*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(8*a^2*c*f*(1 - sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(8*a^2*c*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - tan(e + f*x)/(2*a^2*c*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]
    @test_int [1/((a + a*sec(e + f*x))^(5/2)*(c - c*sec(e + f*x))^(5/2)), x, 4, cot(e + f*x)/(2*a^2*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) - cot(e + f*x)^3/(4*a^2*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x))) + (log(sin(e + f*x))*tan(e + f*x))/(a^2*c^2*f*sqrt(a + a*sec(e + f*x))*sqrt(c - c*sec(e + f*x)))]

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

    @test_int [(1 + sec(e + f*x))^m*(c - c*sec(e + f*x))^n, x, 2, (2^(1/2 + m)*AppellF1(1/2 + n, 1/2 - m, 1, 3/2 + n, (1/2)*(1 - sec(e + f*x)), 1 - sec(e + f*x))*(c - c*sec(e + f*x))^n*tan(e + f*x))/(f*(1 + 2*n)*sqrt(1 + sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^n, x, 3, (2^(1/2 + n)*c*AppellF1(1/2 + m, 1/2 - n, 1, 3/2 + m, (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(1 - sec(e + f*x))^(1/2 - n)*(a + a*sec(e + f*x))^m*(c - c*sec(e + f*x))^(-1 + n)*tan(e + f*x))/(f*(1 + 2*m))]

    @test_int [(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^n, x, 3, (2^(1/2 + n)*c*AppellF1(7/2, 1/2 - n, 1, 9/2, (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(1 - sec(e + f*x))^(1/2 - n)*(a + a*sec(e + f*x))^3*(c - c*sec(e + f*x))^(-1 + n)*tan(e + f*x))/(7*f)]
    @test_int [(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^n, x, 3, (2^(1/2 + n)*c*AppellF1(5/2, 1/2 - n, 1, 7/2, (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(1 - sec(e + f*x))^(1/2 - n)*(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^(-1 + n)*tan(e + f*x))/(5*f)]
    @test_int [(a + a*sec(e + f*x))^1*(c - c*sec(e + f*x))^n, x, 3, (2^(1/2 + n)*c*AppellF1(3/2, 1/2 - n, 1, 5/2, (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(1 - sec(e + f*x))^(1/2 - n)*(a + a*sec(e + f*x))*(c - c*sec(e + f*x))^(-1 + n)*tan(e + f*x))/(3*f)]
    @test_int [1/(a + a*sec(e + f*x))^1*(c - c*sec(e + f*x))^n, x, 3, -((2^(1/2 + n)*c*AppellF1(-(1/2), 1/2 - n, 1, 1/2, (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(1 - sec(e + f*x))^(1/2 - n)*(c - c*sec(e + f*x))^(-1 + n)*tan(e + f*x))/(f*(a + a*sec(e + f*x))))]
    @test_int [1/(a + a*sec(e + f*x))^2*(c - c*sec(e + f*x))^n, x, 3, -((2^(1/2 + n)*c*AppellF1(-(3/2), 1/2 - n, 1, -(1/2), (1/2)*(1 + sec(e + f*x)), 1 + sec(e + f*x))*(1 - sec(e + f*x))^(1/2 - n)*(c - c*sec(e + f*x))^(-1 + n)*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2))]

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

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

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

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

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

    @test_int [sqrt(a + a*sec(e + f*x))/(c + c*sec(e + f*x)), x, 6, (2*sqrt(a)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c*f) - (sqrt(2)*sqrt(a)*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(c*f)]

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

    @test_int [(c + d*sec(e + f*x))^(3/2)/(a + a*sec(e + f*x)), x, 3, -((2*c*cot(e + f*x)*Elliptic.Pi(c/(c + d), asin(sqrt(c + d)/sqrt(c + d*sec(e + f*x))), (c - d)/(c + d))*sqrt(-((d*(1 - sec(e + f*x)))/(c + d*sec(e + f*x))))*sqrt((d*(1 + sec(e + f*x)))/(c + d*sec(e + f*x)))*(c + d*sec(e + f*x)))/(a*sqrt(c + d)*f)) - ((c - d)*Elliptic.E(asin(tan(e + f*x)/(1 + sec(e + f*x))), (c - d)/(c + d))*sqrt(1/(1 + sec(e + f*x)))*sqrt(c + d*sec(e + f*x)))/(a*f*sqrt((c + d*sec(e + f*x))/((c + d)*(1 + sec(e + f*x)))))]
    @test_int [(c + d*sec(e + f*x))^(1/2)/(a + a*sec(e + f*x)), x, 3, -((2*cot(e + f*x)*Elliptic.Pi(c/(c + d), asin(sqrt(c + d)/sqrt(c + d*sec(e + f*x))), (c - d)/(c + d))*sqrt(-((d*(1 - sec(e + f*x)))/(c + d*sec(e + f*x))))*sqrt((d*(1 + sec(e + f*x)))/(c + d*sec(e + f*x)))*(c + d*sec(e + f*x)))/(a*sqrt(c + d)*f)) - (Elliptic.E(asin(tan(e + f*x)/(1 + sec(e + f*x))), (c - d)/(c + d))*sqrt(1/(1 + sec(e + f*x)))*sqrt(c + d*sec(e + f*x)))/(a*f*sqrt((c + d*sec(e + f*x))/((c + d)*(1 + sec(e + f*x)))))]
    @test_int [1/((c + d*sec(e + f*x))^(1/2)*(a + a*sec(e + f*x))), x, 5, (2*sqrt(c + d)*cot(e + f*x)*Elliptic.F(asin(sqrt(c + d*sec(e + f*x))/sqrt(c + d)), (c + d)/(c - d))*sqrt((d*(1 - sec(e + f*x)))/(c + d))*sqrt(-((d*(1 + sec(e + f*x)))/(c - d))))/(a*(c - d)*f) - (2*sqrt(c + d)*cot(e + f*x)*Elliptic.Pi((c + d)/c, asin(sqrt(c + d*sec(e + f*x))/sqrt(c + d)), (c + d)/(c - d))*sqrt((d*(1 - sec(e + f*x)))/(c + d))*sqrt(-((d*(1 + sec(e + f*x)))/(c - d))))/(a*c*f) - (Elliptic.E(asin(tan(e + f*x)/(1 + sec(e + f*x))), (c - d)/(c + d))*sqrt(1/(1 + sec(e + f*x)))*sqrt(c + d*sec(e + f*x)))/(a*(c - d)*f*sqrt((c + d*sec(e + f*x))/((c + d)*(1 + sec(e + f*x)))))]

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

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

    @test_int [sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^4, x, 5, (2*a*d*(2*c + d)*(2*c^2 + 2*c*d + d^2)*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) + (2*a^(3/2)*c^4*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*d^2*(6*c^2 + 8*c*d + 3*d^2)*(a - a*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x))) + (2*d^3*(4*c + 3*d)*(a - a*sec(e + f*x))^2*tan(e + f*x))/(5*a*f*sqrt(a + a*sec(e + f*x))) - (2*d^4*(a - a*sec(e + f*x))^3*tan(e + f*x))/(7*a^2*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^3, x, 5, (2*a*d*(3*c^2 + 3*c*d + d^2)*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) + (2*a^(3/2)*c^3*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*d^2*(3*c + 2*d)*(a - a*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x))) + (2*d^3*(a - a*sec(e + f*x))^2*tan(e + f*x))/(5*a*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2, x, 5, (2*a*d*(2*c + d)*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) + (2*a^(3/2)*c^2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*d^2*(a - a*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^1, x, 4, (2*sqrt(a)*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/f + (2*a*d*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))/(c + d*sec(e + f*x))^1, x, 5, (2*sqrt(a)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c*f) - (2*sqrt(a)*sqrt(d)*atan((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(c + d)*sqrt(a + a*sec(e + f*x)))))/(c*sqrt(c + d)*f)]
    @test_int [sqrt(a + a*sec(e + f*x))/(c + d*sec(e + f*x))^2, x, 7, (2*a^(3/2)*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(c^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a^(3/2)*sqrt(d)*(3*c + 2*d)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(c^2*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a*d*tan(e + f*x))/(c*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [sqrt(a + a*sec(e + f*x))/(c + d*sec(e + f*x))^3, x, 8, (2*a^(3/2)*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(c^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a^(3/2)*sqrt(d)*(15*c^2 + 20*c*d + 8*d^2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(4*c^3*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a*d*tan(e + f*x))/(2*c*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) - (a*d*(7*c + 4*d)*tan(e + f*x))/(4*c^2*(c + d)^2*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]

    @test_int [(a + a*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^3, x, 6, (2*a^(5/2)*c^3*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*a^2*(6*c + 13*d)*(c + d*sec(e + f*x))^2*tan(e + f*x))/(35*f*sqrt(a + a*sec(e + f*x))) + (2*a^2*(c + d*sec(e + f*x))^3*tan(e + f*x))/(7*f*sqrt(a + a*sec(e + f*x))) + (2*a^2*(2*(36*c^3 + 243*c^2*d + 189*c*d^2 + 52*d^3) + d*(24*c^2 + 111*c*d + 52*d^2)*sec(e + f*x))*tan(e + f*x))/(105*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^2, x, 5, (2*a^(5/2)*c^2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*a^2*(c + d*sec(e + f*x))^2*tan(e + f*x))/(5*f*sqrt(a + a*sec(e + f*x))) + (2*a^2*(2*(6*c^2 + 25*c*d + 9*d^2) + d*(4*c + 9*d)*sec(e + f*x))*tan(e + f*x))/(15*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^1, x, 5, (2*a^(3/2)*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/f + (2*a^2*(3*c + 4*d)*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x))) + (2*a*d*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(3*f)]
    @test_int [(a + a*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^1, x, 5, (2*a^(3/2)*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(c*f) + (2*a^(3/2)*(c - d)*atan((sqrt(a)*sqrt(d)*tan(e + f*x))/(sqrt(c + d)*sqrt(a + a*sec(e + f*x)))))/(c*sqrt(d)*sqrt(c + d)*f)]
    @test_int [(a + a*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^2, x, 7, (2*a^(5/2)*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(c^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a^(5/2)*(c^2 - 3*c*d - 2*d^2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(c^2*sqrt(d)*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a^2*(c - d)*tan(e + f*x))/(c*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^3, x, 8, (2*a^(5/2)*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(c^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a^(5/2)*(3*c^3 - 15*c^2*d - 20*c*d^2 - 8*d^3)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(4*c^3*sqrt(d)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a^2*(c - d)*tan(e + f*x))/(2*c*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) + (a^2*(3*c^2 - 7*c*d - 4*d^2)*tan(e + f*x))/(4*c^2*(c + d)^2*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]

    @test_int [(a + a*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^3, x, 5, (2*a^3*(3*c^3 + 12*c^2*d + 12*c*d^2 + 4*d^3)*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) + (2*a^(7/2)*c^3*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*a*d*(3*c^2 + 15*c*d + 13*d^2)*(a - a*sec(e + f*x))^2*tan(e + f*x))/(5*f*sqrt(a + a*sec(e + f*x))) - (6*d^2*(c + 2*d)*(a - a*sec(e + f*x))^3*tan(e + f*x))/(7*f*sqrt(a + a*sec(e + f*x))) + (2*d^3*(a - a*sec(e + f*x))^4*tan(e + f*x))/(9*a*f*sqrt(a + a*sec(e + f*x))) - (2*(c^3 + 12*c^2*d + 24*c*d^2 + 12*d^3)*(a^3 - a^3*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^2, x, 5, (2*a^3*(c + 2*d)*(3*c + 2*d)*tan(e + f*x))/(f*sqrt(a + a*sec(e + f*x))) + (2*a^(7/2)*c^2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*a*d*(2*c + 5*d)*(a - a*sec(e + f*x))^2*tan(e + f*x))/(5*f*sqrt(a + a*sec(e + f*x))) - (2*d^2*(a - a*sec(e + f*x))^3*tan(e + f*x))/(7*f*sqrt(a + a*sec(e + f*x))) - (2*(c^2 + 8*c*d + 8*d^2)*(a^3 - a^3*sec(e + f*x))*tan(e + f*x))/(3*f*sqrt(a + a*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^1, x, 6, (2*a^(5/2)*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/f + (2*a^3*(35*c + 32*d)*tan(e + f*x))/(15*f*sqrt(a + a*sec(e + f*x))) + (2*a^2*(5*c + 8*d)*sqrt(a + a*sec(e + f*x))*tan(e + f*x))/(15*f) + (2*a*d*(a + a*sec(e + f*x))^(3/2)*tan(e + f*x))/(5*f)]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^1, x, 7, (2*a^3*tan(e + f*x))/(d*f*sqrt(a + a*sec(e + f*x))) + (2*a^(7/2)*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(c*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*a^(7/2)*(c - d)^2*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(c*d^(3/2)*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^2, x, 10, (2*a^(7/2)*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(c^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a^(7/2)*(c - d)^2*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(c*d^(3/2)*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*a^(7/2)*(c - d)*sqrt(c + d)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(c^2*d^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a^3*(c - d)^2*tan(e + f*x))/(c*d*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [(a + a*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^3, x, 14, (2*a^(7/2)*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(c^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*a^(7/2)*(c - d)^2*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(4*c*d^(3/2)*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (a^(7/2)*(c - d)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(c^2*d^(3/2)*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*a^(7/2)*sqrt(d)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(c^3*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (a^3*(c - d)^2*tan(e + f*x))/(2*c*d*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) + (a^3*(c - d)*tan(e + f*x))/(c^2*d*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))) - (3*a^3*(c - d)^2*tan(e + f*x))/(4*c*d*(c + d)^2*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]

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

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

    @test_int [1/(a + a*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^3, x, 10, (2*d^3*tan(e + f*x))/(a*f*sqrt(a + a*sec(e + f*x))) - ((c - d)^3*tan(e + f*x))/(2*a*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*c^3*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(sqrt(a)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - d)^3*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*sqrt(a)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c - d)^2*(c + 2*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(sqrt(a)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^2, x, 10, -(((c - d)^2*tan(e + f*x))/(2*a*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) + (2*c^2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(sqrt(a)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - d)^2*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*sqrt(a)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c^2 - d^2)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(sqrt(a)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^1, x, 6, (2*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(3/2)*f) - ((5*c - d)*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(2*sqrt(2)*a^(3/2)*f) - ((c - d)*tan(e + f*x))/(2*f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^1, x, 12, -(tan(e + f*x)/(2*a*(c - d)*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) + (2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(sqrt(a)*c*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c - 2*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(sqrt(a)*(c - d)^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*sqrt(a)*(c - d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*d^(5/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(sqrt(a)*c*(c - d)^2*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^2, x, 15, -(tan(e + f*x)/(2*a*(c - d)^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) + (2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(sqrt(a)*c^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c - 3*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(sqrt(a)*(c - d)^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*sqrt(a)*(c - d)^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d^(5/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(sqrt(a)*c*(c - d)^2*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*(3*c - d)*d^(5/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(sqrt(a)*c^2*(c - d)^3*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d^3*tan(e + f*x))/(a*c*(c - d)^2*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^3, x, 19, If(13>=8, -(tan(e + f*x)/(2*a*(c - d)^3*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) + (2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(sqrt(a)*c^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c - 4*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(sqrt(a)*(c - d)^4*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*sqrt(a)*(c - d)^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*d^(5/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(4*sqrt(a)*c*(c - d)^2*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((3*c - d)*d^(5/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(sqrt(a)*c^2*(c - d)^3*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*d^(5/2)*(6*c^2 - 4*c*d + d^2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(sqrt(a)*c^3*(c - d)^4*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d^3*tan(e + f*x))/(2*a*c*(c - d)^2*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) - ((3*c - d)*d^3*tan(e + f*x))/(a*c^2*(c - d)^3*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))) - (3*d^3*tan(e + f*x))/(4*a*c*(c^2 - d^2)^2*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))), -(tan(e + f*x)/(2*a*(c - d)^3*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x)))) + (2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(sqrt(a)*c^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c - 4*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(sqrt(a)*(c - d)^4*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*sqrt(a)*(c - d)^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*d^(5/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(4*sqrt(a)*c*(c - d)^2*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((3*c - d)*d^(5/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(sqrt(a)*c^2*(c - d)^3*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (2*d^(5/2)*(6*c^2 - 4*c*d + d^2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(sqrt(a)*c^3*(c - d)^4*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (d^3*tan(e + f*x))/(2*a*c*(c - d)^2*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) - ((3*c - d)*d^3*tan(e + f*x))/(a*c^2*(c - d)^3*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))) - (3*d^3*tan(e + f*x))/(4*a*c*(c^2 - d^2)^2*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))))]

    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^3, x, 14, -(((c - d)^3*tan(e + f*x))/(4*a^2*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x)))) - (3*(c - d)^3*tan(e + f*x))/(16*a^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - d)^2*(c + 2*d)*tan(e + f*x))/(2*a^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*c^3*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*(c - d)^3*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(16*sqrt(2)*a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - d)^2*(c + 2*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c^3 - d^3)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^2, x, 14, -(((c - d)^2*tan(e + f*x))/(4*a^2*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x)))) - (3*(c - d)^2*tan(e + f*x))/(16*a^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c^2 - d^2)*tan(e + f*x))/(2*a^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*c^2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*c^2*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*(c - d)^2*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(16*sqrt(2)*a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c^2 - d^2)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*a^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^1, x, 7, (2*c*atan((sqrt(a)*tan(e + f*x))/sqrt(a + a*sec(e + f*x))))/(a^(5/2)*f) - ((43*c - 3*d)*atan((sqrt(a)*tan(e + f*x))/(sqrt(2)*sqrt(a + a*sec(e + f*x)))))/(16*sqrt(2)*a^(5/2)*f) - ((c - d)*tan(e + f*x))/(4*f*(a + a*sec(e + f*x))^(5/2)) - ((11*c - 3*d)*tan(e + f*x))/(16*a*f*(a + a*sec(e + f*x))^(3/2))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^1, x, 16, -(tan(e + f*x)/(4*a^2*(c - d)*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x)))) - ((c - 2*d)*tan(e + f*x))/(2*a^2*(c - d)^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*tan(e + f*x))/(16*a^2*(c - d)*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(a^(3/2)*c*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - 2*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*a^(3/2)*(c - d)^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(16*sqrt(2)*a^(3/2)*(c - d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c^2 - 3*c*d + 3*d^2)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(a^(3/2)*(c - d)^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*d^(7/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(a^(3/2)*c*(c - d)^3*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^2, x, 19, -(tan(e + f*x)/(4*a^2*(c - d)^2*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x)))) - ((c - 3*d)*tan(e + f*x))/(2*a^2*(c - d)^3*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*tan(e + f*x))/(16*a^2*(c - d)^2*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(a^(3/2)*c^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - 3*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*a^(3/2)*(c - d)^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(16*sqrt(2)*a^(3/2)*(c - d)^2*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c^2 - 4*c*d + 6*d^2)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(a^(3/2)*(c - d)^4*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (d^(7/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(a^(3/2)*c*(c - d)^3*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*(4*c - d)*d^(7/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(a^(3/2)*c^2*(c - d)^4*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (d^4*tan(e + f*x))/(a^2*c*(c - d)^3*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]
    @test_int [1/(a + a*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^3, x, 23, -(tan(e + f*x)/(4*a^2*(c - d)^3*f*(1 + sec(e + f*x))^2*sqrt(a + a*sec(e + f*x)))) - ((c - 4*d)*tan(e + f*x))/(2*a^2*(c - d)^4*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*tan(e + f*x))/(16*a^2*(c - d)^3*f*(1 + sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*atanh(sqrt(a - a*sec(e + f*x))/sqrt(a))*tan(e + f*x))/(a^(3/2)*c^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - ((c - 4*d)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(2*sqrt(2)*a^(3/2)*(c - d)^4*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (3*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(16*sqrt(2)*a^(3/2)*(c - d)^3*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) - (sqrt(2)*(c^2 - 5*c*d + 10*d^2)*atanh(sqrt(a - a*sec(e + f*x))/(sqrt(2)*sqrt(a)))*tan(e + f*x))/(a^(3/2)*(c - d)^5*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (3*d^(7/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(4*a^(3/2)*c*(c - d)^3*(c + d)^(5/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + ((4*c - d)*d^(7/2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(a^(3/2)*c^2*(c - d)^4*(c + d)^(3/2)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (2*d^(7/2)*(10*c^2 - 5*c*d + d^2)*atanh((sqrt(d)*sqrt(a - a*sec(e + f*x)))/(sqrt(a)*sqrt(c + d)))*tan(e + f*x))/(a^(3/2)*c^3*(c - d)^5*sqrt(c + d)*f*sqrt(a - a*sec(e + f*x))*sqrt(a + a*sec(e + f*x))) + (d^4*tan(e + f*x))/(2*a^2*c*(c - d)^3*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))^2) + (3*d^4*tan(e + f*x))/(4*a^2*c*(c - d)^3*(c + d)^2*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x))) + ((4*c - d)*d^4*tan(e + f*x))/(a^2*c^2*(c - d)^4*(c + d)*f*sqrt(a + a*sec(e + f*x))*(c + d*sec(e + f*x)))]

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

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

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

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

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

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

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

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

    @test_int [(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^1, x, 4, (a*x)/c + (2*(b*c - a*d)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c*sqrt(c - d)*sqrt(c + d)*f)]
    @test_int [(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^2, x, 5, (a*x)/c^2 + (2*(b*c^3 - 2*a*c^2*d + a*d^3)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*f) - (d*(b*c - a*d)*tan(e + f*x))/(c*(c^2 - d^2)*f*(c + d*sec(e + f*x)))]
    @test_int [(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^3, x, 6, (a*x)/c^3 + ((b*c^3*(2*c^2 + d^2) - a*d*(6*c^4 - 5*c^2*d^2 + 2*d^4))*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) - (d*(b*c - a*d)*tan(e + f*x))/(2*c*(c^2 - d^2)*f*(c + d*sec(e + f*x))^2) - (d*(3*b*c^3 - 5*a*c^2*d + 2*a*d^3)*tan(e + f*x))/(2*c^2*(c^2 - d^2)^2*f*(c + d*sec(e + f*x)))]

    @test_int [(a + b*sec(e + f*x))^2/(c + d*sec(e + f*x))^2, x, 5, (a^2*x)/c^2 + (2*(b*c - a*d)*(2*a*c^2 - b*c*d - a*d^2)*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^2*(c - d)^(3/2)*(c + d)^(3/2)*f) + ((b*c - a*d)^2*sin(e + f*x))/(c*(c^2 - d^2)*f*(d + c*cos(e + f*x)))]
    @test_int [(a + b*sec(e + f*x))^2/(c + d*sec(e + f*x))^3, x, 6, (a^2*x)/c^3 - ((3*b^2*c^4*d - 2*a*b*c^3*(2*c^2 + d^2) + a^2*(6*c^4*d - 5*c^2*d^3 + 2*d^5))*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) - (d*(b*c - a*d)^2*sin(e + f*x))/(2*c^2*(c^2 - d^2)*f*(d + c*cos(e + f*x))^2) - ((b*c - a*d)*(3*a*d*(2*c^2 - d^2) - b*c*(2*c^2 + d^2))*sin(e + f*x))/(2*c^2*(c^2 - d^2)^2*f*(d + c*cos(e + f*x)))]
    @test_int [(a + b*sec(e + f*x))^2/(c + d*sec(e + f*x))^4, x, 7, (a^2*x)/c^4 - ((b^2*c^4*d*(4*c^2 + d^2) - a*b*(4*c^7 + 6*c^5*d^2) + a^2*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^4*(c - d)^(7/2)*(c + d)^(7/2)*f) + (d^2*(b + a*cos(e + f*x))^2*sin(e + f*x))/(3*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^3) - (d*(b*c - a*d)*(6*b*c^3 - 8*a*c^2*d - b*c*d^2 + 3*a*d^3)*sin(e + f*x))/(6*c^3*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))^2) - ((2*a*b*c*d*(18*c^4 - 5*c^2*d^2 + 2*d^4) - a^2*d^2*(34*c^4 - 28*c^2*d^2 + 9*d^4) - b^2*(6*c^6 + 10*c^4*d^2 - c^2*d^4))*sin(e + f*x))/(6*c^3*(c^2 - d^2)^3*f*(d + c*cos(e + f*x)))]

    @test_int [(a + b*sec(e + f*x))^3/(c + d*sec(e + f*x))^3, x, 6, (a^3*x)/c^3 - ((b*c - a*d)*(2*a*b*c*d*(4*c^2 - d^2) - b^2*c^2*(c^2 + 2*d^2) - a^2*(6*c^4 - 5*c^2*d^2 + 2*d^4))*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^3*(c - d)^(5/2)*(c + d)^(5/2)*f) + ((b*c - a*d)^2*(b + a*cos(e + f*x))*sin(e + f*x))/(2*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^2) + ((b*c - a*d)^2*(5*a*c^2 - 3*b*c*d - 2*a*d^2)*sin(e + f*x))/(2*c^2*(c^2 - d^2)^2*f*(d + c*cos(e + f*x)))]
    @test_int [(a + b*sec(e + f*x))^3/(c + d*sec(e + f*x))^4, x, 7, If(13>=8, (a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^4*sqrt(c - d)*sqrt(c + d)*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*cos(e + f*x))^2*sin(e + f*x))/(3*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*sin(e + f*x))/(6*c^3*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*(34*c^4*d - 28*c^2*d^3 + 9*d^5))*sin(e + f*x))/(6*c^3*(c^2 - d^2)^3*f*(d + c*cos(e + f*x))), (a^3*x)/c^4 - ((3*a*b^2*c^4*d*(4*c^2 + d^2) - b^3*c^5*(c^2 + 4*d^2) - a^2*b*(6*c^7 + 9*c^5*d^2) + a^3*(8*c^6*d - 8*c^4*d^3 + 7*c^2*d^5 - 2*d^7))*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(c^4*sqrt(c - d)*sqrt(c + d)*(c^2 - d^2)^3*f) - (d*(b*c - a*d)*(b + a*cos(e + f*x))^2*sin(e + f*x))/(3*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^3) + ((b*c - a*d)^2*(3*b*c^3 - 8*a*c^2*d + 2*b*c*d^2 + 3*a*d^3)*sin(e + f*x))/(6*c^3*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))^2) - ((b*c - a*d)*(b^2*c^2*d*(13*c^2 + 2*d^2) - a*b*c*(18*c^4 + 17*c^2*d^2 - 5*d^4) + a^2*(34*c^4*d - 28*c^2*d^3 + 9*d^5))*sin(e + f*x))/(6*c^3*(c^2 - d^2)^3*f*(d + c*cos(e + f*x))))]
    @test_int [(a + b*sec(e + f*x))^3/(c + d*sec(e + f*x))^5, x, 8, (a^3*x)/c^5 - ((15*a*b^2*c^6*d*(4*c^2 + 3*d^2) - 3*a^2*b*c^5*(8*c^4 + 24*c^2*d^2 + 3*d^4) - b^3*c^5*(4*c^4 + 27*c^2*d^2 + 4*d^4) + a^3*(40*c^8*d - 40*c^6*d^3 + 63*c^4*d^5 - 36*c^2*d^7 + 8*d^9))*atanh((sqrt(c - d)*tan((1/2)*(e + f*x)))/sqrt(c + d)))/(4*c^5*sqrt(c - d)*sqrt(c + d)*(c^2 - d^2)^4*f) + (d^2*(b + a*cos(e + f*x))^3*sin(e + f*x))/(4*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^4) - (d*(8*b*c^3 - 11*a*c^2*d - b*c*d^2 + 4*a*d^3)*(b + a*cos(e + f*x))^2*sin(e + f*x))/(12*c^2*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))^3) - ((b*c - a*d)*(2*a*b*c*d*(32*c^4 + c^2*d^2 + 2*d^4) - a^2*d^2*(58*c^4 - 35*c^2*d^2 + 12*d^4) - b^2*(12*c^6 + 25*c^4*d^2 - 2*c^2*d^4))*sin(e + f*x))/(24*c^4*(c^2 - d^2)^3*f*(d + c*cos(e + f*x))^2) - ((b^3*c^3*d*(68*c^4 + 39*c^2*d^2 - 2*d^4) + a^2*b*c*d*(272*c^6 + 10*c^4*d^2 + 49*c^2*d^4 - 16*d^6) - 3*a*b^2*c^2*(24*c^6 + 84*c^4*d^2 - 5*c^2*d^4 + 2*d^6) - a^3*(212*c^6*d^2 - 210*c^4*d^4 + 139*c^2*d^6 - 36*d^8))*sin(e + f*x))/(24*c^4*(c^2 - d^2)^4*f*(d + c*cos(e + f*x)))]

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

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

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

    #= [sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^3, x, 0, (2*(a - b)*sqrt(a + b)*d*(2*a*d^2 + 3*b*(15*c^2 - 5*c*d + 3*d^2))*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt(-((b*(-1 + sec(e + f*x)))/(a + b)))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(15*b^2*f) + (2*sqrt(a + b)*d*(15*a*b*c*d - 2*a^2*d^2 + 9*b^2*(5*c^2 + d^2))*cot((1/2)*(e + f*x))*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt(-((b*(-1 + sec(e + f*x)))/(a + b))))/(15*b^2*f*sqrt((b*(1 + sec(e + f*x)))/(-a + b))) - (2*c^3*cot(e + f*x)*Elliptic.Pi(a/(a + b), asin(sqrt(a + b)/sqrt(a + b*sec(e + f*x))), (a - b)/(a + b))*sqrt(-((b*(1 - sec(e + f*x)))/(a + b*sec(e + f*x))))*sqrt((b*(1 + sec(e + f*x)))/(a + b*sec(e + f*x)))*(a + b*sec(e + f*x)))/(sqrt(a + b)*f) + (2*c*d^2*sqrt(a + b*sec(e + f*x))*tan(e + f*x))/f - (4*a*d^3*sqrt(a + b*sec(e + f*x))*tan(e + f*x))/(15*b*f) + (2*d^3*(a + b*sec(e + f*x))^(3/2)*tan(e + f*x))/(5*b*f)]
    @test_int [sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^2, x, 0, (2*sqrt(a + b)*d*(6*b*c + a*d)*cot((1/2)*(e + f*x))*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt(-((b*(-1 + sec(e + f*x)))/(a + b))))/(3*b*f*sqrt((b*(1 + sec(e + f*x)))/(-a + b))) + (2*(a - b)*sqrt(a + b)*(6*c - d)*d*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt(-((b*(-1 + sec(e + f*x)))/(a + b)))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(3*b*f) - (2*c^2*cot(e + f*x)*Elliptic.Pi(a/(a + b), asin(sqrt(a + b)/sqrt(a + b*sec(e + f*x))), (a - b)/(a + b))*sqrt(-((b*(1 - sec(e + f*x)))/(a + b*sec(e + f*x))))*sqrt((b*(1 + sec(e + f*x)))/(a + b*sec(e + f*x)))*(a + b*sec(e + f*x)))/(sqrt(a + b)*f) + (2*d^2*sqrt(a + b*sec(e + f*x))*tan(e + f*x))/(3*f)] =#
    @test_int [sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^1, x, 5, -((2*(a - b)*sqrt(a + b)*d*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(b*f)) + (2*sqrt(a + b)*(b*(c - d) + a*d)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(b*f) - (2*sqrt(a + b)*c*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f]
    @test_int [sqrt(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^1, x, 3, -((2*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(c*f)) + (2*(b*c - a*d)*Elliptic.Pi((2*d)/(c + d), asin(sqrt(1 - sec(e + f*x))/sqrt(2)), (2*b)/(a + b))*sqrt((a + b*sec(e + f*x))/(a + b))*tan(e + f*x))/(c*(c + d)*f*sqrt(a + b*sec(e + f*x))*sqrt(-tan(e + f*x)^2))]
    #= [sqrt(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^2, x, 0, 0]
    @test_int [sqrt(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^3, x, 0, 0] =#

    #= [(a + b*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^3, x, 0, 0]
    @test_int [(a + b*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^2, x, 0, 0] =#
    @test_int [(a + b*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^1, x, 6, -((2*(a - b)*sqrt(a + b)*(3*b*c + 4*a*d)*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(3*b*f)) + (2*sqrt(a + b)*(a*b*(6*c - 4*d) - b^2*(3*c - d) + 3*a^2*d)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(3*b*f) - (2*a*sqrt(a + b)*c*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f + (2*b*d*sqrt(a + b*sec(e + f*x))*tan(e + f*x))/(3*f)]
    @test_int [(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^1, x, 5, (2*b*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(d*f) - (2*a*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(c*f) - (2*(b*c - a*d)^2*Elliptic.Pi((2*d)/(c + d), asin(sqrt(1 - sec(e + f*x))/sqrt(2)), (2*b)/(a + b))*sqrt((a + b*sec(e + f*x))/(a + b))*tan(e + f*x))/(c*d*(c + d)*f*sqrt(a + b*sec(e + f*x))*sqrt(-tan(e + f*x)^2))]
    #= [(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^2, x, 0, 0]
    @test_int [(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^3, x, 0, 0] =#

    #= [(a + b*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^3, x, 0, 0]
    @test_int [(a + b*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^2, x, 0, 0] =#
    @test_int [(a + b*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^1, x, 7, -((2*(a - b)*sqrt(a + b)*(35*a*b*c + 23*a^2*d + 9*b^2*d)*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(15*b*f)) + (2*sqrt(a + b)*(a^2*b*(45*c - 23*d) - a*b^2*(35*c - 17*d) + b^3*(5*c - 9*d) + 15*a^3*d)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(15*b*f) - (2*a^2*sqrt(a + b)*c*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/f + (2*b*(5*b*c + 8*a*d)*sqrt(a + b*sec(e + f*x))*tan(e + f*x))/(15*f) + (2*b*d*(a + b*sec(e + f*x))^(3/2)*tan(e + f*x))/(5*f)]
    #= [(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^1, x, 0, 0]
    @test_int [(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^2, x, 0, 0]
    @test_int [(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^3, x, 0, 0] =#

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

    #= [1/sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^3, x, 0, -((2*sqrt(a + b)*c^3*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*f*sqrt(a + b*sec(e + f*x)))) + (2*(a - b)*sqrt(a + b)*d^2*(-9*b*c + 2*a*d)*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt(-((b*(-1 + sec(e + f*x)))/(a + b)))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(3*b^3*f*sqrt(a + b*sec(e + f*x))) + (2*sqrt(a + b)*d*(2*a*d^2 + b*(9*c^2 - 9*c*d + d^2))*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt(-((b*(-1 + sec(e + f*x)))/(a + b)))*sqrt((b*(1 + sec(e + f*x)))/(-a + b)))/(3*b^2*f*sqrt(a + b*sec(e + f*x))) + (2*d^3*sqrt(a + b*sec(e + f*x))*tan(e + f*x))/(3*b*f)]
    @test_int [1/sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^2, x, 0, -((2*(a - b)*sqrt(a + b)*d^2*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.E(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(b^2*f*sqrt(a + b*sec(e + f*x)))) - (2*sqrt(a + b)*c^2*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*f*sqrt(a + b*sec(e + f*x))) + (2*sqrt(a + b)*(2*c - d)*d*sqrt(cos(e + f*x))*sqrt(b + a*cos(e + f*x))*csc(e + f*x)*Elliptic.F(asin(sqrt(b + a*cos(e + f*x))/(sqrt(a + b)*sqrt(cos(e + f*x)))), (a + b)/(a - b))*sqrt(-((b*(-1 + sec(e + f*x)))/(a + b)))*sqrt((b*(1 + sec(e + f*x)))/(-a + b)))/(b*f*sqrt(a + b*sec(e + f*x)))] =#
    @test_int [1/sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^1, x, 3, (2*sqrt(a + b)*d*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(b*f) - (2*sqrt(a + b)*c*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*f)]
    @test_int [1/sqrt(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^1, x, 3, -((2*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*c*f)) - (2*d*Elliptic.Pi((2*d)/(c + d), asin(sqrt(1 - sec(e + f*x))/sqrt(2)), (2*b)/(a + b))*sqrt((a + b*sec(e + f*x))/(a + b))*tan(e + f*x))/(c*(c + d)*f*sqrt(a + b*sec(e + f*x))*sqrt(-tan(e + f*x)^2))]
    #= [1/sqrt(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^2, x, 0, 0]
    @test_int [1/sqrt(a + b*sec(e + f*x))/(c + d*sec(e + f*x))^3, x, 0, 0] =#

    #= [1/(a + b*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^3, x, 0, 0]
    @test_int [1/(a + b*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^2, x, 0, 0] =#
    @test_int [1/(a + b*sec(e + f*x))^(3/2)*(c + d*sec(e + f*x))^1, x, 6, (2*(b*c - a*d)*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*b*sqrt(a + b)*f) - (2*(b*c - a*d)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a*b*sqrt(a + b)*f) - (2*sqrt(a + b)*c*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a^2*f) + (2*b*(b*c - a*d)*tan(e + f*x))/(a*(a^2 - b^2)*f*sqrt(a + b*sec(e + f*x)))]
    #= [1/(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^1, x, 0, 0]
    @test_int [1/(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^2, x, 0, 0]
    @test_int [1/(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^3, x, 0, 0] =#

    #= [1/(a + b*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^3, x, 0, 0]
    @test_int [1/(a + b*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^2, x, 0, 0] =#
    @test_int [1/(a + b*sec(e + f*x))^(5/2)*(c + d*sec(e + f*x))^1, x, 7, (2*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*cot(e + f*x)*Elliptic.E(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(3*a^2*(a - b)*b*(a + b)^(3/2)*f) - (2*(6*a^2*b*c - a*b^2*c - 3*b^3*c - 3*a^3*d + a^2*b*d)*cot(e + f*x)*Elliptic.F(asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(3*a^2*(a - b)*b*(a + b)^(3/2)*f) - (2*sqrt(a + b)*c*cot(e + f*x)*Elliptic.Pi((a + b)/a, asin(sqrt(a + b*sec(e + f*x))/sqrt(a + b)), (a + b)/(a - b))*sqrt((b*(1 - sec(e + f*x)))/(a + b))*sqrt(-((b*(1 + sec(e + f*x)))/(a - b))))/(a^3*f) + (2*b*(b*c - a*d)*tan(e + f*x))/(3*a*(a^2 - b^2)*f*(a + b*sec(e + f*x))^(3/2)) + (2*b*(7*a^2*b*c - 3*b^3*c - 4*a^3*d)*tan(e + f*x))/(3*a^2*(a^2 - b^2)^2*f*sqrt(a + b*sec(e + f*x)))]
    #= [1/(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^1, x, 0, 0]
    @test_int [1/(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^2, x, 0, 0]
    @test_int [1/(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^3, x, 0, 0] =#

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

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

    #= [(a + b*sec(e + f*x))^(1/2)*(c + d*sec(e + f*x))^(3/2), x, 1, 0] =#
    @test_int [(a + b*sec(e + f*x))^(1/2)*(c + d*sec(e + f*x))^(1/2), x, 3, -((2*sqrt(c + d)*cot(e + f*x)*Elliptic.Pi((a*(c + d))/((a + b)*c), asin((sqrt(a + b)*sqrt(c + d*sec(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sec(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(-(((b*c - a*d)*(1 - sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x)))/(sqrt(a + b)*f)) + (2*cot(e + f*x)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin((sqrt((a + b)/(c + d))*sqrt(c + d*sec(e + f*x)))/sqrt(a + b*sec(e + f*x))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(-(((b*c - a*d)*(1 - sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x)))/(sqrt((a + b)/(c + d))*f)]
    @test_int [(a + b*sec(e + f*x))^(1/2)/(c + d*sec(e + f*x))^(1/2), x, 1, -((2*sqrt(c + d)*cot(e + f*x)*Elliptic.Pi((a*(c + d))/((a + b)*c), asin((sqrt(a + b)*sqrt(c + d*sec(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sec(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(-(((b*c - a*d)*(1 - sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x)))/(sqrt(a + b)*c*f))]
    @test_int [(a + b*sec(e + f*x))^(1/2)/(c + d*sec(e + f*x))^(3/2), x, 5, -((2*sqrt(c + d)*cot(e + f*x)*Elliptic.Pi((a*(c + d))/((a + b)*c), asin((sqrt(a + b)*sqrt(c + d*sec(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sec(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(-(((b*c - a*d)*(1 - sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x)))/(sqrt(a + b)*c^2*f)) - (2*sqrt(a + b)*d*cot(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*(1 + sec(e + f*x))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x)))))/(c*(c - d)*sqrt(c + d)*f*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))) - (2*(a - b)*sqrt(a + b)*d*cot(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(c*(c - d)*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [(a + b*sec(e + f*x))^(1/2)/(c + d*sec(e + f*x))^(5/2), x, 7, (2*(a - b)*sqrt(a + b)*d*(6*b*c^3 - 7*a*c^2*d - 2*b*c*d^2 + 3*a*d^3)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(3*c^2*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)^2*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*sqrt(a + b)*(b*c^2*(3*c^2 + 3*c*d - 2*d^2) - a*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(3*c^3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^3*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*d^2*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(3*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))]

    @test_int [(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^(3/2), x, 6, -((2*(a - b)*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c*(c - d)*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))) - (2*sqrt(a + b)*(b*c - a*(2*c - d))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^2*(c - d)*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*a*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^2*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))]
    @test_int [(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^(5/2), x, 7, -((2*(a - b)*sqrt(a + b)*(3*b*c^3 - 7*a*c^2*d + b*c*d^2 + 3*a*d^3)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(3*c^2*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))) - (2*sqrt(a + b)*(b^2*c^3*(3*c + d) - 2*a*b*c^2*(3*c^2 + 2*c*d - d^2) + a^2*d*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(3*c^3*(c - d)^2*(c + d)^(3/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*a*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^3*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*d*(b*c - a*d)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(3*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))]
    @test_int [(a + b*sec(e + f*x))^(3/2)/(c + d*sec(e + f*x))^(7/2), x, 8, (2*(a - b)*sqrt(a + b)*(2*a*b*c*d*(35*c^4 - 8*c^2*d^2 + 5*d^4) - a^2*d^2*(58*c^4 - 41*c^2*d^2 + 15*d^4) - b^2*(15*c^6 + 19*c^4*d^2 - 2*c^2*d^4))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)^2*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*sqrt(a + b)*(b^2*c^3*(15*c^3 + 10*c^2*d + 9*c*d^2 - 2*d^3) - 2*a*b*c^2*(15*c^4 + 20*c^3*d - 4*c^2*d^2 - 4*c*d^3 + 5*d^4) + a^2*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*a*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^4*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*d^2*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(5*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^2*sqrt(c + d*sec(e + f*x))) - (2*d*(10*b*c^3 - 13*a*c^2*d - 2*b*c*d^2 + 5*a*d^3)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(15*c^2*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))]

    @test_int [(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^(5/2), x, 7, -((2*(a - b)*sqrt(a + b)*(7*a*c^2 - 4*b*c*d - 3*a*d^2)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(3*c^2*(c - d)^2*(c + d)^(3/2)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))) + (2*sqrt(a + b)*(b^2*c^2*(c + 3*d) - a*b*c*(7*c^2 + 4*c*d - 3*d^2) + a^2*(9*c^3 - 2*c^2*d - 6*c*d^2 + 3*d^3))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(3*c^3*(c - d)^2*(c + d)^(3/2)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*a^2*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^3*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*(b*c - a*d)^2*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(3*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))]
    @test_int [(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^(7/2), x, 8, If(13>=8, (2*(a - b)*sqrt(a + b)*(b^2*c^2*d*(29*c^2 + 3*d^2) - a*b*c*(35*c^4 + 34*c^2*d^2 - 5*d^4) + a^2*(58*c^4*d - 41*c^2*d^3 + 15*d^5))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*sqrt(a + b)*(b^3*c^4*(5*c^2 + 24*c*d + 3*d^2) - a*b^2*c^3*(35*c^3 + 42*c^2*d + 21*c*d^2 - 2*d^3) + a^2*b*c^2*(45*c^4 + 48*c^3*d + c^2*d^2 - 8*c*d^3 + 10*d^4) - a^3*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*a^2*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^4*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*d*(b*c - a*d)*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(5*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^2*sqrt(c + d*sec(e + f*x))) + (2*(b*c - a*d)*(5*b*c^3 - 13*a*c^2*d + 3*b*c*d^2 + 5*a*d^3)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(15*c^2*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))*sqrt(c + d*sec(e + f*x))), (2*(a - b)*sqrt(a + b)*(b^2*c^2*d*(29*c^2 + 3*d^2) - a*b*c*(35*c^4 + 34*c^2*d^2 - 5*d^4) + a^2*(58*c^4*d - 41*c^2*d^3 + 15*d^5))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(15*c^3*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*sqrt(a + b)*(b^3*c^4*(5*c^2 + 24*c*d + 3*d^2) - a*b^2*c^3*(35*c^3 + 42*c^2*d + 21*c*d^2 - 2*d^3) + a^2*b*c^2*(45*c^4 + 48*c^3*d + c^2*d^2 - 8*c*d^3 + 10*d^4) - a^3*d*(60*c^5 - 2*c^4*d - 66*c^3*d^2 + 25*c^2*d^3 + 30*c*d^4 - 15*d^5))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(15*c^4*(c - d)^3*(c + d)^(5/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*a^2*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^4*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*d*(b*c - a*d)*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(5*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^2*sqrt(c + d*sec(e + f*x))) + (2*(b*c - a*d)*(5*b*c^3 - 13*a*c^2*d + 3*b*c*d^2 + 5*a*d^3)*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(15*c^2*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))*sqrt(c + d*sec(e + f*x))))]
    @test_int [(a + b*sec(e + f*x))^(5/2)/(c + d*sec(e + f*x))^(9/2), x, 9, (2*(a - b)*sqrt(a + b)*(2*b^3*c^3*d*(133*c^4 + 62*c^2*d^2 - 3*d^4) + 2*a^2*b*c*d*(406*c^6 + 73*c^4*d^2 + 132*c^2*d^4 - 35*d^6) - a*b^2*c^2*(245*c^6 + 852*c^4*d^2 + 41*c^2*d^4 + 14*d^6) - a^3*(582*c^6*d^2 - 485*c^4*d^4 + 392*c^2*d^6 - 105*d^8))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(105*c^4*(c - d)^4*(c + d)^(7/2)*(b*c - a*d)^2*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*sqrt(a + b)*(b^3*c^4*(35*c^4 + 231*c^3*d + 67*c^2*d^2 + 57*c*d^3 - 6*d^4) - a*b^2*c^3*(245*c^5 + 413*c^4*d + 439*c^3*d^2 + 53*c^2*d^3 - 12*c*d^4 + 14*d^5) + a^2*b*c^2*(315*c^6 + 497*c^5*d + 219*c^4*d^2 - 73*c^3*d^3 + 208*c^2*d^4 + 56*c*d^5 - 70*d^6) - a^3*d*(525*c^7 + 57*c^6*d - 699*c^5*d^2 + 214*c^4*d^3 + 672*c^3*d^4 - 280*c^2*d^5 - 210*c*d^6 + 105*d^7))*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(105*c^5*(c - d)^4*(c + d)^(7/2)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*a^2*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^5*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) + (2*d^2*(b + a*cos(e + f*x))^2*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(7*c*(c^2 - d^2)*f*(d + c*cos(e + f*x))^3*sqrt(c + d*sec(e + f*x))) - (2*d*(14*b*c^3 - 19*a*c^2*d - 2*b*c*d^2 + 7*a*d^3)*(b + a*cos(e + f*x))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(35*c^2*(c^2 - d^2)^2*f*(d + c*cos(e + f*x))^2*sqrt(c + d*sec(e + f*x))) - (2*(2*a*b*c*d*(91*c^4 - 2*c^2*d^2 + 7*d^4) - a^2*d^2*(162*c^4 - 101*c^2*d^2 + 35*d^4) - b^2*(35*c^6 + 67*c^4*d^2 - 6*c^2*d^4))*sqrt(a + b*sec(e + f*x))*sin(e + f*x))/(105*c^3*(c^2 - d^2)^3*f*(d + c*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))]

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

    @test_int [(c + d*sec(e + f*x))^(3/2)/sqrt(a + b*sec(e + f*x)), x, -1, -((2*c*(c + d)*cot(e + f*x)*Elliptic.Pi((a*(c + d))/((a + b)*c), asin(sqrt(((a + b)*(c + d*sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x))))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x))^(3/2)*sqrt(((a + b)*(b*c - a*d)*(-1 + sec(e + f*x))*(c + d*sec(e + f*x)))/((c + d)^2*(a + b*sec(e + f*x))^2)))/(a*(a + b)*f*sqrt(c + d*sec(e + f*x)))) + (2*d*(c + d)*cot(e + f*x)*Elliptic.Pi((b*(c + d))/((a + b)*d), asin(sqrt(((a + b)*(c + d*sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x))))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x))^(3/2)*sqrt(-(((a + b)*((-b)*c + a*d)*(-1 + sec(e + f*x))*(c + d*sec(e + f*x)))/((c + d)^2*(a + b*sec(e + f*x))^2))))/(b*(a + b)*f*sqrt(c + d*sec(e + f*x))) + (2*(b*c - a*d)*cot(e + f*x)*Elliptic.F(asin(sqrt(((a + b)*(c + d*sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x))))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(((b*c - a*d)*(-1 + sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*sqrt(a + b*sec(e + f*x))*sqrt(c + d*sec(e + f*x)))/(a*b*f*sqrt(((a + b)*(c + d*sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))]
    @test_int [(c + d*sec(e + f*x))^(1/2)/sqrt(a + b*sec(e + f*x)), x, 1, -((2*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(a*sqrt(c + d)*f))]
    @test_int [1/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^(1/2)), x, 3, -((2*sqrt(c + d)*cot(e + f*x)*Elliptic.Pi((a*(c + d))/((a + b)*c), asin((sqrt(a + b)*sqrt(c + d*sec(e + f*x)))/(sqrt(c + d)*sqrt(a + b*sec(e + f*x)))), ((a - b)*(c + d))/((a + b)*(c - d)))*sqrt(-(((b*c - a*d)*(1 - sec(e + f*x)))/((c + d)*(a + b*sec(e + f*x)))))*sqrt(((b*c - a*d)*(1 + sec(e + f*x)))/((c - d)*(a + b*sec(e + f*x))))*(a + b*sec(e + f*x)))/(a*sqrt(a + b)*c*f)) - (2*b*sqrt(a + b)*cot(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(a*sqrt(c + d)*(b*c - a*d)*f)]
    @test_int [1/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^(3/2)), x, 6, -((2*(a - b)*sqrt(a + b)*d^2*cot(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(c*(c - d)*sqrt(c + d)*(b*c - a*d)^2*f)) - (2*sqrt(a + b)*(2*c - d)*d*cot(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(c^2*(c - d)*sqrt(c + d)*(b*c - a*d)*f) - (2*sqrt(a + b)*cot(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(a + b*sec(e + f*x)))/(sqrt(a + b)*sqrt(c + d*sec(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(((b*c - a*d)*(1 - sec(e + f*x)))/((a + b)*(c + d*sec(e + f*x))))*sqrt(-(((b*c - a*d)*(1 + sec(e + f*x)))/((a - b)*(c + d*sec(e + f*x)))))*(c + d*sec(e + f*x)))/(a*c^2*sqrt(c + d)*f), -((2*(a - b)*sqrt(a + b)*d^2*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.E(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c*(c - d)*sqrt(c + d)*(b*c - a*d)^2*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))) - (2*sqrt(a + b)*(2*c - d)*d*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.F(asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(c^2*(c - d)*sqrt(c + d)*(b*c - a*d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x))) - (2*sqrt(a + b)*sqrt(-(((b*c - a*d)*(1 - cos(e + f*x)))/((a + b)*(d + c*cos(e + f*x)))))*sqrt(-(((b*c - a*d)*(1 + cos(e + f*x)))/((a - b)*(d + c*cos(e + f*x)))))*(d + c*cos(e + f*x))^(3/2)*csc(e + f*x)*Elliptic.Pi(((a + b)*c)/(a*(c + d)), asin((sqrt(c + d)*sqrt(b + a*cos(e + f*x)))/(sqrt(a + b)*sqrt(d + c*cos(e + f*x)))), ((a + b)*(c - d))/((a - b)*(c + d)))*sqrt(a + b*sec(e + f*x)))/(a*c^2*sqrt(c + d)*f*sqrt(b + a*cos(e + f*x))*sqrt(c + d*sec(e + f*x)))]
    #= [1/(sqrt(a + b*sec(e + f*x))*(c + d*sec(e + f*x))^(5/2)), x, 0, 0] =#

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

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

    @test_int [(a + b*sec(e + f*x))^(1/3)/(c + d*sec(e + f*x))^(1/3), x, 1, ((d + c*cos(e + f*x))^(1/3)*(a + b*sec(e + f*x))^(1/3)*Unintegrable((b + a*cos(e + f*x))^(1/3)/(d + c*cos(e + f*x))^(1/3), x))/((b + a*cos(e + f*x))^(1/3)*(c + d*sec(e + f*x))^(1/3))]
    @test_int [(a + b*sec(e + f*x))^(1/3)/(c + d*sec(e + f*x))^(4/3), x, 0, Unintegrable((a + b*sec(e + f*x))^(1/3)/(c + d*sec(e + f*x))^(4/3), x)]
    @test_int [(a + b*sec(e + f*x))^(1/3)/(c + d*sec(e + f*x))^(7/3), x, 0, Unintegrable((a + b*sec(e + f*x))^(1/3)/(c + d*sec(e + f*x))^(7/3), x)]

    @test_int [(a + b*sec(e + f*x))^(2/3)/(c + d*sec(e + f*x))^(2/3), x, 1, ((d + c*cos(e + f*x))^(2/3)*(a + b*sec(e + f*x))^(2/3)*Unintegrable((b + a*cos(e + f*x))^(2/3)/(d + c*cos(e + f*x))^(2/3), x))/((b + a*cos(e + f*x))^(2/3)*(c + d*sec(e + f*x))^(2/3))]
    @test_int [(a + b*sec(e + f*x))^(2/3)/(c + d*sec(e + f*x))^(5/3), x, 0, Unintegrable((a + b*sec(e + f*x))^(2/3)/(c + d*sec(e + f*x))^(5/3), x)]
    @test_int [(a + b*sec(e + f*x))^(2/3)/(c + d*sec(e + f*x))^(8/3), x, 0, Unintegrable((a + b*sec(e + f*x))^(2/3)/(c + d*sec(e + f*x))^(8/3), x)]

    @test_int [(a + b*sec(e + f*x))^(4/3)/(c + d*sec(e + f*x))^(4/3), x, 1, ((d + c*cos(e + f*x))^(4/3)*(a + b*sec(e + f*x))^(4/3)*Unintegrable((b + a*cos(e + f*x))^(4/3)/(d + c*cos(e + f*x))^(4/3), x))/((b + a*cos(e + f*x))^(4/3)*(c + d*sec(e + f*x))^(4/3))]
    @test_int [(a + b*sec(e + f*x))^(4/3)/(c + d*sec(e + f*x))^(7/3), x, 0, Unintegrable((a + b*sec(e + f*x))^(4/3)/(c + d*sec(e + f*x))^(7/3), x)]
    @test_int [(a + b*sec(e + f*x))^(4/3)/(c + d*sec(e + f*x))^(10/3), x, 0, Unintegrable((a + b*sec(e + f*x))^(4/3)/(c + d*sec(e + f*x))^(10/3), x)]

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

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

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

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

    @test_int [(a + a*sec(e + f*x))^m*(c*(d*sec(e + f*x))^p)^n, x, 4, -((AppellF1(n*p, 1/2, 1/2 - m, 1 + n*p, sec(e + f*x), -sec(e + f*x))*(c*(d*sec(e + f*x))^p)^n*(1 + sec(e + f*x))^(-(1/2) - m)*(a + a*sec(e + f*x))^m*tan(e + f*x))/(f*n*p*sqrt(1 - sec(e + f*x))))]

    @test_int [(c*(d*sec(e + f*x))^p)^n*(a + a*sec(e + f*x))^3, x, 8, (a^3*(7 + 4*n*p)*HypergeometricFunctions._₂F₁(1/2, -((n*p)/2), (1/2)*(2 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*n*p*(2 + n*p)*sqrt(sin(e + f*x)^2)) - (a^3*(1 + 4*n*p)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*(1 - n^2*p^2)*sqrt(sin(e + f*x)^2)) + (a^3*(5 + 2*n*p)*(c*(d*sec(e + f*x))^p)^n*tan(e + f*x))/(f*(1 + n*p)*(2 + n*p)) + ((c*(d*sec(e + f*x))^p)^n*(a^3 + a^3*sec(e + f*x))*tan(e + f*x))/(f*(2 + n*p))]
    @test_int [(c*(d*sec(e + f*x))^p)^n*(a + a*sec(e + f*x))^2, x, 7, (2*a^2*HypergeometricFunctions._₂F₁(1/2, -((n*p)/2), (1/2)*(2 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*n*p*sqrt(sin(e + f*x)^2)) - (a^2*(1 + 2*n*p)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*(1 - n^2*p^2)*sqrt(sin(e + f*x)^2)) + (a^2*(c*(d*sec(e + f*x))^p)^n*tan(e + f*x))/(f*(1 + n*p))]
    @test_int [(c*(d*sec(e + f*x))^p)^n*(a + a*sec(e + f*x))^1, x, 6, (a*HypergeometricFunctions._₂F₁(1/2, -((n*p)/2), (1/2)*(2 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*n*p*sqrt(sin(e + f*x)^2)) - (a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*(1 - n*p)*sqrt(sin(e + f*x)^2))]
    @test_int [(c*(d*sec(e + f*x))^p)^n/(a + a*sec(e + f*x))^1, x, 7, ((c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*(a + a*sec(e + f*x))) - (cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(a*f*sqrt(sin(e + f*x)^2)) + ((1 - n*p)*cos(e + f*x)^2*HypergeometricFunctions._₂F₁(1/2, (1/2)*(2 - n*p), (1/2)*(4 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(a*f*(2 - n*p)*sqrt(sin(e + f*x)^2))]
    @test_int [(c*(d*sec(e + f*x))^p)^n/(a + a*sec(e + f*x))^2, x, 8, (2*(2 - n*p)*HypergeometricFunctions._₂F₁(1/2, -((n*p)/2), (1/2)*(2 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(3*a^2*f*sqrt(sin(e + f*x)^2)) - ((3 - 2*n*p)*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(3*a^2*f*sqrt(sin(e + f*x)^2)) - (2*(2 - n*p)*(c*(d*sec(e + f*x))^p)^n*tan(e + f*x))/(3*a^2*f*(1 + sec(e + f*x))) - ((c*(d*sec(e + f*x))^p)^n*tan(e + f*x))/(3*f*(a + a*sec(e + f*x))^2)]

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

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

    @test_int [(a + b*sec(e + f*x))^m*(c*(d*sec(e + f*x))^p)^n, x, 1, ((c*(d*sec(e + f*x))^p)^n*Unintegrable((d*sec(e + f*x))^(n*p)*(a + b*sec(e + f*x))^m, x))/(d*sec(e + f*x))^(n*p)]

    @test_int [(c*(d*sec(e + f*x))^p)^n*(a + b*sec(e + f*x))^3, x, 8, (b*(b^2*(1 + n*p) + 3*a^2*(2 + n*p))*HypergeometricFunctions._₂F₁(1/2, -((n*p)/2), (1/2)*(2 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*n*p*(2 + n*p)*sqrt(sin(e + f*x)^2)) - (a*(3*b^2*n*p + a^2*(1 + n*p))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*(1 - n^2*p^2)*sqrt(sin(e + f*x)^2)) + (a*b^2*(5 + 2*n*p)*(c*(d*sec(e + f*x))^p)^n*tan(e + f*x))/(f*(1 + n*p)*(2 + n*p)) + (b^2*(c*(d*sec(e + f*x))^p)^n*(a + b*sec(e + f*x))*tan(e + f*x))/(f*(2 + n*p))]
    @test_int [(c*(d*sec(e + f*x))^p)^n*(a + b*sec(e + f*x))^2, x, 7, (2*a*b*HypergeometricFunctions._₂F₁(1/2, -((n*p)/2), (1/2)*(2 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*n*p*sqrt(sin(e + f*x)^2)) - ((b^2*n*p + a^2*(1 + n*p))*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*(1 - n^2*p^2)*sqrt(sin(e + f*x)^2)) + (b^2*(c*(d*sec(e + f*x))^p)^n*tan(e + f*x))/(f*(1 + n*p))]
    @test_int [(c*(d*sec(e + f*x))^p)^n*(a + b*sec(e + f*x))^1, x, 6, (b*HypergeometricFunctions._₂F₁(1/2, -((n*p)/2), (1/2)*(2 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*n*p*sqrt(sin(e + f*x)^2)) - (a*cos(e + f*x)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(1 - n*p), (1/2)*(3 - n*p), cos(e + f*x)^2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/(f*(1 - n*p)*sqrt(sin(e + f*x)^2))]
    @test_int [(c*(d*sec(e + f*x))^p)^n/(a + b*sec(e + f*x))^1, x, 7, -((b*AppellF1(1/2, (n*p)/2, 1, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*(cos(e + f*x)^2)^((n*p)/2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/((a^2 - b^2)*f)) + (a*AppellF1(1/2, (1/2)*(-1 + n*p), 1, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(cos(e + f*x)^2)^((1/2)*(-1 + n*p))*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/((a^2 - b^2)*f)]
    @test_int [(c*(d*sec(e + f*x))^p)^n/(a + b*sec(e + f*x))^2, x, 10, -((2*a*b*AppellF1(1/2, (1/2)*(-2 + n*p), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*(cos(e + f*x)^2)^((n*p)/2)*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/((a^2 - b^2)^2*f)) + (a^2*AppellF1(1/2, (1/2)*(-3 + n*p), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(cos(e + f*x)^2)^((1/2)*(-1 + n*p))*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/((a^2 - b^2)^2*f) + (b^2*AppellF1(1/2, (1/2)*(-1 + n*p), 2, 3/2, sin(e + f*x)^2, (a^2*sin(e + f*x)^2)/(a^2 - b^2))*cos(e + f*x)*(cos(e + f*x)^2)^((1/2)*(-1 + n*p))*(c*(d*sec(e + f*x))^p)^n*sin(e + f*x))/((a^2 - b^2)^2*f)]
    end

test/4 Trig functions/4.5 Secant/4.5.2.1 (a+b sec)^m (c+d sec)^n.jl