text:
@testset "7.4.2 Exponentials of inverse hyperbolic cotangent functions" begin
    (a, c, m, n, p, x, ) = @variables a c m n p x

    #= ::Package:: =#

    #= ::Title:: =#
    #=Integrands*involving*exponentials*of*inverse*hyperbolic*cotangents=#

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*x^m*ℯ^(n*acoth(a*x))=#

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

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

    @test_int [ℯ^acoth(a*x)*x^3, x, 8, (2*sqrt(1 - 1/(a^2*x^2))*x)/(3*a^3) + (3*sqrt(1 - 1/(a^2*x^2))*x^2)/(8*a^2) + (sqrt(1 - 1/(a^2*x^2))*x^3)/(3*a) + (1/4)*sqrt(1 - 1/(a^2*x^2))*x^4 + (3*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a^4)]
    @test_int [ℯ^acoth(a*x)*x^2, x, 7, (2*sqrt(1 - 1/(a^2*x^2))*x)/(3*a^2) + (sqrt(1 - 1/(a^2*x^2))*x^2)/(2*a) + (1/3)*sqrt(1 - 1/(a^2*x^2))*x^3 + atanh(sqrt(1 - 1/(a^2*x^2)))/(2*a^3)]
    @test_int [ℯ^acoth(a*x)*x^1, x, 6, (sqrt(1 - 1/(a^2*x^2))*x)/a + (1/2)*sqrt(1 - 1/(a^2*x^2))*x^2 + atanh(sqrt(1 - 1/(a^2*x^2)))/(2*a^2)]
    @test_int [ℯ^acoth(a*x), x, 5, sqrt(1 - 1/(a^2*x^2))*x + atanh(sqrt(1 - 1/(a^2*x^2)))/a]
    @test_int [ℯ^acoth(a*x)/x^1, x, 6, -acsc(a*x) + atanh(sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)/x^2, x, 3, a*sqrt(1 - 1/(a^2*x^2)) - a*acsc(a*x)]
    @test_int [ℯ^acoth(a*x)/x^3, x, 3, (1/2)*a*sqrt(1 - 1/(a^2*x^2))*(2*a + 1/x) - (1/2)*a^2*acsc(a*x)]
    @test_int [ℯ^acoth(a*x)/x^4, x, 7, a^3*sqrt(1 - 1/(a^2*x^2)) - (1/3)*a^3*(1 - 1/(a^2*x^2))^(3/2) + (a^2*sqrt(1 - 1/(a^2*x^2)))/(2*x) - (1/2)*a^3*acsc(a*x)]
    @test_int [ℯ^acoth(a*x)/x^5, x, 5, (1/24)*a^3*sqrt(1 - 1/(a^2*x^2))*(16*a + 9/x) + (a*sqrt(1 - 1/(a^2*x^2)))/(4*x^3) + (a^2*sqrt(1 - 1/(a^2*x^2)))/(3*x^2) - (3/8)*a^4*acsc(a*x)]

    @test_int [ℯ^(2*acoth(a*x))*x^3, x, 4, (2*x)/a^3 + x^2/a^2 + (2*x^3)/(3*a) + x^4/4 + (2*log(1 - a*x))/a^4]
    @test_int [ℯ^(2*acoth(a*x))*x^2, x, 4, (2*x)/a^2 + x^2/a + x^3/3 + (2*log(1 - a*x))/a^3]
    @test_int [ℯ^(2*acoth(a*x))*x^1, x, 4, (2*x)/a + x^2/2 + (2*log(1 - a*x))/a^2]
    @test_int [ℯ^(2*acoth(a*x)), x, 4, x + (2*log(1 - a*x))/a]
    @test_int [ℯ^(2*acoth(a*x))/x^1, x, 4, -log(x) + 2*log(1 - a*x)]
    @test_int [ℯ^(2*acoth(a*x))/x^2, x, 4, x^(-1) - 2*a*log(x) + 2*a*log(1 - a*x)]
    @test_int [ℯ^(2*acoth(a*x))/x^3, x, 4, 1/(2*x^2) + (2*a)/x - 2*a^2*log(x) + 2*a^2*log(1 - a*x)]
    @test_int [ℯ^(2*acoth(a*x))/x^4, x, 4, 1/(3*x^3) + a/x^2 + (2*a^2)/x - 2*a^3*log(x) + 2*a^3*log(1 - a*x)]

    @test_int [ℯ^(3*acoth(a*x))*x^2, x, 14, -((4*sqrt(1 - 1/(a^2*x^2)))/(a^2*(a - 1/x))) + (14*sqrt(1 - 1/(a^2*x^2))*x)/(3*a^2) + (3*sqrt(1 - 1/(a^2*x^2))*x^2)/(2*a) + (1/3)*sqrt(1 - 1/(a^2*x^2))*x^3 + (11*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a^3)]
    @test_int [ℯ^(3*acoth(a*x))*x^1, x, 12, -((4*sqrt(1 - 1/(a^2*x^2)))/(a*(a - 1/x))) + (3*sqrt(1 - 1/(a^2*x^2))*x)/a + (1/2)*sqrt(1 - 1/(a^2*x^2))*x^2 + (9*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a^2)]
    @test_int [ℯ^(3*acoth(a*x)), x, 8, -((4*sqrt(1 - 1/(a^2*x^2)))/(a - 1/x)) + sqrt(1 - 1/(a^2*x^2))*x + (3*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [ℯ^(3*acoth(a*x))/x^1, x, 8, -((4*a*sqrt(1 - 1/(a^2*x^2)))/(a - 1/x)) + acsc(a*x) + atanh(sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))/x^2, x, 5, -3*a*sqrt(1 - 1/(a^2*x^2)) - (2*(a + 1/x)^2)/(a*sqrt(1 - 1/(a^2*x^2))) + 3*a*acsc(a*x)]
    @test_int [ℯ^(3*acoth(a*x))/x^3, x, 9, (-(9/2))*a^2*sqrt(1 - 1/(a^2*x^2)) - (a^5*(1 - 1/(a^2*x^2))^(5/2))/(a - 1/x)^3 - (3*a^3*(1 - 1/(a^2*x^2))^(3/2))/(2*(a - 1/x)) + (9/2)*a^2*acsc(a*x)]
    @test_int [ℯ^(3*acoth(a*x))/x^4, x, 10, -((a + 1/x)^3/sqrt(1 - 1/(a^2*x^2))) - (1/3)*a*sqrt(1 - 1/(a^2*x^2))*(3*a + 1/x)^2 - (1/6)*a^2*sqrt(1 - 1/(a^2*x^2))*(28*a + 3/x) + (11/2)*a^3*acsc(a*x)]

    @test_int [ℯ^(4*acoth(a*x))*x^3, x, 4, (12*x)/a^3 + (4*x^2)/a^2 + (4*x^3)/(3*a) + x^4/4 + 4/(a^4*(1 - a*x)) + (16*log(1 - a*x))/a^4]
    @test_int [ℯ^(4*acoth(a*x))*x^2, x, 4, (8*x)/a^2 + (2*x^2)/a + x^3/3 + 4/(a^3*(1 - a*x)) + (12*log(1 - a*x))/a^3]
    @test_int [ℯ^(4*acoth(a*x))*x^1, x, 4, (4*x)/a + x^2/2 + 4/(a^2*(1 - a*x)) + (8*log(1 - a*x))/a^2]
    @test_int [ℯ^(4*acoth(a*x)), x, 4, x + 4/(a*(1 - a*x)) + (4*log(1 - a*x))/a]
    @test_int [ℯ^(4*acoth(a*x))/x^1, x, 4, 4/(1 - a*x) + log(x)]
    @test_int [ℯ^(4*acoth(a*x))/x^2, x, 4, -x^(-1) + (4*a)/(1 - a*x) + 4*a*log(x) - 4*a*log(1 - a*x)]
    @test_int [ℯ^(4*acoth(a*x))/x^3, x, 4, -1/(2*x^2) - (4*a)/x + (4*a^2)/(1 - a*x) + 8*a^2*log(x) - 8*a^2*log(1 - a*x)]
    @test_int [ℯ^(4*acoth(a*x))/x^4, x, 4, -1/(3*x^3) - (2*a)/x^2 - (8*a^2)/x + (4*a^3)/(1 - a*x) + 12*a^3*log(x) - 12*a^3*log(1 - a*x)]

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

    @test_int [x^3/ℯ^acoth(a*x), x, 8, -((2*sqrt(1 - 1/(a^2*x^2))*x)/(3*a^3)) + (3*sqrt(1 - 1/(a^2*x^2))*x^2)/(8*a^2) - (sqrt(1 - 1/(a^2*x^2))*x^3)/(3*a) + (1/4)*sqrt(1 - 1/(a^2*x^2))*x^4 + (3*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a^4)]
    @test_int [x^2/ℯ^acoth(a*x), x, 7, (2*sqrt(1 - 1/(a^2*x^2))*x)/(3*a^2) - (sqrt(1 - 1/(a^2*x^2))*x^2)/(2*a) + (1/3)*sqrt(1 - 1/(a^2*x^2))*x^3 - atanh(sqrt(1 - 1/(a^2*x^2)))/(2*a^3)]
    @test_int [x^1/ℯ^acoth(a*x), x, 6, -((sqrt(1 - 1/(a^2*x^2))*x)/a) + (1/2)*sqrt(1 - 1/(a^2*x^2))*x^2 + atanh(sqrt(1 - 1/(a^2*x^2)))/(2*a^2)]
    @test_int [ℯ^(-acoth(a*x)), x, 5, sqrt(1 - 1/(a^2*x^2))*x - atanh(sqrt(1 - 1/(a^2*x^2)))/a]
    @test_int [1/(ℯ^acoth(a*x)*x^1), x, 6, acsc(a*x) + atanh(sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*x^2), x, 3, (-a)*sqrt(1 - 1/(a^2*x^2)) - a*acsc(a*x)]
    @test_int [1/(ℯ^acoth(a*x)*x^3), x, 3, (1/2)*a*sqrt(1 - 1/(a^2*x^2))*(2*a - 1/x) + (1/2)*a^2*acsc(a*x)]
    @test_int [1/(ℯ^acoth(a*x)*x^4), x, 7, (-a^3)*sqrt(1 - 1/(a^2*x^2)) + (1/3)*a^3*(1 - 1/(a^2*x^2))^(3/2) + (a^2*sqrt(1 - 1/(a^2*x^2)))/(2*x) - (1/2)*a^3*acsc(a*x)]
    @test_int [1/(ℯ^acoth(a*x)*x^5), x, 5, (1/24)*a^3*sqrt(1 - 1/(a^2*x^2))*(16*a - 9/x) - (a*sqrt(1 - 1/(a^2*x^2)))/(4*x^3) + (a^2*sqrt(1 - 1/(a^2*x^2)))/(3*x^2) + (3/8)*a^4*acsc(a*x)]

    @test_int [x^3/ℯ^(2*acoth(a*x)), x, 4, (-2*x)/a^3 + x^2/a^2 - (2*x^3)/(3*a) + x^4/4 + (2*log(1 + a*x))/a^4]
    @test_int [x^2/ℯ^(2*acoth(a*x)), x, 4, (2*x)/a^2 - x^2/a + x^3/3 - (2*log(1 + a*x))/a^3]
    @test_int [x^1/ℯ^(2*acoth(a*x)), x, 4, (-2*x)/a + x^2/2 + (2*log(1 + a*x))/a^2]
    @test_int [ℯ^(-2*acoth(a*x)), x, 4, x - (2*log(1 + a*x))/a]
    @test_int [1/(ℯ^(2*acoth(a*x))*x^1), x, 4, -log(x) + 2*log(1 + a*x)]
    @test_int [1/(ℯ^(2*acoth(a*x))*x^2), x, 4, x^(-1) + 2*a*log(x) - 2*a*log(1 + a*x)]
    @test_int [1/(ℯ^(2*acoth(a*x))*x^3), x, 4, 1/(2*x^2) - (2*a)/x - 2*a^2*log(x) + 2*a^2*log(1 + a*x)]
    @test_int [1/(ℯ^(2*acoth(a*x))*x^4), x, 4, 1/(3*x^3) - a/x^2 + (2*a^2)/x + 2*a^3*log(x) - 2*a^3*log(1 + a*x)]

    @test_int [x^3/ℯ^(3*acoth(a*x)), x, 19, -((4*sqrt(1 - 1/(a^2*x^2)))/(a^3*(a + 1/x))) - (6*sqrt(1 - 1/(a^2*x^2))*x)/a^3 + (19*sqrt(1 - 1/(a^2*x^2))*x^2)/(8*a^2) - (sqrt(1 - 1/(a^2*x^2))*x^3)/a + (1/4)*sqrt(1 - 1/(a^2*x^2))*x^4 + (51*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a^4)]
    @test_int [x^2/ℯ^(3*acoth(a*x)), x, 14, (4*sqrt(1 - 1/(a^2*x^2)))/(a^2*(a + 1/x)) + (14*sqrt(1 - 1/(a^2*x^2))*x)/(3*a^2) - (3*sqrt(1 - 1/(a^2*x^2))*x^2)/(2*a) + (1/3)*sqrt(1 - 1/(a^2*x^2))*x^3 - (11*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a^3)]
    @test_int [x^1/ℯ^(3*acoth(a*x)), x, 12, -((4*sqrt(1 - 1/(a^2*x^2)))/(a*(a + 1/x))) - (3*sqrt(1 - 1/(a^2*x^2))*x)/a + (1/2)*sqrt(1 - 1/(a^2*x^2))*x^2 + (9*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a^2)]
    @test_int [ℯ^(-3*acoth(a*x)), x, 8, (4*sqrt(1 - 1/(a^2*x^2)))/(a + 1/x) + sqrt(1 - 1/(a^2*x^2))*x - (3*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [1/(ℯ^(3*acoth(a*x))*x^1), x, 8, -((4*a*sqrt(1 - 1/(a^2*x^2)))/(a + 1/x)) - acsc(a*x) + atanh(sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*x^2), x, 5, 3*a*sqrt(1 - 1/(a^2*x^2)) + (2*(a - 1/x)^2)/(a*sqrt(1 - 1/(a^2*x^2))) + 3*a*acsc(a*x)]
    @test_int [1/(ℯ^(3*acoth(a*x))*x^3), x, 9, (-(9/2))*a^2*sqrt(1 - 1/(a^2*x^2)) - (a^5*(1 - 1/(a^2*x^2))^(5/2))/(a + 1/x)^3 - (3*a^3*(1 - 1/(a^2*x^2))^(3/2))/(2*(a + 1/x)) - (9/2)*a^2*acsc(a*x)]
    @test_int [1/(ℯ^(3*acoth(a*x))*x^4), x, 10, (1/6)*a^2*sqrt(1 - 1/(a^2*x^2))*(28*a - 3/x) + (a - 1/x)^3/sqrt(1 - 1/(a^2*x^2)) + (1/3)*a*sqrt(1 - 1/(a^2*x^2))*(3*a - 1/x)^2 + (11/2)*a^3*acsc(a*x)]
    @test_int [1/(ℯ^(3*acoth(a*x))*x^5), x, 14, (-(27/4))*a^4*sqrt(1 - 1/(a^2*x^2)) - (9/8)*a^3*sqrt(1 - 1/(a^2*x^2))*(2*a - 3/x) - (a*(a - 1/x)^3)/sqrt(1 - 1/(a^2*x^2)) + (a*sqrt(1 - 1/(a^2*x^2)))/(4*x^3) - (a^2*sqrt(1 - 1/(a^2*x^2)))/x^2 - (51/8)*a^4*acsc(a*x)]

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

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

    @test_int [ℯ^(acoth(a*x)/2)*x^4, x, 11, (611*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(1920*a^4) + (269*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(960*a^3) + (11*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(48*a^2) + (9*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^5 + (31*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5) + (31*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5)]
    @test_int [ℯ^(acoth(a*x)/2)*x^3, x, 10, (83*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(192*a^3) + (29*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(96*a^2) + (7*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(24*a) + (1/4)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4 + (11*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4) + (11*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4)]
    @test_int [ℯ^(acoth(a*x)/2)*x^2, x, 9, (11*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(24*a^2) + (5*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3 + (3*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3) + (3*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3)]
    @test_int [ℯ^(acoth(a*x)/2)*x, x, 7, ((1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(4*a) + (1/2)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4)*x^2 + atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(4*a^2) + atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(4*a^2)]
    @test_int [ℯ^(acoth(a*x)/2), x, 6, (1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x + atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/a + atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/a]
    @test_int [ℯ^(acoth(a*x)/2)/x, x, 17, (-sqrt(2))*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) + sqrt(2)*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) + 2*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + 2*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2) - log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2)]
    @test_int [ℯ^(acoth(a*x)/2)/x^2, x, 13, a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4) - (a*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) + (a*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) + (a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2)) - (a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2))]
    @test_int [ℯ^(acoth(a*x)/2)/x^3, x, 14, (a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/4 + (a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/2 - (a^2*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) + (a^2*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) + (a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) - (a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2))]
    @test_int [ℯ^(acoth(a*x)/2)/x^4, x, 15, (3*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/8 + (a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/12 + (a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/(3*x) - (3*a^3*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (3*a^3*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (3*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2)) - (3*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2))]

    @test_int [ℯ^((3*acoth(a*x))/2)*x^4, x, 11, (557*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(640*a^4) + (157*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(320*a^3) + (5*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(16*a^2) + (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5 - (237*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5) + (237*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5)]
    @test_int [ℯ^((3*acoth(a*x))/2)*x^3, x, 10, (63*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(64*a^3) + (15*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(32*a^2) + (3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(8*a) + (1/4)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4 - (123*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4) + (123*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4)]
    @test_int [ℯ^((3*acoth(a*x))/2)*x^2, x, 9, (23*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(24*a^2) + (7*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3 - (17*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3) + (17*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3)]
    @test_int [ℯ^((3*acoth(a*x))/2)*x, x, 7, (3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(4*a) + (1/2)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4)*x^2 - (9*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2) + (9*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2)]
    @test_int [ℯ^((3*acoth(a*x))/2), x, 6, (1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x - (3*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a + (3*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a]
    @test_int [ℯ^((3*acoth(a*x))/2)/x, x, 17, (-sqrt(2))*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) + sqrt(2)*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - 2*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + 2*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) - log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2) + log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2)]
    @test_int [ℯ^((3*acoth(a*x))/2)/x^2, x, 13, a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4) - (3*a*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) + (3*a*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) - (3*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2)) + (3*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2))]
    @test_int [ℯ^((3*acoth(a*x))/2)/x^3, x, 14, (3*a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/4 + (a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4))/2 - (9*a^2*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) + (9*a^2*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) - (9*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (9*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2))]
    @test_int [ℯ^((3*acoth(a*x))/2)/x^4, x, 15, (17*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/24 + (a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4))/4 + (a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(7/4))/(3*x) - (17*a^3*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (17*a^3*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) - (17*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2)) + (17*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2))]

    @test_int [ℯ^((5*acoth(a*x))/2)*x^4, x, 12, -((26111*(1 + 1/(a*x))^(1/4))/(1920*a^5*(1 - 1/(a*x))^(1/4))) + (5533*(1 + 1/(a*x))^(1/4)*x)/(1920*a^4*(1 - 1/(a*x))^(1/4)) + (1189*(1 + 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 - 1/(a*x))^(1/4)) + (181*(1 + 1/(a*x))^(1/4)*x^3)/(240*a^2*(1 - 1/(a*x))^(1/4)) + (21*(1 + 1/(a*x))^(1/4)*x^4)/(40*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^5)/(5*(1 - 1/(a*x))^(1/4)) + (1003*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5) + (1003*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5)]
    @test_int [ℯ^((5*acoth(a*x))/2)*x^3, x, 11, -((2467*(1 + 1/(a*x))^(1/4))/(192*a^4*(1 - 1/(a*x))^(1/4))) + (521*(1 + 1/(a*x))^(1/4)*x)/(192*a^3*(1 - 1/(a*x))^(1/4)) + (113*(1 + 1/(a*x))^(1/4)*x^2)/(96*a^2*(1 - 1/(a*x))^(1/4)) + (17*(1 + 1/(a*x))^(1/4)*x^3)/(24*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^4)/(4*(1 - 1/(a*x))^(1/4)) + (475*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4) + (475*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4)]
    @test_int [ℯ^((5*acoth(a*x))/2)*x^2, x, 10, -((287*(1 + 1/(a*x))^(1/4))/(24*a^3*(1 - 1/(a*x))^(1/4))) + (61*(1 + 1/(a*x))^(1/4)*x)/(24*a^2*(1 - 1/(a*x))^(1/4)) + (13*(1 + 1/(a*x))^(1/4)*x^2)/(12*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(1/4)*x^3)/(3*(1 - 1/(a*x))^(1/4)) + (55*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3) + (55*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3)]
    @test_int [ℯ^((5*acoth(a*x))/2)*x, x, 8, -((25*(1 + 1/(a*x))^(1/4))/(2*a^2*(1 - 1/(a*x))^(1/4))) + (5*(1 + 1/(a*x))^(5/4)*x)/(4*a*(1 - 1/(a*x))^(1/4)) + ((1 + 1/(a*x))^(9/4)*x^2)/(2*(1 - 1/(a*x))^(1/4)) + (25*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2) + (25*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2)]
    @test_int [ℯ^((5*acoth(a*x))/2), x, 7, -((10*(1 + 1/(a*x))^(1/4))/(a*(1 - 1/(a*x))^(1/4))) + ((1 + 1/(a*x))^(5/4)*x)/(1 - 1/(a*x))^(1/4) + (5*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a + (5*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a]
    @test_int [ℯ^((5*acoth(a*x))/2)/x, x, 19, -((8*(1 + 1/(a*x))^(1/4))/(1 - 1/(a*x))^(1/4)) + sqrt(2)*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - sqrt(2)*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) + 2*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + 2*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) - log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2) + log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2)]
    @test_int [ℯ^((5*acoth(a*x))/2)/x^2, x, 14, -5*a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4) - (4*a*(1 + 1/(a*x))^(5/4))/(1 - 1/(a*x))^(1/4) + (5*a*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) - (5*a*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) - (5*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2)) + (5*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2))]
    @test_int [ℯ^((5*acoth(a*x))/2)/x^3, x, 15, (-25*a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/4 - (5*a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/2 - (2*a^2*(1 + 1/(a*x))^(9/4))/(1 - 1/(a*x))^(1/4) + (25*a^2*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) - (25*a^2*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) - (25*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (25*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2))]
    @test_int [ℯ^((5*acoth(a*x))/2)/x^4, x, 16, (-55*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/8 - (11*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(5/4))/4 - (2*a^3*(1 + 1/(a*x))^(9/4))/(1 - 1/(a*x))^(1/4) - (a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(9/4))/3 + (55*a^3*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) - (55*a^3*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) - (55*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2)) + (55*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2))]

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

    @test_int [x^4/ℯ^(acoth(a*x)/2), x, 11, (611*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(1920*a^4) - (269*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(960*a^3) + (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(48*a^2) - (9*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^5 + (31*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5) - (31*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5)]
    @test_int [x^3/ℯ^(acoth(a*x)/2), x, 10, -((83*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(192*a^3)) + (29*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(96*a^2) - (7*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3)/(24*a) + (1/4)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^4 - (11*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4) + (11*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4)]
    @test_int [x^2/ℯ^(acoth(a*x)/2), x, 9, (11*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(24*a^2) - (5*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x^3 + (3*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3) - (3*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3)]
    @test_int [x/ℯ^(acoth(a*x)/2), x, 7, -(((1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x)/(4*a)) + (1/2)*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4)*x^2 - atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(4*a^2) + atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(4*a^2)]
    @test_int [ℯ^(-acoth(a*x)/2), x, 6, (1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)*x + atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/a - atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/a]
    @test_int [1/(ℯ^(acoth(a*x)/2)*x), x, 17, sqrt(2)*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - sqrt(2)*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - 2*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + 2*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2) - log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2)]
    @test_int [1/(ℯ^(acoth(a*x)/2)*x^2), x, 13, -(a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4)) - (a*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) + (a*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) - (a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2)) + (a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2))]
    @test_int [1/(ℯ^(acoth(a*x)/2)*x^3), x, 14, (a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/4 + (a^2*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/2 + (a^2*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) - (a^2*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) + (a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) - (a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2))]
    @test_int [1/(ℯ^(acoth(a*x)/2)*x^4), x, 15, (-3*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/8 - (a^3*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/12 + (a^2*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/(3*x) - (3*a^3*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (3*a^3*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) - (3*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2)) + (3*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2))]

    @test_int [x^4/ℯ^((3*acoth(a*x))/2), x, 11, (557*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(640*a^4) - (157*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(320*a^3) + (5*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(16*a^2) - (11*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4)/(40*a) + (1/5)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^5 - (237*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5) - (237*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5)]
    @test_int [x^3/ℯ^((3*acoth(a*x))/2), x, 10, -((63*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(64*a^3)) + (15*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(32*a^2) - (3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3)/(8*a) + (1/4)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^4 + (123*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4) + (123*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4)]
    @test_int [x^2/ℯ^((3*acoth(a*x))/2), x, 9, (23*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(24*a^2) - (7*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^2)/(12*a) + (1/3)*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x^3 - (17*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3) - (17*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3)]
    @test_int [x/ℯ^((3*acoth(a*x))/2), x, 7, -((3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x)/(4*a)) + (1/2)*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4)*x^2 + (9*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2) + (9*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2)]
    @test_int [ℯ^((-3*acoth(a*x))/2), x, 6, (1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)*x - (3*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a - (3*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a]
    @test_int [1/(ℯ^((3*acoth(a*x))/2)*x), x, 17, sqrt(2)*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - sqrt(2)*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) + 2*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + 2*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) - log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2) + log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2)]
    @test_int [1/(ℯ^((3*acoth(a*x))/2)*x^2), x, 13, -(a*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4)) - (3*a*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) + (3*a*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) + (3*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2)) - (3*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2))]
    @test_int [1/(ℯ^((3*acoth(a*x))/2)*x^3), x, 14, (3*a^2*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/4 + (a^2*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4))/2 + (9*a^2*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) - (9*a^2*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) - (9*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (9*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2))]
    @test_int [1/(ℯ^((3*acoth(a*x))/2)*x^4), x, 15, (-17*a^3*(1 - 1/(a*x))^(3/4)*(1 + 1/(a*x))^(1/4))/24 - (a^3*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4))/4 + (a^2*(1 - 1/(a*x))^(7/4)*(1 + 1/(a*x))^(1/4))/(3*x) - (17*a^3*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (17*a^3*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (17*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2)) - (17*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2))]

    @test_int [x^4/ℯ^((5*acoth(a*x))/2), x, 12, (26111*(1 - 1/(a*x))^(1/4))/(1920*a^5*(1 + 1/(a*x))^(1/4)) + (5533*(1 - 1/(a*x))^(1/4)*x)/(1920*a^4*(1 + 1/(a*x))^(1/4)) - (1189*(1 - 1/(a*x))^(1/4)*x^2)/(960*a^3*(1 + 1/(a*x))^(1/4)) + (181*(1 - 1/(a*x))^(1/4)*x^3)/(240*a^2*(1 + 1/(a*x))^(1/4)) - (21*(1 - 1/(a*x))^(1/4)*x^4)/(40*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(1/4)*x^5)/(5*(1 + 1/(a*x))^(1/4)) + (1003*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5) - (1003*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*a^5)]
    @test_int [x^3/ℯ^((5*acoth(a*x))/2), x, 11, -((2467*(1 - 1/(a*x))^(1/4))/(192*a^4*(1 + 1/(a*x))^(1/4))) - (521*(1 - 1/(a*x))^(1/4)*x)/(192*a^3*(1 + 1/(a*x))^(1/4)) + (113*(1 - 1/(a*x))^(1/4)*x^2)/(96*a^2*(1 + 1/(a*x))^(1/4)) - (17*(1 - 1/(a*x))^(1/4)*x^3)/(24*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(1/4)*x^4)/(4*(1 + 1/(a*x))^(1/4)) - (475*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4) + (475*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(64*a^4)]
    @test_int [x^2/ℯ^((5*acoth(a*x))/2), x, 10, (287*(1 - 1/(a*x))^(1/4))/(24*a^3*(1 + 1/(a*x))^(1/4)) + (61*(1 - 1/(a*x))^(1/4)*x)/(24*a^2*(1 + 1/(a*x))^(1/4)) - (13*(1 - 1/(a*x))^(1/4)*x^2)/(12*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(1/4)*x^3)/(3*(1 + 1/(a*x))^(1/4)) + (55*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3) - (55*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(8*a^3)]
    @test_int [x/ℯ^((5*acoth(a*x))/2), x, 8, -((25*(1 - 1/(a*x))^(1/4))/(2*a^2*(1 + 1/(a*x))^(1/4))) - (5*(1 - 1/(a*x))^(5/4)*x)/(4*a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(9/4)*x^2)/(2*(1 + 1/(a*x))^(1/4)) - (25*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2) + (25*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(4*a^2)]
    @test_int [ℯ^((-5*acoth(a*x))/2), x, 7, (10*(1 - 1/(a*x))^(1/4))/(a*(1 + 1/(a*x))^(1/4)) + ((1 - 1/(a*x))^(5/4)*x)/(1 + 1/(a*x))^(1/4) + (5*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a - (5*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/a]
    @test_int [1/(ℯ^((5*acoth(a*x))/2)*x), x, 19, -((8*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - sqrt(2)*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) + sqrt(2)*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)) - 2*atan((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) + 2*atanh((1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)) - log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2) + log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4))/sqrt(2)]
    @test_int [1/(ℯ^((5*acoth(a*x))/2)*x^2), x, 14, (4*a*(1 - 1/(a*x))^(5/4))/(1 + 1/(a*x))^(1/4) + 5*a*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4) + (5*a*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) - (5*a*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/sqrt(2) + (5*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2)) - (5*a*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(2*sqrt(2))]
    @test_int [1/(ℯ^((5*acoth(a*x))/2)*x^3), x, 15, (-2*a^2*(1 - 1/(a*x))^(9/4))/(1 + 1/(a*x))^(1/4) - (25*a^2*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/4 - (5*a^2*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/2 - (25*a^2*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) + (25*a^2*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(4*sqrt(2)) - (25*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (25*a^2*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2))]
    @test_int [1/(ℯ^((5*acoth(a*x))/2)*x^4), x, 16, (2*a^3*(1 - 1/(a*x))^(9/4))/(1 + 1/(a*x))^(1/4) + (55*a^3*(1 - 1/(a*x))^(1/4)*(1 + 1/(a*x))^(3/4))/8 + (11*a^3*(1 - 1/(a*x))^(5/4)*(1 + 1/(a*x))^(3/4))/4 + (a^3*(1 - 1/(a*x))^(9/4)*(1 + 1/(a*x))^(3/4))/3 + (55*a^3*atan(1 - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) - (55*a^3*atan(1 + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(8*sqrt(2)) + (55*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) - (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2)) - (55*a^3*log(1 + sqrt(1 - 1/(a*x))/sqrt(1 + 1/(a*x)) + (sqrt(2)*(1 - 1/(a*x))^(1/4))/(1 + 1/(a*x))^(1/4)))/(16*sqrt(2))]

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

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

    @test_int [ℯ^(acoth(x)/3)*x^2, x, 16, (11/27)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x + (7/18)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x^2 + (1/3)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x^3 - (19*atan((1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3)))/(54*sqrt(3)) + (19*atan((1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3)))/(54*sqrt(3)) + (19/81)*atanh((1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) - (19/324)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) + (19/324)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6))]
    @test_int [ℯ^(acoth(x)/3)*x, x, 14, (1/6)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x + (1/2)*(1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6)*x^2 - atan((1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3))/(6*sqrt(3)) + atan((1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3))/(6*sqrt(3)) + (1/9)*atanh((1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) - (1/36)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) + (1/36)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6))]
    @test_int [ℯ^(acoth(x)/3), x, 13, (1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6)*x - atan((1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3))/sqrt(3) + atan((1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3))/sqrt(3) + (2/3)*atanh((1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) - (1/6)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) + (1/6)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6))]
    @test_int [ℯ^(acoth(x)/3)/x, x, 25, (-sqrt(3))*atan((1 - (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3)) + sqrt(3)*atan((1 + (2*(1 + 1/x)^(1/6))/((-1 + x)/x)^(1/6))/sqrt(3)) - atan(sqrt(3) - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + atan(sqrt(3) + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + 2*atan(((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)) + 2*atanh((1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) - (1/2)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) - (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) + (1/2)*log(1 + (1 + 1/x)^(1/3)/((-1 + x)/x)^(1/3) + (1 + 1/x)^(1/6)/((-1 + x)/x)^(1/6)) + (1/2)*sqrt(3)*log(1 - (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)) - (1/2)*sqrt(3)*log(1 + (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3))]
    @test_int [ℯ^(acoth(x)/3)/x^2, x, 14, (1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6) - (1/3)*atan(sqrt(3) - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + (1/3)*atan(sqrt(3) + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + (2/3)*atan(((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)) + log(1 - (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3))/(2*sqrt(3)) - log(1 + (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3))/(2*sqrt(3))]
    @test_int [ℯ^(acoth(x)/3)/x^3, x, 15, (1/6)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6) + (1/2)*(1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6) - (1/18)*atan(sqrt(3) - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + (1/18)*atan(sqrt(3) + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + (1/9)*atan(((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)) + log(1 - (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3))/(12*sqrt(3)) - log(1 + (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3))/(12*sqrt(3))]
    @test_int [ℯ^(acoth(x)/3)/x^4, x, 16, (19/54)*(1 + 1/x)^(1/6)*((-1 + x)/x)^(5/6) + (1/18)*(1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6) + ((1 + 1/x)^(7/6)*((-1 + x)/x)^(5/6))/(3*x) - (19/162)*atan(sqrt(3) - (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + (19/162)*atan(sqrt(3) + (2*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6)) + (19/81)*atan(((-1 + x)/x)^(1/6)/(1 + 1/x)^(1/6)) + (19*log(1 - (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)))/(108*sqrt(3)) - (19*log(1 + (sqrt(3)*((-1 + x)/x)^(1/6))/(1 + 1/x)^(1/6) + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)))/(108*sqrt(3))]

    @test_int [ℯ^((2*acoth(x))/3)*x^2, x, 6, (14/27)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x + (4/9)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x^2 + (1/3)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x^3 - (22*atan(1/sqrt(3) + (2*((-1 + x)/x)^(1/3))/(sqrt(3)*(1 + 1/x)^(1/3))))/(27*sqrt(3)) - (11/27)*log((1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)) - (11*log(x))/81]
    @test_int [ℯ^((2*acoth(x))/3)*x, x, 4, (1/3)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x + (1/2)*(1 + 1/x)^(4/3)*((-1 + x)/x)^(2/3)*x^2 - (2*atan(1/sqrt(3) + (2*((-1 + x)/x)^(1/3))/(sqrt(3)*(1 + 1/x)^(1/3))))/(3*sqrt(3)) - (1/3)*log((1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)) - log(x)/9]
    @test_int [ℯ^((2*acoth(x))/3), x, 3, (1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3)*x - (2*atan(1/sqrt(3) + (2*((-1 + x)/x)^(1/3))/(sqrt(3)*(1 + 1/x)^(1/3))))/sqrt(3) - log((1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)) - log(x)/3]
    @test_int [ℯ^((2*acoth(x))/3)/x, x, 4, (-sqrt(3))*atan(1/sqrt(3) - (2*((-1 + x)/x)^(1/3))/(sqrt(3)*(1 + 1/x)^(1/3))) - sqrt(3)*atan(1/sqrt(3) + (2*((-1 + x)/x)^(1/3))/(sqrt(3)*(1 + 1/x)^(1/3))) - (3/2)*log((1 + 1/x)^(1/3) - ((-1 + x)/x)^(1/3)) - (3/2)*log(1 + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)) - (1/2)*log(1 + 1/x) - log(x)/2]
    @test_int [ℯ^((2*acoth(x))/3)/x^2, x, 3, (1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3) - (2*atan(1/sqrt(3) - (2*((-1 + x)/x)^(1/3))/(sqrt(3)*(1 + 1/x)^(1/3))))/sqrt(3) - log(1 + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)) - (1/3)*log(1 + 1/x)]
    @test_int [ℯ^((2*acoth(x))/3)/x^3, x, 4, (1/3)*(1 + 1/x)^(1/3)*((-1 + x)/x)^(2/3) + (1/2)*(1 + 1/x)^(4/3)*((-1 + x)/x)^(2/3) - (2*atan(1/sqrt(3) - (2*((-1 + x)/x)^(1/3))/(sqrt(3)*(1 + 1/x)^(1/3))))/(3*sqrt(3)) - (1/3)*log(1 + ((-1 + x)/x)^(1/3)/(1 + 1/x)^(1/3)) - (1/9)*log(1 + 1/x)]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*ℯ^(n/4*acoth(a*x))=#

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

    @test_int [ℯ^(acoth(a*x)/4)*x^2, x, 19, (37*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(96*a^2) + (3*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^2)/(8*a) + (1/3)*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x^3 - (11*atan(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)))/(64*sqrt(2)*a^3) + (11*atan(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)))/(64*sqrt(2)*a^3) + (11*atan((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)))/(64*a^3) + (11*atanh((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)))/(64*a^3) - (11*log(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*sqrt(2)*a^3) + (11*log(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4)))/(128*sqrt(2)*a^3)]
    @test_int [ℯ^(acoth(a*x)/4)*x^1, x, 17, ((1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x)/(8*a) + (1/2)*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8)*x^2 - atan(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8))/(16*sqrt(2)*a^2) + atan(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8))/(16*sqrt(2)*a^2) + atan((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8))/(16*a^2) + atanh((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8))/(16*a^2) - log(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(32*sqrt(2)*a^2) + log(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(32*sqrt(2)*a^2)]
    @test_int [ℯ^(acoth(a*x)/4)*x^0, x, 16, (1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8)*x - atan(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8))/(2*sqrt(2)*a) + atan(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8))/(2*sqrt(2)*a) + atan((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8))/(2*a) + atanh((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8))/(2*a) - log(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(4*sqrt(2)*a) + log(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/(4*sqrt(2)*a)]
    @test_int [ℯ^(acoth(a*x)/4)/x^1, x, 39, (-sqrt(2 + sqrt(2)))*atan((sqrt(2 - sqrt(2)) - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 + sqrt(2))) - sqrt(2 - sqrt(2))*atan((sqrt(2 + sqrt(2)) - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 - sqrt(2))) + sqrt(2 + sqrt(2))*atan((sqrt(2 - sqrt(2)) + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 + sqrt(2))) + sqrt(2 - sqrt(2))*atan((sqrt(2 + sqrt(2)) + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 - sqrt(2))) - sqrt(2)*atan(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)) + sqrt(2)*atan(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8)) + 2*atan((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)) + 2*atanh((1 + 1/(a*x))^(1/8)/(1 - 1/(a*x))^(1/8)) + (1/2)*sqrt(2 - sqrt(2))*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (sqrt(2 - sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) - (1/2)*sqrt(2 - sqrt(2))*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (sqrt(2 - sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) + (1/2)*sqrt(2 + sqrt(2))*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (sqrt(2 + sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) - (1/2)*sqrt(2 + sqrt(2))*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (sqrt(2 + sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) - log(1 - (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/sqrt(2) + log(1 + (sqrt(2)*(1 + 1/(a*x))^(1/8))/(1 - 1/(a*x))^(1/8) + (1 + 1/(a*x))^(1/4)/(1 - 1/(a*x))^(1/4))/sqrt(2)]
    @test_int [ℯ^(acoth(a*x)/4)/x^2, x, 25, a*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8) - (1/4)*sqrt(2 + sqrt(2))*a*atan((sqrt(2 - sqrt(2)) - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 + sqrt(2))) - (1/4)*sqrt(2 - sqrt(2))*a*atan((sqrt(2 + sqrt(2)) - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 - sqrt(2))) + (1/4)*sqrt(2 + sqrt(2))*a*atan((sqrt(2 - sqrt(2)) + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 + sqrt(2))) + (1/4)*sqrt(2 - sqrt(2))*a*atan((sqrt(2 + sqrt(2)) + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 - sqrt(2))) + (1/8)*sqrt(2 - sqrt(2))*a*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (sqrt(2 - sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) - (1/8)*sqrt(2 - sqrt(2))*a*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (sqrt(2 - sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) + (1/8)*sqrt(2 + sqrt(2))*a*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (sqrt(2 + sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) - (1/8)*sqrt(2 + sqrt(2))*a*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (sqrt(2 + sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))]
    @test_int [ℯ^(acoth(a*x)/4)/x^3, x, 26, (1/8)*a^2*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(1/8) + (1/2)*a^2*(1 - 1/(a*x))^(7/8)*(1 + 1/(a*x))^(9/8) - (1/32)*sqrt(2 + sqrt(2))*a^2*atan((sqrt(2 - sqrt(2)) - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 + sqrt(2))) - (1/32)*sqrt(2 - sqrt(2))*a^2*atan((sqrt(2 + sqrt(2)) - (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 - sqrt(2))) + (1/32)*sqrt(2 + sqrt(2))*a^2*atan((sqrt(2 - sqrt(2)) + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 + sqrt(2))) + (1/32)*sqrt(2 - sqrt(2))*a^2*atan((sqrt(2 + sqrt(2)) + (2*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))/sqrt(2 - sqrt(2))) + (1/64)*sqrt(2 - sqrt(2))*a^2*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (sqrt(2 - sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) - (1/64)*sqrt(2 - sqrt(2))*a^2*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (sqrt(2 - sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) + (1/64)*sqrt(2 + sqrt(2))*a^2*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) - (sqrt(2 + sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8)) - (1/64)*sqrt(2 + sqrt(2))*a^2*log(1 + (1 - 1/(a*x))^(1/4)/(1 + 1/(a*x))^(1/4) + (sqrt(2 + sqrt(2))*(1 - 1/(a*x))^(1/8))/(1 + 1/(a*x))^(1/8))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*ℯ^(n*acoth(a*x))*with*m*symbolic=#

    @test_int [ℯ^(4*acoth(a*x))*x^m, x, 5, x^(1 + m)/(1 + m) + (4*x^(1 + m))/(1 - a*x) - 4*x^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, a*x)]
    @test_int [ℯ^(3*acoth(a*x))*x^m, x, 9, -((3*x^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)))/(1 + m)) - (x^m*HypergeometricFunctions._₂F₁(1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)))/(a*m) + (4*x^(1 + m)*HypergeometricFunctions._₂F₁(3/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)))/(1 + m) + (4*x^m*HypergeometricFunctions._₂F₁(3/2, -(m/2), 1 - m/2, 1/(a^2*x^2)))/(a*m)]
    @test_int [ℯ^(2*acoth(a*x))*x^m, x, 4, x^(1 + m)/(1 + m) - (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, a*x))/(1 + m)]
    @test_int [ℯ^(1*acoth(a*x))*x^m, x, 4, (x^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)))/(1 + m) + (x^m*HypergeometricFunctions._₂F₁(1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)))/(a*m)]
    @test_int [x^m/ℯ^(1*acoth(a*x)), x, 4, (x^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)))/(1 + m) - (x^m*HypergeometricFunctions._₂F₁(1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)))/(a*m)]
    @test_int [x^m/ℯ^(2*acoth(a*x)), x, 4, x^(1 + m)/(1 + m) - (2*x^(1 + m)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (-a)*x))/(1 + m)]
    @test_int [x^m/ℯ^(3*acoth(a*x)), x, 9, -((3*x^(1 + m)*HypergeometricFunctions._₂F₁(1/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)))/(1 + m)) + (x^m*HypergeometricFunctions._₂F₁(1/2, -(m/2), 1 - m/2, 1/(a^2*x^2)))/(a*m) + (4*x^(1 + m)*HypergeometricFunctions._₂F₁(3/2, (1/2)*(-1 - m), (1 - m)/2, 1/(a^2*x^2)))/(1 + m) - (4*x^m*HypergeometricFunctions._₂F₁(3/2, -(m/2), 1 - m/2, 1/(a^2*x^2)))/(a*m)]

    @test_int [ℯ^(5*acoth(a*x)/2)*x^m, x, 2, (x^(1 + m)*AppellF1(-1 - m, 5/4, -(5/4), -m, 1/(a*x), -(1/(a*x))))/(1 + m)]
    @test_int [ℯ^(3*acoth(a*x)/2)*x^m, x, 2, (x^(1 + m)*AppellF1(-1 - m, 3/4, -(3/4), -m, 1/(a*x), -(1/(a*x))))/(1 + m)]
    @test_int [ℯ^(1*acoth(a*x)/2)*x^m, x, 2, (x^(1 + m)*AppellF1(-1 - m, 1/4, -(1/4), -m, 1/(a*x), -(1/(a*x))))/(1 + m)]
    @test_int [x^m/ℯ^(1*acoth(a*x)/2), x, 2, (x^(1 + m)*AppellF1(-1 - m, -(1/4), 1/4, -m, 1/(a*x), -(1/(a*x))))/(1 + m)]
    @test_int [x^m/ℯ^(3*acoth(a*x)/2), x, 2, (x^(1 + m)*AppellF1(-1 - m, -(3/4), 3/4, -m, 1/(a*x), -(1/(a*x))))/(1 + m)]
    @test_int [x^m/ℯ^(5*acoth(a*x)/2), x, 2, (x^(1 + m)*AppellF1(-1 - m, -(5/4), 5/4, -m, 1/(a*x), -(1/(a*x))))/(1 + m)]

    @test_int [ℯ^(2*acoth(x)/3)*x^m, x, 2, (x^(1 + m)*AppellF1(-1 - m, 1/3, -(1/3), -m, 1/x, -(1/x)))/(1 + m)]
    @test_int [ℯ^(1*acoth(x)/3)*x^m, x, 2, (x^(1 + m)*AppellF1(-1 - m, 1/6, -(1/6), -m, 1/x, -(1/x)))/(1 + m)]

    @test_int [ℯ^(acoth(a*x)/4)*x^m, x, 2, (x^(1 + m)*AppellF1(-1 - m, 1/8, -(1/8), -m, 1/(a*x), -(1/(a*x))))/(1 + m)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*ℯ^(n*acoth(a*x))*with*n*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*x^m, x, 2, (x^(1 + m)*AppellF1(-1 - m, n/2, -(n/2), -m, 1/(a*x), -(1/(a*x))))/(1 + m)]

    @test_int [ℯ^(n*acoth(a*x))*x^2, x, 5, (n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/(6*a) + (1/3)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3 + (2*(2 + n^2)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*HypergeometricFunctions._₂F₁(2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)))/(3*a^3*(2 - n))]
    @test_int [ℯ^(n*acoth(a*x))*x^1, x, 3, (1/2)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2 + (2*n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*HypergeometricFunctions._₂F₁(2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)))/(a^2*(2 - n))]
    @test_int [ℯ^(n*acoth(a*x))*x^0, x, 2, (4*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*HypergeometricFunctions._₂F₁(2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)))/(a*(2 - n))]
    @test_int [ℯ^(n*acoth(a*x))/x^1, x, 4, -((2*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)))/((1 - 1/(a*x))^(n/2)*n)) + (2^(1 + n/2)*HypergeometricFunctions._₂F₁(-(n/2), -(n/2), 1 - n/2, (a - 1/x)/(2*a)))/((1 - 1/(a*x))^(n/2)*n)]
    @test_int [ℯ^(n*acoth(a*x))/x^2, x, 2, (2^(1 + n/2)*a*(1 - 1/(a*x))^(1 - n/2)*HypergeometricFunctions._₂F₁(1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)))/(2 - n)]
    @test_int [ℯ^(n*acoth(a*x))/x^3, x, 3, (1/2)*a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2) + (2^(n/2)*a^2*n*(1 - 1/(a*x))^(1 - n/2)*HypergeometricFunctions._₂F₁(1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)))/(2 - n)]
    @test_int [ℯ^(n*acoth(a*x))/x^4, x, 4, (1/6)*a^3*n*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2) + (a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(3*x) + (2^(n/2)*a^3*(2 + n^2)*(1 - 1/(a*x))^(1 - n/2)*HypergeometricFunctions._₂F₁(1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)))/(3*(2 - n))]
    @test_int [ℯ^(n*acoth(a*x))/x^5, x, 4, (1/24)*a^3*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(a*(6 + n^2) + (2*n)/x) + (a^2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(4*x^2) + (2^(-2 + n/2)*a^4*n*(8 + n^2)*(1 - 1/(a*x))^(1 - n/2)*HypergeometricFunctions._₂F₁(1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)))/(3*(2 - n))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*ℯ^(n*acoth(a*x))*(c-c*a*x)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))  (c-c*a*x)^p=#

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

    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^p, x, 4, (sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x*(c - a*c*x)^p)/(1 + p) + (((a - 1/x)/(a + 1/x))^(1/2 - p)*sqrt(1 + 1/(a*x))*(c - a*c*x)^p*HypergeometricFunctions._₂F₁(1/2 - p, -p, 1 - p, 2/((a + 1/x)*x)))/(a*p*(1 + p)*sqrt(1 - 1/(a*x)))]

    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^4, x, 9, (-(7/8))*a*c^4*sqrt(1 - 1/(a^2*x^2))*x^2 + (17/15)*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^3 - (3/4)*a^3*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^4 + (1/5)*a^4*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^5 + (7*c^4*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a)]
    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^3, x, 8, (-(5/8))*a*c^3*sqrt(1 - 1/(a^2*x^2))*x^2 + (2/3)*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^3 - (1/4)*a^3*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^4 + (5*c^3*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a)]
    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^2, x, 7, (-(1/2))*a*c^2*sqrt(1 - 1/(a^2*x^2))*x^2 + (1/3)*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3 + (c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [ℯ^acoth(a*x)*(c - a*c*x), x, 6, (-(1/2))*a*c*sqrt(1 - 1/(a^2*x^2))*x^2 + (c*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x), x, 7, (2*(a + 1/x))/(a^2*c*sqrt(1 - 1/(a^2*x^2))) - atanh(sqrt(1 - 1/(a^2*x^2)))/(a*c)]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^2, x, 3, -((a^2*(1 - 1/(a^2*x^2))^(3/2))/(3*c^2*(a - 1/x)^3))]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^3, x, 4, (a^3*(1 - 1/(a^2*x^2))^(3/2))/(5*c^3*(a - 1/x)^4) - (4*a^2*(1 - 1/(a^2*x^2))^(3/2))/(15*c^3*(a - 1/x)^3)]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^4, x, 6, -((a^4*(1 - 1/(a^2*x^2))^(3/2))/(7*c^4*(a - 1/x)^5)) + (12*a^3*(1 - 1/(a^2*x^2))^(3/2))/(35*c^4*(a - 1/x)^4) - (23*a^2*(1 - 1/(a^2*x^2))^(3/2))/(105*c^4*(a - 1/x)^3)]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^5, x, 8, (a^5*(1 - 1/(a^2*x^2))^(3/2))/(9*c^5*(a - 1/x)^6) - (8*a^4*(1 - 1/(a^2*x^2))^(3/2))/(21*c^5*(a - 1/x)^5) + (47*a^3*(1 - 1/(a^2*x^2))^(3/2))/(105*c^5*(a - 1/x)^4) - (58*a^2*(1 - 1/(a^2*x^2))^(3/2))/(315*c^5*(a - 1/x)^3)]

    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^p, x, 5, (2*(c - a*c*x)^p)/(a*p) - (c - a*c*x)^(1 + p)/(a*c*(1 + p))]

    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^5, x, 4, (2*c^5*(1 - a*x)^5)/(5*a) - (c^5*(1 - a*x)^6)/(6*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^4, x, 4, (c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^3, x, 4, (2*c^3*(1 - a*x)^3)/(3*a) - (c^3*(1 - a*x)^4)/(4*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^2, x, 4, -(c^2*x) + (a^2*c^2*x^3)/3]
    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x), x, 1, -(c*x) - (a*c*x^2)/2, (c*ℯ^(2*acoth(a*x))*(1 - a^2*x^2))/(2*a)]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x), x, 4, -2/(a*c*(1 - a*x)) - log(1 - a*x)/(a*c)]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x)^2, x, 3, -(x/(c^2*(1 - a*x)^2))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x)^3, x, 4, -2/(3*a*c^3*(1 - a*x)^3) + 1/(2*a*c^3*(1 - a*x)^2)]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x)^4, x, 4, -1/(2*a*c^4*(1 - a*x)^4) + 1/(3*a*c^4*(1 - a*x)^3)]

    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^p, x, 5, (3*sqrt(1 + 1/(a*x))*(c - a*c*x)^p)/(a*p*(1 + p)*sqrt(1 - 1/(a*x))) + ((1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^p)/((1 + p)*sqrt(1 - 1/(a*x))) - (3*((a - 1/x)/(a + 1/x))^(3/2 - p)*sqrt(1 + 1/(a*x))*(c - a*c*x)^p*HypergeometricFunctions._₂F₁(1 - p, 3/2 - p, 2 - p, 2/((a + 1/x)*x)))/(a^2*p*(1 - p^2)*(1 - 1/(a*x))^(3/2)*x)]

    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^4, x, 8, (3/8)*a*c^4*sqrt(1 - 1/(a^2*x^2))*x^2 - (1/4)*a^3*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^4 + (1/5)*a^4*c^4*(1 - 1/(a^2*x^2))^(5/2)*x^5 - (3*c^4*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^3, x, 7, (3/8)*a*c^3*sqrt(1 - 1/(a^2*x^2))*x^2 - (1/4)*a^3*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^4 - (3*c^3*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^2, x, 8, (1/2)*a*c^2*sqrt(1 - 1/(a^2*x^2))*x^2 + (1/3)*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3 - (c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x), x, 8, -2*c*sqrt(1 - 1/(a^2*x^2))*x - (1/2)*a*c*sqrt(1 - 1/(a^2*x^2))*x^2 - (3*c*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x), x, 9, (8*(a + 1/x))/(3*a^2*c*(1 - 1/(a^2*x^2))^(3/2)) + 4/(3*a^2*c*sqrt(1 - 1/(a^2*x^2))*x) - atanh(sqrt(1 - 1/(a^2*x^2)))/(a*c)]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^2, x, 3, -((a^4*(1 - 1/(a^2*x^2))^(5/2))/(5*c^2*(a - 1/x)^5))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^3, x, 4, (a^5*(1 - 1/(a^2*x^2))^(5/2))/(7*c^3*(a - 1/x)^6) - (6*a^4*(1 - 1/(a^2*x^2))^(5/2))/(35*c^3*(a - 1/x)^5)]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^4, x, 6, -((47*(a + 1/x)^5)/(315*a^6*c^4*(1 - 1/(a^2*x^2))^(5/2))) + (16*(a + 1/x)^6)/(63*a^7*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (a + 1/x)^7/(9*a^8*c^4*(1 - 1/(a^2*x^2))^(9/2))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^5, x, 7, -((152*(a + 1/x)^5)/(1155*a^6*c^5*(1 - 1/(a^2*x^2))^(5/2))) + (79*(a + 1/x)^6)/(231*a^7*c^5*(1 - 1/(a^2*x^2))^(7/2)) - (10*(a + 1/x)^7)/(33*a^8*c^5*(1 - 1/(a^2*x^2))^(9/2)) + (a + 1/x)^8/(11*a^9*c^5*(1 - 1/(a^2*x^2))^(11/2))]

    @test_int [ℯ^(4*acoth(a*x))*(c - a*c*x)^p, x, 5, (4*c*(c - a*c*x)^(-1 + p))/(a*(1 - p)) + (4*(c - a*c*x)^p)/(a*p) - (c - a*c*x)^(1 + p)/(a*c*(1 + p))]

    @test_int [ℯ^(4*acoth(a*x))*(c - a*c*x)^5, x, 4, -((c^5*(1 - a*x)^4)/a) + (4*c^5*(1 - a*x)^5)/(5*a) - (c^5*(1 - a*x)^6)/(6*a)]
    @test_int [ℯ^(4*acoth(a*x))*(c - a*c*x)^4, x, 5, c^4*x - (2*a^2*c^4*x^3)/3 + (a^4*c^4*x^5)/5]
    @test_int [ℯ^(4*acoth(a*x))*(c - a*c*x)^3, x, 4, (2*c^3*(1 + a*x)^3)/(3*a) - (c^3*(1 + a*x)^4)/(4*a)]
    @test_int [ℯ^(4*acoth(a*x))*(c - a*c*x)^2, x, 3, (c^2*(1 + a*x)^3)/(3*a)]
    @test_int [ℯ^(4*acoth(a*x))*(c - a*c*x), x, 4, -3*c*x - (1/2)*a*c*x^2 - (4*c*log(1 - a*x))/a]
    @test_int [ℯ^(4*acoth(a*x))/(c - a*c*x), x, 4, 2/(a*c*(1 - a*x)^2) - 4/(a*c*(1 - a*x)) - log(1 - a*x)/(a*c)]
    @test_int [ℯ^(4*acoth(a*x))/(c - a*c*x)^2, x, 3, (1 + a*x)^3/(6*a*c^2*(1 - a*x)^3)]
    @test_int [ℯ^(4*acoth(a*x))/(c - a*c*x)^3, x, 4, 1/(a*c^3*(1 - a*x)^4) - 4/(3*a*c^3*(1 - a*x)^3) + 1/(2*a*c^3*(1 - a*x)^2)]
    @test_int [ℯ^(4*acoth(a*x))/(c - a*c*x)^4, x, 4, 4/(5*a*c^4*(1 - a*x)^5) - 1/(a*c^4*(1 - a*x)^4) + 1/(3*a*c^4*(1 - a*x)^3)]

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

    @test_int [(c - a*c*x)^p/ℯ^acoth(a*x), x, 3, (((a - 1/x)/(a + 1/x))^(-(1/2) - p)*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x*(c - a*c*x)^p*HypergeometricFunctions._₂F₁(-1 - p, -(1/2) - p, -p, 2/((a + 1/x)*x)))/(1 + p)]

    @test_int [(c - a*c*x)^3/ℯ^acoth(a*x), x, 9, (20/3)*c^3*sqrt(1 - 1/(a^2*x^2))*x - (27/8)*a*c^3*sqrt(1 - 1/(a^2*x^2))*x^2 + (4/3)*a^2*c^3*sqrt(1 - 1/(a^2*x^2))*x^3 - (1/4)*a^3*c^3*sqrt(1 - 1/(a^2*x^2))*x^4 - (35*c^3*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a)]
    @test_int [(c - a*c*x)^2/ℯ^acoth(a*x), x, 8, (11/3)*c^2*sqrt(1 - 1/(a^2*x^2))*x - (3/2)*a*c^2*sqrt(1 - 1/(a^2*x^2))*x^2 + (1/3)*a^2*c^2*sqrt(1 - 1/(a^2*x^2))*x^3 - (5*c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [(c - a*c*x)/ℯ^acoth(a*x), x, 7, 2*c*sqrt(1 - 1/(a^2*x^2))*x - (1/2)*a*c*sqrt(1 - 1/(a^2*x^2))*x^2 - (3*c*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)), x, 5, -(atanh(sqrt(1 - 1/(a^2*x^2)))/(a*c))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^2), x, 3, -(sqrt(1 - 1/(a^2*x^2))/(c^2*(a - 1/x)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^3), x, 4, (a*sqrt(1 - 1/(a^2*x^2)))/(3*c^3*(a - 1/x)^2) - (2*sqrt(1 - 1/(a^2*x^2)))/(3*c^3*(a - 1/x))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^4), x, 6, -((a^2*sqrt(1 - 1/(a^2*x^2)))/(5*c^4*(a - 1/x)^3)) + (8*a*sqrt(1 - 1/(a^2*x^2)))/(15*c^4*(a - 1/x)^2) - (7*sqrt(1 - 1/(a^2*x^2)))/(15*c^4*(a - 1/x))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^5), x, 7, (a^3*sqrt(1 - 1/(a^2*x^2)))/(7*c^5*(a - 1/x)^4) - (18*a^2*sqrt(1 - 1/(a^2*x^2)))/(35*c^5*(a - 1/x)^3) + (23*a*sqrt(1 - 1/(a^2*x^2)))/(35*c^5*(a - 1/x)^2) - (12*sqrt(1 - 1/(a^2*x^2)))/(35*c^5*(a - 1/x))]

    @test_int [(c - a*c*x)^p/ℯ^(2*acoth(a*x)), x, 4, ((c - a*c*x)^(2 + p)*HypergeometricFunctions._₂F₁(1, 2 + p, 3 + p, (1/2)*(1 - a*x)))/(2*a*c^2*(2 + p))]

    @test_int [(c - a*c*x)^4/ℯ^(2*acoth(a*x)), x, 4, 16*c^4*x - (4*c^4*(1 - a*x)^2)/a - (4*c^4*(1 - a*x)^3)/(3*a) - (c^4*(1 - a*x)^4)/(2*a) - (c^4*(1 - a*x)^5)/(5*a) - (32*c^4*log(1 + a*x))/a]
    @test_int [(c - a*c*x)^3/ℯ^(2*acoth(a*x)), x, 4, 8*c^3*x - (2*c^3*(1 - a*x)^2)/a - (2*c^3*(1 - a*x)^3)/(3*a) - (c^3*(1 - a*x)^4)/(4*a) - (16*c^3*log(1 + a*x))/a]
    @test_int [(c - a*c*x)^2/ℯ^(2*acoth(a*x)), x, 4, 4*c^2*x - (c^2*(1 - a*x)^2)/a - (c^2*(1 - a*x)^3)/(3*a) - (8*c^2*log(1 + a*x))/a]
    @test_int [(c - a*c*x)/ℯ^(2*acoth(a*x)), x, 4, 3*c*x - (1/2)*a*c*x^2 - (4*c*log(1 + a*x))/a]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)), x, 3, -(log(1 + a*x)/(a*c))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^2), x, 4, -(atanh(a*x)/(a*c^2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^3), x, 5, -(1/(2*a*c^3*(1 - a*x))) - atanh(a*x)/(2*a*c^3)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^4), x, 5, -(1/(4*a*c^4*(1 - a*x)^2)) - 1/(4*a*c^4*(1 - a*x)) - atanh(a*x)/(4*a*c^4)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^5), x, 5, -(1/(6*a*c^5*(1 - a*x)^3)) - 1/(8*a*c^5*(1 - a*x)^2) - 1/(8*a*c^5*(1 - a*x)) - atanh(a*x)/(8*a*c^5)]

    @test_int [(c - a*c*x)^p/ℯ^(3*acoth(a*x)), x, 3, (((a - 1/x)/(a + 1/x))^(-(3/2) - p)*(1 - 1/(a*x))^(3/2)*x*(c - a*c*x)^p*HypergeometricFunctions._₂F₁(-(3/2) - p, -1 - p, -p, 2/((a + 1/x)*x)))/((1 + p)*sqrt(1 + 1/(a*x)))]

    @test_int [(c - a*c*x)^3/ℯ^(3*acoth(a*x)), x, 10, (32*c^3*(a - 1/x))/(a^2*sqrt(1 - 1/(a^2*x^2))) + 30*c^3*sqrt(1 - 1/(a^2*x^2))*x - (67/8)*a*c^3*sqrt(1 - 1/(a^2*x^2))*x^2 + 2*a^2*c^3*sqrt(1 - 1/(a^2*x^2))*x^3 - (1/4)*a^3*c^3*sqrt(1 - 1/(a^2*x^2))*x^4 - (315*c^3*atanh(sqrt(1 - 1/(a^2*x^2))))/(8*a)]
    @test_int [(c - a*c*x)^2/ℯ^(3*acoth(a*x)), x, 9, (16*c^2*(a - 1/x))/(a^2*sqrt(1 - 1/(a^2*x^2))) + (35/3)*c^2*sqrt(1 - 1/(a^2*x^2))*x - (5/2)*a*c^2*sqrt(1 - 1/(a^2*x^2))*x^2 + (1/3)*a^2*c^2*sqrt(1 - 1/(a^2*x^2))*x^3 - (35*c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [(c - a*c*x)/ℯ^(3*acoth(a*x)), x, 8, (8*c*(a - 1/x))/(a^2*sqrt(1 - 1/(a^2*x^2))) + 4*c*sqrt(1 - 1/(a^2*x^2))*x - (1/2)*a*c*sqrt(1 - 1/(a^2*x^2))*x^2 - (15*c*atanh(sqrt(1 - 1/(a^2*x^2))))/(2*a)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)), x, 6, (2*(a - 1/x))/(a^2*c*sqrt(1 - 1/(a^2*x^2))) - atanh(sqrt(1 - 1/(a^2*x^2)))/(a*c)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^2), x, 3, (a - 1/x)/(a^2*c^2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^3), x, 3, 1/(a*c^3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^4), x, 5, 2/(3*a*c^4*sqrt(1 - 1/(a^2*x^2))) - 1/(3*a^2*c^4*sqrt(1 - 1/(a^2*x^2))*(a - 1/x)*x^2)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^5), x, 6, -((4*(a + 1/x))/(5*a^2*c^5*(1 - 1/(a^2*x^2))^(3/2))) + (a + 1/x)^2/(5*a^3*c^5*(1 - 1/(a^2*x^2))^(5/2)) + (5*a + 2/x)/(5*a^2*c^5*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^6), x, 7, -((46*(a + 1/x))/(35*a^2*c^6*(1 - 1/(a^2*x^2))^(3/2))) + (24*(a + 1/x)^2)/(35*a^3*c^6*(1 - 1/(a^2*x^2))^(5/2)) - (a + 1/x)^3/(7*a^4*c^6*(1 - 1/(a^2*x^2))^(7/2)) + (35*a + 13/x)/(35*a^2*c^6*sqrt(1 - 1/(a^2*x^2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))  (c-c*a*x)^(p/2)=#

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

    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^(9/2), x, 7, -((32*(a - 1/x)^3*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(99*a^4*(1 - 1/(a*x))^(9/2))) + (9088*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(3465*a^4*(1 - 1/(a*x))^(9/2)*x^3) - (768*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(385*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (128*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(9/2))/(231*a^2*(1 - 1/(a*x))^(9/2)*x) + (2*(a - 1/x)^4*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(9/2))/(11*a^4*(1 - 1/(a*x))^(9/2))]
    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^(7/2), x, 6, -((8*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(21*a*(1 - 1/(a*x))^(7/2))) - (568*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(315*a^3*(1 - 1/(a*x))^(7/2)*x^2) + (48*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(7/2))/(35*a^2*(1 - 1/(a*x))^(7/2)*x) + (2*(a - 1/x)^3*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(7/2))/(9*a^3*(1 - 1/(a*x))^(7/2))]
    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^(5/2), x, 5, (64*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(105*(c - a*c*x)^(3/2)) + (16*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(35*sqrt(c - a*c*x)) + (2/7)*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*sqrt(c - a*c*x), -((36*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(5/2))/(35*a*(1 - 1/(a*x))^(5/2))) + (142*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(5/2))/(105*a^2*(1 - 1/(a*x))^(5/2)*x) + (2*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(5/2))/(7*(1 - 1/(a*x))^(5/2))]
    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^(3/2), x, 4, (8*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(15*(c - a*c*x)^(3/2)) + (2*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(5*sqrt(c - a*c*x)), -((14*(1 + 1/(a*x))^(3/2)*(c - a*c*x)^(3/2))/(15*a*(1 - 1/(a*x))^(3/2))) + (2*(1 + 1/(a*x))^(3/2)*x*(c - a*c*x)^(3/2))/(5*(1 - 1/(a*x))^(3/2))]
    @test_int [ℯ^acoth(a*x)*(c - a*c*x)^(1/2), x, 1, (2*ℯ^acoth(a*x)*(1 + a*x)*sqrt(c - a*c*x))/(3*a)]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^(1/2), x, 5, (2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x)/sqrt(c - a*c*x) - (2*sqrt(2)*sqrt(1 - 1/(a*x))*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(sqrt(a)*sqrt(1/x)*sqrt(c - a*c*x))]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^(3/2), x, 5, -((a*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x)/((a - 1/x)*(c - a*c*x)^(3/2))) - (sqrt(a)*(1 - 1/(a*x))^(3/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(sqrt(2)*(1/x)^(3/2)*(c - a*c*x)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^(5/2), x, 6, (a^2*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x^2)/(8*(a - 1/x)*(c - a*c*x)^(5/2)) - (a^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x^2)/(4*(a - 1/x)^2*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))^(5/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(8*sqrt(2)*(1/x)^(5/2)*(c - a*c*x)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a*c*x)^(7/2), x, 7, -((a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^2)/(6*(a - 1/x)^3*(c - a*c*x)^(7/2))) - (a^3*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x^3)/(32*(a - 1/x)*(c - a*c*x)^(7/2)) + (a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^3)/(16*(a - 1/x)^2*(c - a*c*x)^(7/2)) - (a^(5/2)*(1 - 1/(a*x))^(7/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(32*sqrt(2)*(1/x)^(7/2)*(c - a*c*x)^(7/2))]

    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^(7/2), x, 5, (4*(c - a*c*x)^(7/2))/(7*a) - (2*(c - a*c*x)^(9/2))/(9*a*c)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^(5/2), x, 5, (4*(c - a*c*x)^(5/2))/(5*a) - (2*(c - a*c*x)^(7/2))/(7*a*c)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^(3/2), x, 5, (4*(c - a*c*x)^(3/2))/(3*a) - (2*(c - a*c*x)^(5/2))/(5*a*c)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a*c*x)^(1/2), x, 5, (4*sqrt(c - a*c*x))/a - (2*(c - a*c*x)^(3/2))/(3*a*c)]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x)^(1/2), x, 5, -(4/(a*sqrt(c - a*c*x))) - (2*sqrt(c - a*c*x))/(a*c)]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x)^(3/2), x, 5, -(4/(3*a*(c - a*c*x)^(3/2))) + 2/(a*c*sqrt(c - a*c*x))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x)^(5/2), x, 5, -(4/(5*a*(c - a*c*x)^(5/2))) + 2/(3*a*c*(c - a*c*x)^(3/2))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a*c*x)^(7/2), x, 5, -(4/(7*a*(c - a*c*x)^(7/2))) + 2/(5*a*c*(c - a*c*x)^(5/2))]

    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^(9/2), x, 6, (-8*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(9/2))/(33*a*(1 - 1/(a*x))^(9/2)) - (856*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(9/2))/(1155*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (16*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(9/2))/(21*a^2*(1 - 1/(a*x))^(9/2)*x) + (2*(a - x^(-1))^3*(1 + 1/(a*x))^(5/2)*x*(c - a*c*x)^(9/2))/(11*a^3*(1 - 1/(a*x))^(9/2))]
    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^(7/2), x, 5, (-44*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(7/2))/(63*a*(1 - 1/(a*x))^(7/2)) + (214*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(7/2))/(315*a^2*(1 - 1/(a*x))^(7/2)*x) + (2*(1 + 1/(a*x))^(5/2)*x*(c - a*c*x)^(7/2))/(9*(1 - 1/(a*x))^(7/2))]
    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^(5/2), x, 4, (-18*(1 + 1/(a*x))^(5/2)*(c - a*c*x)^(5/2))/(35*a*(1 - 1/(a*x))^(5/2)) + (2*(1 + 1/(a*x))^(5/2)*x*(c - a*c*x)^(5/2))/(7*(1 - 1/(a*x))^(5/2))]
    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^(3/2), x, 1, (2*ℯ^(3*acoth(a*x))*(1 + a*x)*(c - a*c*x)^(3/2))/(5*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a*c*x)^(1/2), x, 6, (4*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(a*sqrt(1 - 1/(a*x))) + (2*(1 + 1/(a*x))^(3/2)*x*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(a^(3/2)*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^(1/2), x, 6, -((6*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/((a - 1/x)*sqrt(c - a*c*x))) + (2*a*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x)/((a - 1/x)*sqrt(c - a*c*x)) - (3*sqrt(2)*sqrt(1 - 1/(a*x))*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(sqrt(a)*sqrt(1/x)*sqrt(c - a*c*x))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^(3/2), x, 6, -((3*a*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x)/(4*(a - 1/x)*(c - a*c*x)^(3/2))) - (a^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)*x)/(2*(a - 1/x)^2*(c - a*c*x)^(3/2)) - (3*sqrt(a)*(1 - 1/(a*x))^(3/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(4*sqrt(2)*(1/x)^(3/2)*(c - a*c*x)^(3/2))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^(5/2), x, 7, (a^2*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x^2)/(16*(a - 1/x)*(c - a*c*x)^(5/2)) + (a^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x^2)/(24*(a - 1/x)^2*(c - a*c*x)^(5/2)) - (a^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^2)/(6*(a - 1/x)^3*(c - a*c*x)^(5/2)) + (a^(3/2)*(1 - 1/(a*x))^(5/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(16*sqrt(2)*(1/x)^(5/2)*(c - a*c*x)^(5/2))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a*c*x)^(7/2), x, 8, -((a^5*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x^2)/(8*(a - 1/x)^4*(c - a*c*x)^(7/2))) - (3*a^3*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x^3)/(256*(a - 1/x)*(c - a*c*x)^(7/2)) - (a^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^3)/(128*(a - 1/x)^2*(c - a*c*x)^(7/2)) + (a^5*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x^3)/(32*(a - 1/x)^3*(c - a*c*x)^(7/2)) - (3*a^(5/2)*(1 - 1/(a*x))^(7/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(256*sqrt(2)*(1/x)^(7/2)*(c - a*c*x)^(7/2))]

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

    @test_int [(c - a*c*x)^(9/2)/ℯ^acoth(a*x), x, 8, (16384*c^5*sqrt(1 - 1/(a^2*x^2))*x)/(693*sqrt(c - a*c*x)) + (4096/693)*c^4*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x) + (512/231)*c^3*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(3/2) + (640/693)*c^2*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(5/2) + (40/99)*c*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(7/2) + (2/11)*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(9/2), -((40*(a - 1/x)^4*sqrt(1 + 1/(a*x))*(c - a*c*x)^(9/2))/(99*a^5*(1 - 1/(a*x))^(9/2))) - (22016*sqrt(1 + 1/(a*x))*(c - a*c*x)^(9/2))/(693*a^5*(1 - 1/(a*x))^(9/2)*x^4) + (1024*sqrt(1 + 1/(a*x))*(c - a*c*x)^(9/2))/(99*a^4*(1 - 1/(a*x))^(9/2)*x^3) - (512*sqrt(1 + 1/(a*x))*(c - a*c*x)^(9/2))/(231*a^3*(1 - 1/(a*x))^(9/2)*x^2) + (640*(a - 1/x)^3*sqrt(1 + 1/(a*x))*(c - a*c*x)^(9/2))/(693*a^5*(1 - 1/(a*x))^(9/2)*x) + (2*(a - 1/x)^5*sqrt(1 + 1/(a*x))*x*(c - a*c*x)^(9/2))/(11*a^5*(1 - 1/(a*x))^(9/2))]
    @test_int [(c - a*c*x)^(7/2)/ℯ^acoth(a*x), x, 7, (4096*c^4*sqrt(1 - 1/(a^2*x^2))*x)/(315*sqrt(c - a*c*x)) + (1024/315)*c^3*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x) + (128/105)*c^2*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(3/2) + (32/63)*c*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(5/2) + (2/9)*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(7/2), -((32*(a - 1/x)^3*sqrt(1 + 1/(a*x))*(c - a*c*x)^(7/2))/(63*a^4*(1 - 1/(a*x))^(7/2))) + (5504*sqrt(1 + 1/(a*x))*(c - a*c*x)^(7/2))/(315*a^4*(1 - 1/(a*x))^(7/2)*x^3) - (256*sqrt(1 + 1/(a*x))*(c - a*c*x)^(7/2))/(45*a^3*(1 - 1/(a*x))^(7/2)*x^2) + (128*sqrt(1 + 1/(a*x))*(c - a*c*x)^(7/2))/(105*a^2*(1 - 1/(a*x))^(7/2)*x) + (2*(a - 1/x)^4*sqrt(1 + 1/(a*x))*x*(c - a*c*x)^(7/2))/(9*a^4*(1 - 1/(a*x))^(7/2))]
    @test_int [(c - a*c*x)^(5/2)/ℯ^acoth(a*x), x, 6, (256*c^3*sqrt(1 - 1/(a^2*x^2))*x)/(35*sqrt(c - a*c*x)) + (64/35)*c^2*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x) + (24/35)*c*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(3/2) + (2/7)*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(5/2), -((24*sqrt(1 + 1/(a*x))*(c - a*c*x)^(5/2))/(35*a*(1 - 1/(a*x))^(5/2))) - (344*sqrt(1 + 1/(a*x))*(c - a*c*x)^(5/2))/(35*a^3*(1 - 1/(a*x))^(5/2)*x^2) + (16*sqrt(1 + 1/(a*x))*(c - a*c*x)^(5/2))/(5*a^2*(1 - 1/(a*x))^(5/2)*x) + (2*(a - 1/x)^3*sqrt(1 + 1/(a*x))*x*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2))]
    @test_int [(c - a*c*x)^(3/2)/ℯ^acoth(a*x), x, 5, (64*c^2*sqrt(1 - 1/(a^2*x^2))*x)/(15*sqrt(c - a*c*x)) + (16/15)*c*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x) + (2/5)*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(3/2), -((28*sqrt(1 + 1/(a*x))*(c - a*c*x)^(3/2))/(15*a*(1 - 1/(a*x))^(3/2))) + (86*sqrt(1 + 1/(a*x))*(c - a*c*x)^(3/2))/(15*a^2*(1 - 1/(a*x))^(3/2)*x) + (2*sqrt(1 + 1/(a*x))*x*(c - a*c*x)^(3/2))/(5*(1 - 1/(a*x))^(3/2))]
    @test_int [(c - a*c*x)^(1/2)/ℯ^acoth(a*x), x, 4, (8*c*sqrt(1 - 1/(a^2*x^2))*x)/(3*sqrt(c - a*c*x)) + (2/3)*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x), -((10*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(3*a*sqrt(1 - 1/(a*x)))) + (2*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^(1/2)), x, 1, (2*(1 + a*x))/(ℯ^acoth(a*x)*(a*sqrt(c - a*c*x)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^(3/2)), x, 4, -((sqrt(2)*sqrt(a)*(1 - 1/(a*x))^(3/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/((1/x)^(3/2)*(c - a*c*x)^(3/2)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^(5/2)), x, 5, -((a^2*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x^2)/(2*(a - 1/x)*(c - a*c*x)^(5/2))) + (a^(3/2)*(1 - 1/(a*x))^(5/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(2*sqrt(2)*(1/x)^(5/2)*(c - a*c*x)^(5/2))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a*c*x)^(7/2)), x, 6, -((a^3*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x^2)/(4*(a - 1/x)^2*(c - a*c*x)^(7/2))) + (3*a^3*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x^3)/(16*(a - 1/x)*(c - a*c*x)^(7/2)) - (3*a^(5/2)*(1 - 1/(a*x))^(7/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(16*sqrt(2)*(1/x)^(7/2)*(c - a*c*x)^(7/2))]

    @test_int [(c - a*c*x)^(7/2)/ℯ^(2*acoth(a*x)), x, 10, -((32*c^3*sqrt(c - a*c*x))/a) - (16*c^2*(c - a*c*x)^(3/2))/(3*a) - (8*c*(c - a*c*x)^(5/2))/(5*a) - (4*(c - a*c*x)^(7/2))/(7*a) - (2*(c - a*c*x)^(9/2))/(9*a*c) + (32*sqrt(2)*c^(7/2)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a]
    @test_int [(c - a*c*x)^(5/2)/ℯ^(2*acoth(a*x)), x, 9, -((16*c^2*sqrt(c - a*c*x))/a) - (8*c*(c - a*c*x)^(3/2))/(3*a) - (4*(c - a*c*x)^(5/2))/(5*a) - (2*(c - a*c*x)^(7/2))/(7*a*c) + (16*sqrt(2)*c^(5/2)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a]
    @test_int [(c - a*c*x)^(3/2)/ℯ^(2*acoth(a*x)), x, 8, -((8*c*sqrt(c - a*c*x))/a) - (4*(c - a*c*x)^(3/2))/(3*a) - (2*(c - a*c*x)^(5/2))/(5*a*c) + (8*sqrt(2)*c^(3/2)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a]
    @test_int [(c - a*c*x)^(1/2)/ℯ^(2*acoth(a*x)), x, 7, -((4*sqrt(c - a*c*x))/a) - (2*(c - a*c*x)^(3/2))/(3*a*c) + (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^(1/2)), x, 6, -((2*sqrt(c - a*c*x))/(a*c)) + (2*sqrt(2)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/(a*sqrt(c))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^(3/2)), x, 5, (sqrt(2)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/(a*c^(3/2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^(5/2)), x, 6, -(1/(a*c^2*sqrt(c - a*c*x))) + atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))/(sqrt(2)*a*c^(5/2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^(7/2)), x, 7, -(1/(3*a*c^2*(c - a*c*x)^(3/2))) - 1/(2*a*c^3*sqrt(c - a*c*x)) + atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))/(2*sqrt(2)*a*c^(7/2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a*c*x)^(9/2)), x, 8, -(1/(5*a*c^2*(c - a*c*x)^(5/2))) - 1/(6*a*c^3*(c - a*c*x)^(3/2)) - 1/(4*a*c^4*sqrt(c - a*c*x)) + atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))/(4*sqrt(2)*a*c^(9/2))]

    @test_int [(c - a*c*x)^(9/2)/ℯ^(3*acoth(a*x)), x, 9, -((16*(a - 1/x)^5*(c - a*c*x)^(9/2))/(33*a^6*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x)))) - (94208*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x))*x^5) - (40960*(c - a*c*x)^(9/2))/(231*a^5*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x))*x^4) + (4096*(c - a*c*x)^(9/2))/(231*a^4*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x))*x^3) - (1024*(a - 1/x)^3*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x))*x^2) + (320*(a - 1/x)^4*(c - a*c*x)^(9/2))/(231*a^6*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x))*x) + (2*(a - 1/x)^6*x*(c - a*c*x)^(9/2))/(11*a^6*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x)))]
    @test_int [(c - a*c*x)^(7/2)/ℯ^(3*acoth(a*x)), x, 8, -((40*(a - 1/x)^4*(c - a*c*x)^(7/2))/(63*a^5*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x)))) + (11776*(c - a*c*x)^(7/2))/(63*a^5*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x^4) + (5120*(c - a*c*x)^(7/2))/(63*a^4*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x^3) - (512*(c - a*c*x)^(7/2))/(63*a^3*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x^2) + (128*(a - 1/x)^3*(c - a*c*x)^(7/2))/(63*a^5*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x) + (2*(a - 1/x)^5*x*(c - a*c*x)^(7/2))/(9*a^5*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x)))]
    @test_int [(c - a*c*x)^(5/2)/ℯ^(3*acoth(a*x)), x, 7, -((32*(a - 1/x)^3*(c - a*c*x)^(5/2))/(35*a^4*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))) - (2944*(c - a*c*x)^(5/2))/(35*a^4*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x^3) - (256*(c - a*c*x)^(5/2))/(7*a^3*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x^2) + (128*(c - a*c*x)^(5/2))/(35*a^2*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x) + (2*(a - 1/x)^4*x*(c - a*c*x)^(5/2))/(7*a^4*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))]
    @test_int [(c - a*c*x)^(3/2)/ℯ^(3*acoth(a*x)), x, 6, -((8*(c - a*c*x)^(3/2))/(5*a*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x)))) + (184*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x^2) + (16*(c - a*c*x)^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x) + (2*(a - 1/x)^3*x*(c - a*c*x)^(3/2))/(5*a^3*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x)))]
    @test_int [(c - a*c*x)^(1/2)/ℯ^(3*acoth(a*x)), x, 5, -((20*sqrt(c - a*c*x))/(3*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))) - (46*sqrt(c - a*c*x))/(3*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x) + (2*x*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^(1/2)), x, 4, (6*sqrt(1 - 1/(a*x)))/(a*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x)) + (2*sqrt(1 - 1/(a*x))*x)/(sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^(3/2)), x, 1, -((2*(1 + a*x))/(ℯ^(3*acoth(a*x))*(a*(c - a*c*x)^(3/2))))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^(5/2)), x, 5, (a*(1 - 1/(a*x))^(5/2)*x^2)/(sqrt(1 + 1/(a*x))*(c - a*c*x)^(5/2)) - (a^(3/2)*(1 - 1/(a*x))^(5/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(sqrt(2)*(1/x)^(5/2)*(c - a*c*x)^(5/2))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a*c*x)^(7/2)), x, 6, -((a^2*(1 - 1/(a*x))^(7/2)*x^2)/(2*(a - 1/x)*sqrt(1 + 1/(a*x))*(c - a*c*x)^(7/2))) - (3*a^2*(1 - 1/(a*x))^(7/2)*x^3)/(4*sqrt(1 + 1/(a*x))*(c - a*c*x)^(7/2)) + (3*a^(5/2)*(1 - 1/(a*x))^(7/2)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(4*sqrt(2)*(1/x)^(7/2)*(c - a*c*x)^(7/2))]

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

    #= ::Subsubsection::Closed:: =#
    #=p>0=#

    @test_int [ℯ^acoth(x)*x*(1 + x), x, 7, sqrt(1 + 1/x)*sqrt((-1 + x)/x)*x + (1/3)*(1 + 1/x)^(3/2)*sqrt((-1 + x)/x)*x^2 + (1/3)*(1 + 1/x)^(5/2)*sqrt((-1 + x)/x)*x^3 + atanh(sqrt(1 + 1/x)*sqrt((-1 + x)/x))]
    @test_int [ℯ^acoth(x)*(1 + x), x, 6, (3/2)*sqrt(1 + 1/x)*sqrt((-1 + x)/x)*x + (1/2)*(1 + 1/x)^(3/2)*sqrt((-1 + x)/x)*x^2 + (3/2)*atanh(sqrt(1 + 1/x)*sqrt((-1 + x)/x))]

    @test_int [ℯ^acoth(x)*(1 - x)*x, x, 3, (-(1/3))*(1 - 1/x^2)^(3/2)*x^3]
    @test_int [ℯ^acoth(x)*(1 - x), x, 6, (-(1/2))*sqrt(1 - 1/x^2)*x^2 + (1/2)*atanh(sqrt(1 - 1/x^2))]

    @test_int [ℯ^acoth(x)*x*(1 + x)^2, x, 8, (15/8)*sqrt(1 + 1/x)*sqrt((-1 + x)/x)*x + (5/8)*(1 + 1/x)^(3/2)*sqrt((-1 + x)/x)*x^2 + (1/4)*(1 + 1/x)^(5/2)*sqrt((-1 + x)/x)*x^3 + (1/4)*(1 + 1/x)^(7/2)*sqrt((-1 + x)/x)*x^4 + (15/8)*atanh(sqrt(1 + 1/x)*sqrt((-1 + x)/x))]
    @test_int [ℯ^acoth(x)*(1 + x)^2, x, 7, (5/2)*sqrt(1 + 1/x)*sqrt((-1 + x)/x)*x + (5/6)*(1 + 1/x)^(3/2)*sqrt((-1 + x)/x)*x^2 + (1/3)*(1 + 1/x)^(5/2)*sqrt((-1 + x)/x)*x^3 + (5/2)*atanh(sqrt(1 + 1/x)*sqrt((-1 + x)/x))]

    @test_int [ℯ^acoth(x)*(1 - x)^2*x, x, 8, (1/8)*sqrt(1 - 1/x^2)*x^2 - (1/3)*(1 - 1/x^2)^(3/2)*x^3 + (1/4)*(1 - 1/x^2)^(3/2)*x^4 - (1/8)*atanh(sqrt(1 - 1/x^2))]
    @test_int [ℯ^acoth(x)*(1 - x)^2, x, 7, (-(1/2))*sqrt(1 - 1/x^2)*x^2 + (1/3)*(1 - 1/x^2)^(3/2)*x^3 + (1/2)*atanh(sqrt(1 - 1/x^2))]

    #= ::Subsubsection::Closed:: =#
    #=p<0=#

    @test_int [(ℯ^acoth(x)*x)/(1 + x), x, 3, sqrt(1 + 1/x)*sqrt((-1 + x)/x)*x]
    @test_int [ℯ^acoth(x)/(1 + x), x, 4, atanh(sqrt(1 + 1/x)*sqrt((-1 + x)/x))]

    @test_int [(ℯ^acoth(x)*x)/(1 - x), x, 8, (2*(1 + 1/x))/sqrt(1 - 1/x^2) - sqrt(1 - 1/x^2)*x - 2*atanh(sqrt(1 - 1/x^2))]
    @test_int [ℯ^acoth(x)/(1 - x), x, 7, (2*(1 + 1/x))/sqrt(1 - 1/x^2) - atanh(sqrt(1 - 1/x^2))]

    @test_int [(ℯ^acoth(x)*x)/(1 + x)^2, x, 5, -(sqrt((-1 + x)/x)/sqrt(1 + 1/x)) + atanh(sqrt(1 + 1/x)*sqrt((-1 + x)/x))]
    @test_int [ℯ^acoth(x)/(1 + x)^2, x, 3, sqrt((-1 + x)/x)/sqrt(1 + 1/x)]

    @test_int [(ℯ^acoth(x)*x)/(1 - x)^2, x, 9, -((4*(1 + 1/x))/(3*(1 - 1/x^2)^(3/2))) - (3 + 5/x)/(3*sqrt(1 - 1/x^2)) + atanh(sqrt(1 - 1/x^2))]
    @test_int [ℯ^acoth(x)/(1 - x)^2, x, 3, -((1 - 1/x^2)^(3/2)/(3*(1 - 1/x)^3))]

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

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

    @test_int [ℯ^acoth(a*x)*x^m*sqrt(c - a*c*x), x, 3, (2*x^(1 + m)*sqrt(c - a*c*x)*HypergeometricFunctions._₂F₁(-(1/2), -(3/2) - m, -(1/2) - m, -(1/(a*x))))/((3 + 2*m)*sqrt(1 - 1/(a*x)))]

    @test_int [ℯ^acoth(a*x)*x^2*sqrt(c - a*c*x), x, 5, (16*(1 + 1/(a*x))^(3/2)*x*sqrt(c - a*c*x))/(105*a^2*sqrt(1 - 1/(a*x))) - (8*(1 + 1/(a*x))^(3/2)*x^2*sqrt(c - a*c*x))/(35*a*sqrt(1 - 1/(a*x))) + (2*(1 + 1/(a*x))^(3/2)*x^3*sqrt(c - a*c*x))/(7*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^acoth(a*x)*x*sqrt(c - a*c*x), x, 4, -((4*(1 + 1/(a*x))^(3/2)*x*sqrt(c - a*c*x))/(15*a*sqrt(1 - 1/(a*x)))) + (2*(1 + 1/(a*x))^(3/2)*x^2*sqrt(c - a*c*x))/(5*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - a*c*x), x, 1, (2*ℯ^acoth(a*x)*(1 + a*x)*sqrt(c - a*c*x))/(3*a)]
    @test_int [(ℯ^acoth(a*x)*sqrt(c - a*c*x))/x, x, 5, (2*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/sqrt(1 - 1/(a*x)) - (2*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/(sqrt(a)*sqrt(1 - 1/(a*x)))]
    @test_int [(ℯ^acoth(a*x)*sqrt(c - a*c*x))/x^2, x, 5, -((sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*x)) - (sqrt(a)*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/sqrt(1 - 1/(a*x))]

    @test_int [ℯ^(2*acoth(a*x))*x^3*sqrt(c - a*c*x), x, 5, (4*sqrt(c - a*c*x))/a^4 - (14*(c - a*c*x)^(3/2))/(3*a^4*c) + (18*(c - a*c*x)^(5/2))/(5*a^4*c^2) - (10*(c - a*c*x)^(7/2))/(7*a^4*c^3) + (2*(c - a*c*x)^(9/2))/(9*a^4*c^4)]
    @test_int [ℯ^(2*acoth(a*x))*x^2*sqrt(c - a*c*x), x, 5, (4*sqrt(c - a*c*x))/a^3 - (10*(c - a*c*x)^(3/2))/(3*a^3*c) + (8*(c - a*c*x)^(5/2))/(5*a^3*c^2) - (2*(c - a*c*x)^(7/2))/(7*a^3*c^3)]
    @test_int [ℯ^(2*acoth(a*x))*x*sqrt(c - a*c*x), x, 5, (4*sqrt(c - a*c*x))/a^2 - (2*(c - a*c*x)^(3/2))/(a^2*c) + (2*(c - a*c*x)^(5/2))/(5*a^2*c^2)]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - a*c*x), x, 5, (4*sqrt(c - a*c*x))/a - (2*(c - a*c*x)^(3/2))/(3*a*c)]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a*c*x))/x, x, 6, 2*sqrt(c - a*c*x) + 2*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a*c*x))/x^2, x, 6, sqrt(c - a*c*x)/x + 3*a*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a*c*x))/x^3, x, 7, sqrt(c - a*c*x)/(2*x^2) + (7*a*sqrt(c - a*c*x))/(4*x) + (7/4)*a^2*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a*c*x))/x^4, x, 8, sqrt(c - a*c*x)/(3*x^3) + (11*a*sqrt(c - a*c*x))/(12*x^2) + (11*a^2*sqrt(c - a*c*x))/(8*x) + (11/8)*a^3*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a*c*x))/x^5, x, 9, sqrt(c - a*c*x)/(4*x^4) + (5*a*sqrt(c - a*c*x))/(8*x^3) + (25*a^2*sqrt(c - a*c*x))/(32*x^2) + (75*a^3*sqrt(c - a*c*x))/(64*x) + (75/64)*a^4*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c))]

    @test_int [ℯ^(3*acoth(a*x))*x^3*sqrt(c - a*c*x), x, 10, (1576*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(315*a^4*sqrt(1 - 1/(a*x))) + (472*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(315*a^3*sqrt(1 - 1/(a*x))) + (92*sqrt(1 + 1/(a*x))*x^2*sqrt(c - a*c*x))/(105*a^2*sqrt(1 - 1/(a*x))) + (38*sqrt(1 + 1/(a*x))*x^3*sqrt(c - a*c*x))/(63*a*sqrt(1 - 1/(a*x))) + (2*sqrt(1 + 1/(a*x))*x^4*sqrt(c - a*c*x))/(9*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(a^(9/2)*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))*x^2*sqrt(c - a*c*x), x, 9, (104*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(21*a^3*sqrt(1 - 1/(a*x))) + (32*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(21*a^2*sqrt(1 - 1/(a*x))) + (6*sqrt(1 + 1/(a*x))*x^2*sqrt(c - a*c*x))/(7*a*sqrt(1 - 1/(a*x))) + (2*sqrt(1 + 1/(a*x))*x^3*sqrt(c - a*c*x))/(7*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(a^(7/2)*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))*x*sqrt(c - a*c*x), x, 7, (4*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(a^2*sqrt(1 - 1/(a*x))) + (2*(1 + 1/(a*x))^(3/2)*x*sqrt(c - a*c*x))/(3*a*sqrt(1 - 1/(a*x))) + (2*(1 + 1/(a*x))^(5/2)*x^2*sqrt(c - a*c*x))/(5*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(a^(5/2)*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - a*c*x), x, 6, (4*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(a*sqrt(1 - 1/(a*x))) + (2*(1 + 1/(a*x))^(3/2)*x*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(a^(3/2)*sqrt(1 - 1/(a*x)))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a*c*x))/x, x, 8, (2*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/sqrt(1 - 1/(a*x)) + (2*sqrt(1/x)*sqrt(c - a*c*x)*asinh(sqrt(1/x)/sqrt(a)))/(sqrt(a)*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/(sqrt(a)*sqrt(1 - 1/(a*x)))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a*c*x))/x^2, x, 8, (sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*x) + (5*sqrt(a)*sqrt(1/x)*sqrt(c - a*c*x)*asinh(sqrt(1/x)/sqrt(a)))/sqrt(1 - 1/(a*x)) - (4*sqrt(2)*sqrt(a)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/sqrt(1 - 1/(a*x))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a*c*x))/x^3, x, 9, (7*a*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(4*sqrt(1 - 1/(a*x))*x) + (a*(1 + 1/(a*x))^(3/2)*sqrt(c - a*c*x))/(2*sqrt(1 - 1/(a*x))*x) + (23*a^(3/2)*sqrt(1/x)*sqrt(c - a*c*x)*asinh(sqrt(1/x)/sqrt(a)))/(4*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*a^(3/2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/sqrt(1 - 1/(a*x))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a*c*x))/x^4, x, 10, (a*(1 + 1/(a*x))^(3/2)*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x))*x^2) + (13*a^2*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(8*sqrt(1 - 1/(a*x))*x) + (3*a^2*(1 + 1/(a*x))^(3/2)*sqrt(c - a*c*x))/(4*sqrt(1 - 1/(a*x))*x) + (45*a^(5/2)*sqrt(1/x)*sqrt(c - a*c*x)*asinh(sqrt(1/x)/sqrt(a)))/(8*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*a^(5/2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/sqrt(1 - 1/(a*x))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a*c*x))/x^5, x, 11, (a*(1 + 1/(a*x))^(3/2)*sqrt(c - a*c*x))/(4*sqrt(1 - 1/(a*x))*x^3) + (11*a^2*(1 + 1/(a*x))^(3/2)*sqrt(c - a*c*x))/(24*sqrt(1 - 1/(a*x))*x^2) + (107*a^3*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(64*sqrt(1 - 1/(a*x))*x) + (21*a^3*(1 + 1/(a*x))^(3/2)*sqrt(c - a*c*x))/(32*sqrt(1 - 1/(a*x))*x) + (363*a^(7/2)*sqrt(1/x)*sqrt(c - a*c*x)*asinh(sqrt(1/x)/sqrt(a)))/(64*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*a^(7/2)*sqrt(1/x)*sqrt(c - a*c*x)*atanh((sqrt(2)*sqrt(1/x))/(sqrt(a)*sqrt(1 + 1/(a*x)))))/sqrt(1 - 1/(a*x))]

    @test_int [ℯ^acoth(x)*x*(1 + x)^(3/2), x, 6, (46*sqrt(-((1 - x)/x))*(1 + x)^(3/2))/(21*(1 + 1/x)^(3/2)) + (92*sqrt(-((1 - x)/x))*(1 + x)^(3/2))/(21*(1 + 1/x)^(3/2)*x) + (8*sqrt(-((1 - x)/x))*x*(1 + x)^(3/2))/(7*(1 + 1/x)^(3/2)) + (2*sqrt(-((1 - x)/x))*x^2*(1 + x)^(3/2))/(7*(1 + 1/x)^(3/2))]
    @test_int [ℯ^acoth(x)*(1 + x)^(3/2), x, 5, (28*sqrt(-((1 - x)/x))*(1 + x)^(3/2))/(15*(1 + 1/x)^(3/2)) + (86*sqrt(-((1 - x)/x))*(1 + x)^(3/2))/(15*(1 + 1/x)^(3/2)*x) + (2*sqrt(-((1 - x)/x))*x*(1 + x)^(3/2))/(5*(1 + 1/x)^(3/2))]

    @test_int [ℯ^acoth(x)*(1 - x)^(3/2)*x, x, 5, (44*(1 + 1/x)^(3/2)*(1 - x)^(3/2))/(105*(1 - 1/x)^(3/2)) - (22*(1 + 1/x)^(3/2)*(1 - x)^(3/2)*x)/(35*(1 - 1/x)^(3/2)) + (2*(1 + 1/x)^(3/2)*(1 - x)^(3/2)*x^2)/(7*(1 - 1/x)^(3/2))]
    @test_int [ℯ^acoth(x)*(1 - x)^(3/2), x, 4, (-14*(1 + x^(-1))^(3/2)*(1 - x)^(3/2))/(15*(1 - x^(-1))^(3/2)) + (2*(1 + x^(-1))^(3/2)*(1 - x)^(3/2)*x)/(5*(1 - x^(-1))^(3/2))]

    @test_int [ℯ^acoth(x)*x*sqrt(1 + x), x, 5, (12*sqrt(-((1 - x)/x))*sqrt(1 + x))/(5*sqrt(1 + 1/x)) + (6*sqrt(-((1 - x)/x))*x*sqrt(1 + x))/(5*sqrt(1 + 1/x)) + (2*sqrt(-((1 - x)/x))*x^2*sqrt(1 + x))/(5*sqrt(1 + 1/x))]
    @test_int [ℯ^acoth(x)*sqrt(1 + x), x, 4, (10*sqrt(-((1 - x)/x))*sqrt(1 + x))/(3*sqrt(1 + 1/x)) + (2*sqrt(-((1 - x)/x))*x*sqrt(1 + x))/(3*sqrt(1 + 1/x))]

    @test_int [ℯ^acoth(x)*sqrt(1 - x)*x, x, 4, -((4*(1 + 1/x)^(3/2)*sqrt(1 - x)*x)/(15*sqrt(1 - 1/x))) + (2*(1 + 1/x)^(3/2)*sqrt(1 - x)*x^2)/(5*sqrt(1 - 1/x))]
    @test_int [ℯ^acoth(x)*sqrt(1 - x), x, 1, (2/3)*ℯ^acoth(x)*sqrt(1 - x)*(1 + x)]

    @test_int [(ℯ^acoth(x)*x)/sqrt(1 + x), x, 4, (4*sqrt(1 + 1/x)*sqrt(-((1 - x)/x))*x)/(3*sqrt(1 + x)) + (2*sqrt(1 + 1/x)*sqrt(-((1 - x)/x))*x^2)/(3*sqrt(1 + x))]
    @test_int [ℯ^acoth(x)/sqrt(1 + x), x, 3, (2*sqrt(1 + 1/x)*sqrt(-((1 - x)/x))*x)/sqrt(1 + x)]

    @test_int [(ℯ^acoth(x)*x)/sqrt(1 - x), x, 6, (2*sqrt(1 - 1/x)*sqrt(1 + 1/x)*x)/sqrt(1 - x) + (2*sqrt(1 - 1/x)*(1 + 1/x)^(3/2)*x^2)/(3*sqrt(1 - x)) - (2*sqrt(2)*sqrt(1 - 1/x)*atanh((sqrt(2)*sqrt(1/x))/sqrt(1 + 1/x)))/(sqrt(1 - x)*sqrt(1/x))]
    @test_int [ℯ^acoth(x)/sqrt(1 - x), x, 5, (2*sqrt(1 - x^(-1))*sqrt(1 + x^(-1))*x)/sqrt(1 - x) - (2*sqrt(2)*sqrt(1 - x^(-1))*atanh((sqrt(2)*sqrt(x^(-1)))/sqrt(1 + x^(-1))))/(sqrt(1 - x)*sqrt(x^(-1)))]

    @test_int [(ℯ^acoth(x)*x)/(1 + x)^(3/2), x, 5, (2*(1 + 1/x)^(3/2)*sqrt(-((1 - x)/x))*x^2)/(1 + x)^(3/2) + (sqrt(2)*(1 + 1/x)^(3/2)*atan((sqrt(2)*sqrt(1/x))/sqrt(-((1 - x)/x))))/((1/x)^(3/2)*(1 + x)^(3/2))]
    @test_int [ℯ^acoth(x)/(1 + x)^(3/2), x, 4, -((sqrt(2)*(1 + 1/x)^(3/2)*atan((sqrt(2)*sqrt(1/x))/sqrt(-((1 - x)/x))))/((1/x)^(3/2)*(1 + x)^(3/2)))]

    @test_int [(ℯ^acoth(x)*x)/(1 - x)^(3/2), x, 6, (5*(1 - 1/x)^(3/2)*sqrt(1 + 1/x)*x^2)/(2*(1 - x)^(3/2)) - (sqrt(1 - 1/x)*(1 + 1/x)^(3/2)*x^2)/(2*(1 - x)^(3/2)) - (5*(1 - 1/x)^(3/2)*atanh((sqrt(2)*sqrt(1/x))/sqrt(1 + 1/x)))/(sqrt(2)*(1 - x)^(3/2)*(1/x)^(3/2))]
    @test_int [ℯ^acoth(x)/(1 - x)^(3/2), x, 5, -((sqrt(1 - x^(-1))*sqrt(1 + x^(-1))*x)/(1 - x)^(3/2)) - ((1 - x^(-1))^(3/2)*atanh((sqrt(2)*sqrt(x^(-1)))/sqrt(1 + x^(-1))))/(sqrt(2)*(1 - x)^(3/2)*(x^(-1))^(3/2))]

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

    @test_int [(x^m*sqrt(c - a*c*x))/ℯ^acoth(a*x), x, 4, If(13>=8, (2*sqrt(1 + 1/(a*x))*x^(1 + m)*sqrt(c - a*c*x))/((3 + 2*m)*sqrt(1 - 1/(a*x))) - (2*(5 + 4*m)*x^m*sqrt(c - a*c*x)*HypergeometricFunctions._₂F₁(1/2, -(1/2) - m, 1/2 - m, -(1/(a*x))))/(a*(1 + 2*m)*(3 + 2*m)*sqrt(1 - 1/(a*x))), (2*sqrt(1 + 1/(a*x))*x^(1 + m)*sqrt(c - a*c*x))/((3 + 2*m)*sqrt(1 - 1/(a*x))) - (2*(5 + 4*m)*x^m*sqrt(c - a*c*x)*HypergeometricFunctions._₂F₁(1/2, -(1/2) - m, 1/2 - m, -(1/(a*x))))/(a*(3 + 8*m + 4*m^2)*sqrt(1 - 1/(a*x))))]

    @test_int [(x^2*sqrt(c - a*c*x))/ℯ^acoth(a*x), x, 6, (152*c*sqrt(1 - 1/(a^2*x^2))*x)/(105*a^2*sqrt(c - a*c*x)) + (38*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x))/(105*a^2) + (6*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(3/2))/(35*a^2*c) - (2*sqrt(1 - 1/(a^2*x^2))*x^2*(c - a*c*x)^(3/2))/(7*a*c), -((208*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(105*a^3*sqrt(1 - 1/(a*x)))) + (104*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(105*a^2*sqrt(1 - 1/(a*x))) - (26*sqrt(1 + 1/(a*x))*x^2*sqrt(c - a*c*x))/(35*a*sqrt(1 - 1/(a*x))) + (2*sqrt(1 + 1/(a*x))*x^3*sqrt(c - a*c*x))/(7*sqrt(1 - 1/(a*x)))]
    @test_int [(x*sqrt(c - a*c*x))/ℯ^acoth(a*x), x, 5, -((8*c*sqrt(1 - 1/(a^2*x^2))*x)/(5*a*sqrt(c - a*c*x))) - (2*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x))/(5*a) - (2*sqrt(1 - 1/(a^2*x^2))*x*(c - a*c*x)^(3/2))/(5*a*c), (12*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(5*a^2*sqrt(1 - 1/(a*x))) - (6*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(5*a*sqrt(1 - 1/(a*x))) + (2*sqrt(1 + 1/(a*x))*x^2*sqrt(c - a*c*x))/(5*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/ℯ^acoth(a*x), x, 4, (8*c*sqrt(1 - 1/(a^2*x^2))*x)/(3*sqrt(c - a*c*x)) + (2/3)*sqrt(1 - 1/(a^2*x^2))*x*sqrt(c - a*c*x), -((10*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(3*a*sqrt(1 - 1/(a*x)))) + (2*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^acoth(a*x)*x), x, 5, (2*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/sqrt(1 - 1/(a*x)) + (2*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/(sqrt(a)*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^acoth(a*x)*x^2), x, 5, (sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*x) - (3*sqrt(a)*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/sqrt(1 - 1/(a*x))]

    @test_int [(x^3*sqrt(c - a*c*x))/ℯ^(2*acoth(a*x)), x, 9, (4*sqrt(c - a*c*x))/a^4 + (2*(c - a*c*x)^(3/2))/(3*a^4*c) + (2*(c - a*c*x)^(5/2))/(5*a^4*c^2) - (2*(c - a*c*x)^(7/2))/(7*a^4*c^3) + (2*(c - a*c*x)^(9/2))/(9*a^4*c^4) - (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a^4]
    @test_int [(x^2*sqrt(c - a*c*x))/ℯ^(2*acoth(a*x)), x, 9, -((4*sqrt(c - a*c*x))/a^3) - (2*(c - a*c*x)^(3/2))/(3*a^3*c) - (2*(c - a*c*x)^(7/2))/(7*a^3*c^3) + (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a^3]
    @test_int [(x*sqrt(c - a*c*x))/ℯ^(2*acoth(a*x)), x, 8, (4*sqrt(c - a*c*x))/a^2 + (2*(c - a*c*x)^(3/2))/(3*a^2*c) + (2*(c - a*c*x)^(5/2))/(5*a^2*c^2) - (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a^2]
    @test_int [sqrt(c - a*c*x)/ℯ^(2*acoth(a*x)), x, 7, -((4*sqrt(c - a*c*x))/a) - (2*(c - a*c*x)^(3/2))/(3*a*c) + (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c))))/a]
    @test_int [sqrt(c - a*c*x)/(ℯ^(2*acoth(a*x))*x), x, 9, 2*sqrt(c - a*c*x) + 2*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c)) - 4*sqrt(2)*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(2*acoth(a*x))*x^2), x, 9, sqrt(c - a*c*x)/x - 5*a*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c)) + 4*sqrt(2)*a*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(2*acoth(a*x))*x^3), x, 10, sqrt(c - a*c*x)/(2*x^2) - (9*a*sqrt(c - a*c*x))/(4*x) + (23/4)*a^2*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c)) - 4*sqrt(2)*a^2*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(2*acoth(a*x))*x^4), x, 11, sqrt(c - a*c*x)/(3*x^3) - (13*a*sqrt(c - a*c*x))/(12*x^2) + (19*a^2*sqrt(c - a*c*x))/(8*x) - (45/8)*a^3*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c)) + 4*sqrt(2)*a^3*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(2*acoth(a*x))*x^5), x, 12, sqrt(c - a*c*x)/(4*x^4) - (17*a*sqrt(c - a*c*x))/(24*x^3) + (107*a^2*sqrt(c - a*c*x))/(96*x^2) - (149*a^3*sqrt(c - a*c*x))/(64*x) + (363/64)*a^4*sqrt(c)*atanh(sqrt(c - a*c*x)/sqrt(c)) - 4*sqrt(2)*a^4*sqrt(c)*atanh(sqrt(c - a*c*x)/(sqrt(2)*sqrt(c)))]

    @test_int [(x^3*sqrt(c - a*c*x))/ℯ^(3*acoth(a*x)), x, 8, (1312*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(45*a^4*sqrt(1 - 1/(a*x))) - (656*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(45*a^3*sqrt(1 - 1/(a*x))) - (82*x^2*sqrt(c - a*c*x))/(9*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (164*sqrt(1 + 1/(a*x))*x^2*sqrt(c - a*c*x))/(15*a^2*sqrt(1 - 1/(a*x))) - (8*x^3*sqrt(c - a*c*x))/(9*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (2*x^4*sqrt(c - a*c*x))/(9*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))]
    @test_int [(x^2*sqrt(c - a*c*x))/ℯ^(3*acoth(a*x)), x, 7, -((2672*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(105*a^3*sqrt(1 - 1/(a*x)))) - (334*x*sqrt(c - a*c*x))/(35*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (1336*sqrt(1 + 1/(a*x))*x*sqrt(c - a*c*x))/(105*a^2*sqrt(1 - 1/(a*x))) - (44*x^2*sqrt(c - a*c*x))/(35*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (2*x^3*sqrt(c - a*c*x))/(7*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))]
    @test_int [(x^1*sqrt(c - a*c*x))/ℯ^(3*acoth(a*x)), x, 6, -((158*sqrt(c - a*c*x))/(15*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))) + (316*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(15*a^2*sqrt(1 - 1/(a*x))) - (32*x*sqrt(c - a*c*x))/(15*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (2*x^2*sqrt(c - a*c*x))/(5*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/ℯ^(3*acoth(a*x)), x, 5, -((20*sqrt(c - a*c*x))/(3*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))) - (46*sqrt(c - a*c*x))/(3*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x) + (2*x*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(3*acoth(a*x))*x^1), x, 6, (2*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (10*sqrt(c - a*c*x))/(a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x) - (2*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/(sqrt(a)*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(3*acoth(a*x))*x^2), x, 6, (-8*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x) - (sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*x) + (7*sqrt(a)*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/sqrt(1 - 1/(a*x))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(3*acoth(a*x))*x^3), x, 7, (-8*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x^2) - (sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(2*sqrt(1 - 1/(a*x))*x^2) + (47*a*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(4*sqrt(1 - 1/(a*x))*x) - (47*a^(3/2)*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/(4*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(3*acoth(a*x))*x^4), x, 8, (-8*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x^3) - (sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(3*sqrt(1 - 1/(a*x))*x^3) + (119*a*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(12*sqrt(1 - 1/(a*x))*x^2) - (119*a^2*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(8*sqrt(1 - 1/(a*x))*x) + (119*a^(5/2)*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/(8*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - a*c*x)/(ℯ^(3*acoth(a*x))*x^5), x, 9, (-8*sqrt(c - a*c*x))/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x^4) - (sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(4*sqrt(1 - 1/(a*x))*x^4) + (223*a*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(24*sqrt(1 - 1/(a*x))*x^3) - (1115*a^2*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(96*sqrt(1 - 1/(a*x))*x^2) + (1115*a^3*sqrt(1 + 1/(a*x))*sqrt(c - a*c*x))/(64*sqrt(1 - 1/(a*x))*x) - (1115*a^(7/2)*sqrt(x^(-1))*sqrt(c - a*c*x)*asinh(sqrt(x^(-1))/sqrt(a)))/(64*sqrt(1 - 1/(a*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))  (c-c*a*x)^p*with*n*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(n/2 + 2), x, 6, -(((56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a*(4 + n)*(6 + n))) + (2*(56 + 14*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((4 + n)/2))/(a^2*(6 + n)*(8 + 6*n + n^2)*x) + ((8 + n)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 + n)/2))/(6 + n) - ((a - 1/x)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((4 + n)/2))/a]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(n/2 + 1), x, 4, If(13>=8, -((2*(6 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((2 + n)/2))/(a*(2 + n)*(4 + n))) + (2*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((2 + n)/2))/(4 + n), -((2*(6 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((2 + n)/2))/(a*(8 + 6*n + n^2))) + (2*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((2 + n)/2))/(4 + n))]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(n/2 + 0), x, 1, (2*ℯ^(n*acoth(a*x))*(1 + a*x)*(c - a*c*x)^(n/2))/(a*(2 + n))]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(n/2 - 1), x, 3, (2*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^(n/2)*x*(c - a*c*x)^((1/2)*(-2 + n))*HypergeometricFunctions._₂F₁(1, -(n/2), 1 - n/2, 2/((a + 1/x)*x)))/n]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(n/2 - 2), x, 3, -((2*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*x*(c - a*c*x)^((1/2)*(-4 + n))*HypergeometricFunctions._₂F₁(2, 1 - n/2, 2 - n/2, 2/((a + 1/x)*x)))/(2 - n))]

    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^p, x, 3, (((a - 1/x)/(a + 1/x))^((1/2)*(n - 2*p))*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^p*HypergeometricFunctions._₂F₁((1/2)*(n - 2*p), -1 - p, -p, 2/((a + 1/x)*x)))/((1 - 1/(a*x))^(n/2)*(1 + p))]

    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^3, x, 3, -((32*c^3*(1 - 1/(a*x))^(4 - n/2)*(1 + 1/(a*x))^((1/2)*(-8 + n))*HypergeometricFunctions._₂F₁(5, 4 - n/2, 5 - n/2, (a - 1/x)/(a + 1/x)))/(a*(8 - n)))]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^2, x, 3, (16*c^2*(1 - 1/(a*x))^(3 - n/2)*(1 + 1/(a*x))^((1/2)*(-6 + n))*HypergeometricFunctions._₂F₁(4, 3 - n/2, 4 - n/2, (a - 1/x)/(a + 1/x)))/(a*(6 - n))]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^1, x, 3, -((8*c*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((1/2)*(-4 + n))*HypergeometricFunctions._₂F₁(3, 2 - n/2, 3 - n/2, (a - 1/x)/(a + 1/x)))/(a*(4 - n)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^1, x, 3, (2*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)))/((1 - 1/(a*x))^(n/2)*(a*c*n))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^2, x, 3, -(((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^2*(2 + n)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^3, x, 4, If(13>=8, ((1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(4 + n)) - ((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(2 + n)*(4 + n)), -(((3 + n)*(1 - 1/(a*x))^((1/2)*(-2 - n))*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(8 + 6*n + n^2))) + ((1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^3*(4 + n)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^4, x, 6, If(13>=8, ((5 + n)*(1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(6 + n)) - ((14 + 8*n + n^2)*(1 - 1/(a*x))^(-2 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(4 + n)*(6 + n)) - ((14 + 8*n + n^2)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(6 + n)*(8 + 6*n + n^2)) - ((1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a^2*c^4*x), -(((14 + 8*n + n^2)*(1 - 1/(a*x))^((1/2)*(-4 - n))*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(24 + 10*n + n^2))) - ((14 + 8*n + n^2)*(1 - 1/(a*x))^((1/2)*(-2 - n))*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(48 + 44*n + 12*n^2 + n^3)) + ((5 + n)*(1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^4*(6 + n)) - ((1 - 1/(a*x))^(-3 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a^2*c^4*x))]

    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(5/2), x, 3, ((2/7)*((a - 1/x)/(a + 1/x))^((1/2)*(-5 + n))*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^(5/2)*HypergeometricFunctions._₂F₁(-(7/2), (1/2)*(-5 + n), -(5/2), 2/((a + 1/x)*x)))/(1 - 1/(a*x))^(n/2)]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(3/2), x, 3, ((2/5)*((a - 1/x)/(a + 1/x))^((1/2)*(-3 + n))*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^(3/2)*HypergeometricFunctions._₂F₁(-(5/2), (1/2)*(-3 + n), -(3/2), 2/((a + 1/x)*x)))/(1 - 1/(a*x))^(n/2)]
    @test_int [ℯ^(n*acoth(a*x))*(c - a*c*x)^(1/2), x, 3, ((2/3)*((a - 1/x)/(a + 1/x))^((1/2)*(-1 + n))*(1 + 1/(a*x))^((2 + n)/2)*x*sqrt(c - a*c*x)*HypergeometricFunctions._₂F₁(-(3/2), (1/2)*(-1 + n), -(1/2), 2/((a + 1/x)*x)))/(1 - 1/(a*x))^(n/2)]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^(1/2), x, 3, (2*((a - 1/x)/(a + 1/x))^((1 + n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x*HypergeometricFunctions._₂F₁(-(1/2), (1 + n)/2, 1/2, 2/((a + 1/x)*x)))/((1 - 1/(a*x))^(n/2)*sqrt(c - a*c*x))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^(3/2), x, 3, -((2*((a - 1/x)/(a + 1/x))^((3 + n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x*HypergeometricFunctions._₂F₁(1/2, (3 + n)/2, 3/2, 2/((a + 1/x)*x)))/((1 - 1/(a*x))^(n/2)*(c - a*c*x)^(3/2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^(5/2), x, 4, -((a*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/((3 + n)*(c - a*c*x)^(5/2))) + (a*((a - 1/x)/(a + 1/x))^((3 + n)/2)*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2*HypergeometricFunctions._₂F₁(1/2, (3 + n)/2, 3/2, 2/((a + 1/x)*x)))/((3 + n)*(c - a*c*x)^(5/2))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a*c*x)^(7/2), x, 5, -((a*(1 - 1/(a*x))^((2 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^2)/((5 + n)*(c - a*c*x)^(7/2))) + (3*a^2*(1 - 1/(a*x))^((4 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3)/(2*(15 + 8*n + n^2)*(c - a*c*x)^(7/2)) - (3*a^2*((a - 1/x)/(a + 1/x))^((3 + n)/2)*(1 - 1/(a*x))^((4 - n)/2)*(1 + 1/(a*x))^((2 + n)/2)*x^3*HypergeometricFunctions._₂F₁(1/2, (3 + n)/2, 3/2, 2/((a + 1/x)*x)))/(2*(15 + 8*n + n^2)*(c - a*c*x)^(7/2))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*ℯ^(n*acoth(a*x))*(c-c/(a*x))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a*x))^p=#

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

    @test_int [ℯ^acoth(a*x)*(c - c/(a*x))^4, x, 9, -((c^4*(1 - 1/(a^2*x^2))^(3/2))/(3*a)) + (c^4*sqrt(1 - 1/(a^2*x^2))*(6*a - 1/x))/(2*a^2) + c^4*(1 - 1/(a^2*x^2))^(3/2)*x - (c^4*acsc(a*x))/(2*a) - (3*c^4*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [ℯ^acoth(a*x)*(c - c/(a*x))^3, x, 8, (c^3*sqrt(1 - 1/(a^2*x^2))*(4*a + 1/x))/(2*a^2) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (c^3*acsc(a*x))/(2*a) - (2*c^3*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [ℯ^acoth(a*x)*(c - c/(a*x))^2, x, 7, (c^2*sqrt(1 - 1/(a^2*x^2))*(a + 1/x)*x)/a + (c^2*acsc(a*x))/a - (c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [ℯ^acoth(a*x)*(c - c/(a*x)), x, 3, c*sqrt(1 - 1/(a^2*x^2))*x + (c*acsc(a*x))/a]
    @test_int [ℯ^acoth(a*x)/(c - c/(a*x)), x, 7, -((2*(a + 1/x))/(a^2*c*sqrt(1 - 1/(a^2*x^2)))) + (sqrt(1 - 1/(a^2*x^2))*x)/c + (2*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c)]
    @test_int [ℯ^acoth(a*x)/(c - c/(a*x))^2, x, 8, -((4*(a + 1/x))/(3*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))) - (9*a + 11/x)/(3*a^2*c^2*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^2 + (3*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^2)]
    @test_int [ℯ^acoth(a*x)/(c - c/(a*x))^3, x, 9, -((8*(a + 1/x))/(5*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2))) - (4*(5*a + 8/x))/(15*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) - (60*a + 79/x)/(15*a^2*c^3*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^3 + (4*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^3)]
    @test_int [ℯ^acoth(a*x)/(c - c/(a*x))^4, x, 10, -((16*(a + 1/x))/(7*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2))) - (4*(7*a + 17/x))/(35*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2)) - (175*a + 307/x)/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (525*a + 719/x)/(105*a^2*c^4*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^4 + (5*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^4)]

    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^5, x, 5, -c^5/(4*a^5*x^4) + c^5/(a^4*x^3) - c^5/(a^3*x^2) - (2*c^5)/(a^2*x) + c^5*x - (3*c^5*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^4, x, 5, c^4/(3*a^4*x^3) - c^4/(a^3*x^2) + c^4*x - (2*c^4*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^3, x, 5, -c^3/(2*a^3*x^2) + c^3/(a^2*x) + c^3*x - (c^3*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^2, x, 6, c^2/(a^2*x) + c^2*x]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x)), x, 5, c*x + (c*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a*x)), x, 5, x/c + 2/(a*c*(1 - a*x)) + (3*log(1 - a*x))/(a*c)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a*x))^2, x, 5, x/c^2 - 1/(a*c^2*(1 - a*x)^2) + 5/(a*c^2*(1 - a*x)) + (4*log(1 - a*x))/(a*c^2)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a*x))^3, x, 5, x/c^3 + 2/(3*a*c^3*(1 - a*x)^3) - 7/(2*a*c^3*(1 - a*x)^2) + 9/(a*c^3*(1 - a*x)) + (5*log(1 - a*x))/(a*c^3)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a*x))^4, x, 5, x/c^4 - 1/(2*a*c^4*(1 - a*x)^4) + 3/(a*c^4*(1 - a*x)^3) - 8/(a*c^4*(1 - a*x)^2) + 14/(a*c^4*(1 - a*x)) + (6*log(1 - a*x))/(a*c^4)]

    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x))^4, x, 8, (c^4*sqrt(1 - 1/(a^2*x^2))*(2*a + 3/x))/(2*a^2) + (c^4*(1 - 1/(a^2*x^2))^(3/2)*(3*a + 1/x)*x)/(3*a) + (3*c^4*acsc(a*x))/(2*a) - (c^4*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x))^3, x, 4, (3*c^3*sqrt(1 - 1/(a^2*x^2)))/(2*a^2*x) + c^3*(1 - 1/(a^2*x^2))^(3/2)*x + (3*c^3*acsc(a*x))/(2*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x))^2, x, 8, (c^2*sqrt(1 - 1/(a^2*x^2))*(a - 1/x)*x)/a + (c^2*acsc(a*x))/a + (c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x)), x, 8, c*sqrt(1 - 1/(a^2*x^2))*x - (c*acsc(a*x))/a + (2*c*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a*x)), x, 8, -((8*(a + 1/x))/(3*a^2*c*(1 - 1/(a^2*x^2))^(3/2))) - (4*(3*a + 4/x))/(3*a^2*c*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c + (4*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c)]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a*x))^2, x, 9, -((16*(a + 1/x))/(5*a^2*c^2*(1 - 1/(a^2*x^2))^(5/2))) - (4*(5*a + 11/x))/(15*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2)) - (75*a + 103/x)/(15*a^2*c^2*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^2 + (5*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^2)]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a*x))^3, x, 10, -((32*(a + 1/x))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(7/2))) - (2*(7*a + 13/x))/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2)) - (42*a + 59/x)/(7*a^2*c^3*sqrt(1 - 1/(a^2*x^2))) - 16/(7*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2)*x) + (sqrt(1 - 1/(a^2*x^2))*x)/c^3 + (6*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^3)]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a*x))^4, x, 11, (16*(9*a - 5/x))/(63*a^2*c^4*(1 - 1/(a^2*x^2))^(7/2)) - (64*(a + 1/x))/(9*a^2*c^4*(1 - 1/(a^2*x^2))^(9/2)) - (8*(21*a + 41/x))/(105*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2)) - (735*a + 1417/x)/(315*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (2205*a + 3149/x)/(315*a^2*c^4*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^4 + (7*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^4)]

    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a*x))^5, x, 5, c^5/(4*a^5*x^4) - c^5/(3*a^4*x^3) - c^5/(a^3*x^2) + (2*c^5)/(a^2*x) + c^5*x - (c^5*log(x))/a]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a*x))^4, x, 6, -c^4/(3*a^4*x^3) + (2*c^4)/(a^2*x) + c^4*x]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a*x))^3, x, 5, c^3/(2*a^3*x^2) + c^3/(a^2*x) + c^3*x + (c^3*log(x))/a]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a*x))^2, x, 5, -(c^2/(a^2*x)) + c^2*x + (2*c^2*log(x))/a]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a*x)), x, 5, c*x - (c*log(x))/a + (4*c*log(1 - a*x))/a]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a*x)), x, 5, x/c - 2/(a*c*(1 - a*x)^2) + 8/(a*c*(1 - a*x)) + (5*log(1 - a*x))/(a*c)]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a*x))^2, x, 5, x/c^2 + 4/(3*a*c^2*(1 - a*x)^3) - 6/(a*c^2*(1 - a*x)^2) + 13/(a*c^2*(1 - a*x)) + (6*log(1 - a*x))/(a*c^2)]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a*x))^3, x, 5, x/c^3 - 1/(a*c^3*(1 - a*x)^4) + 16/(3*a*c^3*(1 - a*x)^3) - 25/(2*a*c^3*(1 - a*x)^2) + 19/(a*c^3*(1 - a*x)) + (7*log(1 - a*x))/(a*c^3)]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a*x))^4, x, 5, x/c^4 + 4/(5*a*c^4*(1 - a*x)^5) - 5/(a*c^4*(1 - a*x)^4) + 41/(3*a*c^4*(1 - a*x)^3) - 22/(a*c^4*(1 - a*x)^2) + 26/(a*c^4*(1 - a*x)) + (8*log(1 - a*x))/(a*c^4)]

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

    @test_int [(c - c/(a*x))^4/ℯ^acoth(a*x), x, 10, -((32*c^4*sqrt(1 - 1/(a^2*x^2)))/(3*a)) - (c^4*sqrt(1 - 1/(a^2*x^2)))/(3*a^3*x^2) + (5*c^4*sqrt(1 - 1/(a^2*x^2)))/(2*a^2*x) + c^4*sqrt(1 - 1/(a^2*x^2))*x - (25*c^4*acsc(a*x))/(2*a) - (5*c^4*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [(c - c/(a*x))^3/ℯ^acoth(a*x), x, 9, -((4*c^3*sqrt(1 - 1/(a^2*x^2)))/a) + (c^3*sqrt(1 - 1/(a^2*x^2)))/(2*a^2*x) + c^3*sqrt(1 - 1/(a^2*x^2))*x - (13*c^3*acsc(a*x))/(2*a) - (4*c^3*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [(c - c/(a*x))^2/ℯ^acoth(a*x), x, 8, -((c^2*sqrt(1 - 1/(a^2*x^2)))/a) + c^2*sqrt(1 - 1/(a^2*x^2))*x - (3*c^2*acsc(a*x))/a - (3*c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [(c - c/(a*x))^1/ℯ^acoth(a*x), x, 7, c*sqrt(1 - 1/(a^2*x^2))*x - (c*acsc(a*x))/a - (2*c*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a*x))^1), x, 2, (sqrt(1 - 1/(a^2*x^2))*x)/c]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a*x))^2), x, 6, (2*sqrt(1 - 1/(a^2*x^2))*x)/c^2 - (a*sqrt(1 - 1/(a^2*x^2))*x)/(c^2*(a - 1/x)) + atanh(sqrt(1 - 1/(a^2*x^2)))/(a*c^2)]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a*x))^3), x, 8, -((2*(a + 1/x))/(3*a^2*c^3*(1 - 1/(a^2*x^2))^(3/2))) - (6*a + 7/x)/(3*a^2*c^3*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^3 + (2*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^3)]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a*x))^4), x, 9, -((4*(a + 1/x))/(5*a^2*c^4*(1 - 1/(a^2*x^2))^(5/2))) - (5*a + 7/x)/(5*a^2*c^4*(1 - 1/(a^2*x^2))^(3/2)) - (15*a + 19/x)/(5*a^2*c^4*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^4 + (3*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^4)]

    @test_int [(c - c/(a*x))^4/ℯ^(2*acoth(a*x)), x, 5, c^4/(3*a^4*x^3) - (3*c^4)/(a^3*x^2) + (16*c^4)/(a^2*x) + c^4*x + (26*c^4*log(x))/a - (32*c^4*log(1 + a*x))/a]
    @test_int [(c - c/(a*x))^3/ℯ^(2*acoth(a*x)), x, 5, -c^3/(2*a^3*x^2) + (5*c^3)/(a^2*x) + c^3*x + (11*c^3*log(x))/a - (16*c^3*log(1 + a*x))/a]
    @test_int [(c - c/(a*x))^2/ℯ^(2*acoth(a*x)), x, 5, c^2/(a^2*x) + c^2*x + (4*c^2*log(x))/a - (8*c^2*log(1 + a*x))/a]
    @test_int [(c - c/(a*x))^1/ℯ^(2*acoth(a*x)), x, 5, c*x + (c*log(x))/a - (4*c*log(1 + a*x))/a]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^1), x, 5, x/c - log(1 + a*x)/(a*c)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^2), x, 6, x/c^2 - atanh(a*x)/(a*c^2)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^3), x, 5, x/c^3 + 1/(2*a*c^3*(1 - a*x)) + (5*log(1 - a*x))/(4*a*c^3) - log(1 + a*x)/(4*a*c^3)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^4), x, 5, x/c^4 - 1/(4*a*c^4*(1 - a*x)^2) + 7/(4*a*c^4*(1 - a*x)) + (17*log(1 - a*x))/(8*a*c^4) - log(1 + a*x)/(8*a*c^4)]

    @test_int [(c - c/(a*x))^4/ℯ^(3*acoth(a*x)), x, 11, (68*c^4*sqrt(1 - 1/(a^2*x^2)))/(3*a) + (64*c^4*(a - 1/x))/(a^2*sqrt(1 - 1/(a^2*x^2))) + (c^4*sqrt(1 - 1/(a^2*x^2)))/(3*a^3*x^2) - (7*c^4*sqrt(1 - 1/(a^2*x^2)))/(2*a^2*x) + c^4*sqrt(1 - 1/(a^2*x^2))*x + (91*c^4*acsc(a*x))/(2*a) - (7*c^4*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [(c - c/(a*x))^3/ℯ^(3*acoth(a*x)), x, 10, (6*c^3*sqrt(1 - 1/(a^2*x^2)))/a + (32*c^3*(a - 1/x))/(a^2*sqrt(1 - 1/(a^2*x^2))) - (c^3*sqrt(1 - 1/(a^2*x^2)))/(2*a^2*x) + c^3*sqrt(1 - 1/(a^2*x^2))*x + (33*c^3*acsc(a*x))/(2*a) - (6*c^3*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [(c - c/(a*x))^2/ℯ^(3*acoth(a*x)), x, 9, (c^2*sqrt(1 - 1/(a^2*x^2)))/a + (16*c^2*(a - 1/x))/(a^2*sqrt(1 - 1/(a^2*x^2))) + c^2*sqrt(1 - 1/(a^2*x^2))*x + (5*c^2*acsc(a*x))/a - (5*c^2*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [(c - c/(a*x))^1/ℯ^(3*acoth(a*x)), x, 8, (8*c*(a - 1/x))/(a^2*sqrt(1 - 1/(a^2*x^2))) + c*sqrt(1 - 1/(a^2*x^2))*x + (c*acsc(a*x))/a - (4*c*atanh(sqrt(1 - 1/(a^2*x^2))))/a]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^1), x, 6, (2*(a - 1/x))/(a^2*c*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c - (2*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^2), x, 6, (2*sqrt(1 - 1/(a^2*x^2))*x)/c^2 - ((a - 1/x)*x)/(a*c^2*sqrt(1 - 1/(a^2*x^2))) - atanh(sqrt(1 - 1/(a^2*x^2)))/(a*c^2)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^3), x, 3, -(2/(a^2*c^3*sqrt(1 - 1/(a^2*x^2))*x)) + x/(c^3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^4), x, 7, (8*sqrt(1 - 1/(a^2*x^2))*x)/(3*c^4) - (a*x)/(3*c^4*sqrt(1 - 1/(a^2*x^2))*(a - 1/x)) - ((4*a + 3/x)*x)/(3*a*c^4*sqrt(1 - 1/(a^2*x^2))) + atanh(sqrt(1 - 1/(a^2*x^2)))/(a*c^4)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^5), x, 9, -((2*(a + 1/x))/(5*a^2*c^5*(1 - 1/(a^2*x^2))^(5/2))) - (10*a + 13/x)/(15*a^2*c^5*(1 - 1/(a^2*x^2))^(3/2)) - (30*a + 41/x)/(15*a^2*c^5*sqrt(1 - 1/(a^2*x^2))) + (sqrt(1 - 1/(a^2*x^2))*x)/c^5 + (2*atanh(sqrt(1 - 1/(a^2*x^2))))/(a*c^5)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a*x))^(p/2)=#

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

    @test_int [ℯ^acoth(a*x)*(c - c/(a*x))^(9/2), x, 8, (173*c^5*sqrt(1 - 1/(a^2*x^2)))/(105*a*sqrt(c - c/(a*x))) + (227*c^4*sqrt(1 - 1/(a^2*x^2))*sqrt(c - c/(a*x)))/(105*a) + (59*c^3*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(3/2))/(35*a) + (9*c^2*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(5/2))/(7*a) + c*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(7/2)*x - (7*c^(9/2)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a, ((400*a - 227/x)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2))/(105*a^2*(1 - 1/(a*x))^(9/2)) + (59*(a - 1/x)^2*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2))/(35*a^3*(1 - 1/(a*x))^(9/2)) + (9*(a - 1/x)^3*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2))/(7*a^4*(1 - 1/(a*x))^(9/2)) + ((a - 1/x)^4*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2)*x)/(a^4*(1 - 1/(a*x))^(9/2)) - (7*(c - c/(a*x))^(9/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(9/2))]
    @test_int [ℯ^acoth(a*x)*(c - c/(a*x))^(7/2), x, 7, (49*c^4*sqrt(1 - 1/(a^2*x^2)))/(15*a*sqrt(c - c/(a*x))) + (31*c^3*sqrt(1 - 1/(a^2*x^2))*sqrt(c - c/(a*x)))/(15*a) + (7*c^2*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(3/2))/(5*a) + c*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(5/2)*x - (5*c^(7/2)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a, ((80*a - 31/x)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2))/(15*a^2*(1 - 1/(a*x))^(7/2)) + (7*(a - 1/x)^2*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2))/(5*a^3*(1 - 1/(a*x))^(7/2)) + ((a - 1/x)^3*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2)*x)/(a^3*(1 - 1/(a*x))^(7/2)) - (5*(c - c/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(7/2))]
    @test_int [ℯ^acoth(a*x)*(c - c/(a*x))^(5/2), x, 7, -((2*c^4*(1 - 1/(a^2*x^2))^(3/2))/(3*a*(c - c/(a*x))^(3/2))) + (3*c^3*sqrt(1 - 1/(a^2*x^2)))/(a*sqrt(c - c/(a*x))) + (c^4*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2) - (3*c^(5/2)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a, (3*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(5/2))/(a*(1 - 1/(a*x))^(5/2)) - (2*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(5/2))/(3*a*(1 - 1/(a*x))^(5/2)) + ((1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(5/2)*x)/(1 - 1/(a*x))^(5/2) - (3*(c - c/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(5/2))]
    @test_int [ℯ^acoth(a*x)*(c - c/(a*x))^(3/2), x, 5, (c^2*sqrt(1 - 1/(a^2*x^2)))/(a*sqrt(c - c/(a*x))) + (c^3*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2) - (c^(3/2)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x)), x, 4, (c*sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a*x)) + (sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a]
    @test_int [ℯ^acoth(a*x)/sqrt(c - c/(a*x)), x, 8, (sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x)/sqrt(c - c/(a*x)) + (3*sqrt(1 - 1/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(a*sqrt(c - c/(a*x))) - (2*sqrt(2)*sqrt(1 - 1/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a*sqrt(c - c/(a*x)))]
    @test_int [ℯ^acoth(a*x)/(c - c/(a*x))^(3/2), x, 9, (-2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x)))/((a - x^(-1))*(c - c/(a*x))^(3/2)) + (a*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x)/((a - x^(-1))*(c - c/(a*x))^(3/2)) + (5*(1 - 1/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(3/2)) - (7*(1 - 1/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(sqrt(2)*a*(c - c/(a*x))^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - c/(a*x))^(5/2), x, 10, (-3*a*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))/(2*(a - x^(-1))^2*(c - c/(a*x))^(5/2)) - (23*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))/(8*(a - x^(-1))*(c - c/(a*x))^(5/2)) + (a^2*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x)/((a - x^(-1))^2*(c - c/(a*x))^(5/2)) + (7*(1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(5/2)) - (79*(1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(8*sqrt(2)*a*(c - c/(a*x))^(5/2))]

    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^(9/2), x, 12, (5*c^4*sqrt(c - c/(a*x)))/a + (5*c^3*(c - c/(a*x))^(3/2))/(3*a) + (c^2*(c - c/(a*x))^(5/2))/a + (5*c*(c - c/(a*x))^(7/2))/(7*a) + (c - c/(a*x))^(9/2)*x - (5*c^(9/2)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^(7/2), x, 11, (3*c^3*sqrt(c - c/(a*x)))/a + (c^2*(c - c/(a*x))^(3/2))/a + (3*c*(c - c/(a*x))^(5/2))/(5*a) + (c - c/(a*x))^(7/2)*x - (3*c^(7/2)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^(5/2), x, 10, (c^2*sqrt(c - c/(a*x)))/a + (c*(c - c/(a*x))^(3/2))/(3*a) + (c - c/(a*x))^(5/2)*x - (c^(5/2)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a*x))^(3/2), x, 9, -((c*sqrt(c - c/(a*x)))/a) + (c - c/(a*x))^(3/2)*x + (c^(3/2)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x)), x, 8, sqrt(c - c/(a*x))*x + (3*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a]
    @test_int [ℯ^(2*acoth(a*x))/sqrt(c - c/(a*x)), x, 9, -5/(a*sqrt(c - c/(a*x))) + x/sqrt(c - c/(a*x)) + (5*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(a*sqrt(c))]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a*x))^(3/2), x, 10, -7/(3*a*(c - c/(a*x))^(3/2)) - 7/(a*c*sqrt(c - c/(a*x))) + x/(c - c/(a*x))^(3/2) + (7*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(a*c^(3/2))]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a*x))^(5/2), x, 11, -9/(5*a*(c - c/(a*x))^(5/2)) - 3/(a*c*(c - c/(a*x))^(3/2)) - 9/(a*c^2*sqrt(c - c/(a*x))) + x/(c - c/(a*x))^(5/2) + (9*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(a*c^(5/2))]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a*x))^(7/2), x, 12, -11/(7*a*(c - c/(a*x))^(7/2)) - 11/(5*a*c*(c - c/(a*x))^(5/2)) - 11/(3*a*c^2*(c - c/(a*x))^(3/2)) - 11/(a*c^3*sqrt(c - c/(a*x))) + x/(c - c/(a*x))^(7/2) + (11*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(a*c^(7/2))]

    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x))^(9/2), x, 8, (3*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2))/(a*(1 - 1/(a*x))^(9/2)) + (3*(28*a - 17/x)*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2))/(35*a^2*(1 - 1/(a*x))^(9/2)) + (9*(a - 1/x)^2*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2))/(7*a^3*(1 - 1/(a*x))^(9/2)) + ((a - 1/x)^3*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(9/2)*x)/(a^3*(1 - 1/(a*x))^(9/2)) - (3*(c - c/(a*x))^(9/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(9/2))]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x))^(7/2), x, 8, (sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2))/(a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x))^(3/2)*(c - c/(a*x))^(7/2))/(3*a*(1 - 1/(a*x))^(7/2)) - (2*(1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(7/2))/(5*a*(1 - 1/(a*x))^(7/2)) + ((1 + 1/(a*x))^(5/2)*(c - c/(a*x))^(7/2)*x)/(1 - 1/(a*x))^(7/2) - ((c - c/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(7/2))]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x))^(5/2), x, 6, -((c^4*(1 - 1/(a^2*x^2))^(3/2))/(3*a*(c - c/(a*x))^(3/2))) - (c^3*sqrt(1 - 1/(a^2*x^2)))/(a*sqrt(c - c/(a*x))) + (c^5*(1 - 1/(a^2*x^2))^(5/2)*x)/(c - c/(a*x))^(5/2) + (c^(5/2)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a*x))^(3/2), x, 5, -((3*c^2*sqrt(1 - 1/(a^2*x^2)))/(a*sqrt(c - c/(a*x)))) + (c^3*(1 - 1/(a^2*x^2))^(3/2)*x)/(c - c/(a*x))^(3/2) + (3*c^(3/2)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x)), x, 8, (sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x)/sqrt(1 - 1/(a*x)) + (5*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(a*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))/sqrt(c - c/(a*x)), x, 9, -((3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/((a - 1/x)*sqrt(c - c/(a*x)))) + (a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x)/((a - 1/x)*sqrt(c - c/(a*x))) + (7*sqrt(1 - 1/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(a*sqrt(c - c/(a*x))) - (5*sqrt(2)*sqrt(1 - 1/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a*sqrt(c - c/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a*x))^(3/2), x, 10, -((2*a*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x)))/((a - 1/x)^2*(c - c/(a*x))^(3/2))) - (15*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x)))/(4*(a - 1/x)*(c - c/(a*x))^(3/2)) + (a^2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x)/((a - 1/x)^2*(c - c/(a*x))^(3/2)) + (9*(1 - 1/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(3/2)) - (51*(1 - 1/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(4*sqrt(2)*a*(c - c/(a*x))^(3/2))]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a*x))^(5/2), x, 11, -((5*a^2*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))/(3*(a - 1/x)^3*(c - c/(a*x))^(5/2))) - (29*a*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))/(12*(a - 1/x)^2*(c - c/(a*x))^(5/2)) - (73*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))/(16*(a - 1/x)*(c - c/(a*x))^(5/2)) + (a^3*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x)/((a - 1/x)^3*(c - c/(a*x))^(5/2)) + (11*(1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(5/2)) - (249*(1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(16*sqrt(2)*a*(c - c/(a*x))^(5/2))]

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

    @test_int [(c - c/(a*x))^(7/2)/ℯ^acoth(a*x), x, 7, -(((80*a - 7/x)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2))/(5*a^2*(1 - 1/(a*x))^(7/2))) + (3*(a - 1/x)^2*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2))/(5*a^3*(1 - 1/(a*x))^(7/2)) + ((a - 1/x)^3*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2)*x)/(a^3*(1 - 1/(a*x))^(7/2)) - (9*(c - c/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(7/2))]
    @test_int [(c - c/(a*x))^(5/2)/ℯ^acoth(a*x), x, 6, -(((16*a + 1/x)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(5/2))/(3*a^2*(1 - 1/(a*x))^(5/2))) + ((a - 1/x)^2*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(5/2)*x)/(a^2*(1 - 1/(a*x))^(5/2)) - (7*(c - c/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(5/2))]
    @test_int [(c - c/(a*x))^(3/2)/ℯ^acoth(a*x), x, 6, (-2*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(3/2))/(a*(1 - 1/(a*x))^(3/2)) + (sqrt(1 + 1/(a*x))*(c - c/(a*x))^(3/2)*x)/(1 - 1/(a*x))^(3/2) - (5*(c - c/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(3/2))]
    @test_int [sqrt(c - c/(a*x))/ℯ^acoth(a*x), x, 4, (c*sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a*x)) - (3*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a]
    @test_int [1/(ℯ^acoth(a*x)*sqrt(c - c/(a*x))), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a*x)) - atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x)))/(a*sqrt(c))]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a*x))^(3/2)), x, 9, ((1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x)/(c - c/(a*x))^(3/2) + ((1 - 1/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(3/2)) - (sqrt(2)*(1 - 1/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a*(c - c/(a*x))^(3/2))]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a*x))^(5/2)), x, 10, (-3*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))/(2*(a - x^(-1))*(c - c/(a*x))^(5/2)) + (a*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x)/((a - x^(-1))*(c - c/(a*x))^(5/2)) + (3*(1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(5/2)) - (9*(1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(2*sqrt(2)*a*(c - c/(a*x))^(5/2))]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a*x))^(7/2)), x, 11, (-5*a*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x)))/(4*(a - x^(-1))^2*(c - c/(a*x))^(7/2)) - (35*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x)))/(16*(a - x^(-1))*(c - c/(a*x))^(7/2)) + (a^2*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x)/((a - x^(-1))^2*(c - c/(a*x))^(7/2)) + (5*(1 - 1/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(7/2)) - (115*(1 - 1/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(16*sqrt(2)*a*(c - c/(a*x))^(7/2))]

    @test_int [(c - c/(a*x))^(7/2)/ℯ^(2*acoth(a*x)), x, 14, -((21*c^3*sqrt(c - c/(a*x)))/a) - (5*c^2*(c - c/(a*x))^(3/2))/(3*a) + (3*c*(c - c/(a*x))^(5/2))/(5*a) + (c - c/(a*x))^(7/2)*x - (11*c^(7/2)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a + (32*sqrt(2)*c^(7/2)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a]
    @test_int [(c - c/(a*x))^(5/2)/ℯ^(2*acoth(a*x)), x, 13, -((7*c^2*sqrt(c - c/(a*x)))/a) + (c*(c - c/(a*x))^(3/2))/(3*a) + (c - c/(a*x))^(5/2)*x - (9*c^(5/2)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a + (16*sqrt(2)*c^(5/2)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a]
    @test_int [(c - c/(a*x))^(3/2)/ℯ^(2*acoth(a*x)), x, 12, -((c*sqrt(c - c/(a*x)))/a) + (c - c/(a*x))^(3/2)*x - (7*c^(3/2)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a + (8*sqrt(2)*c^(3/2)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a]
    @test_int [sqrt(c - c/(a*x))/ℯ^(2*acoth(a*x)), x, 11, sqrt(c - c/(a*x))*x - (5*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a + (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a]
    @test_int [1/(ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x))), x, 11, (sqrt(c - c/(a*x))*x)/c - (3*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(a*sqrt(c)) + (2*sqrt(2)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/(a*sqrt(c))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^(3/2)), x, 12, (sqrt(c - c/(a*x))*x)/c^2 - atanh(sqrt(c - c/(a*x))/sqrt(c))/(a*c^(3/2)) + (sqrt(2)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/(a*c^(3/2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^(5/2)), x, 12, -(2/(a*c^2*sqrt(c - c/(a*x)))) + x/(c^2*sqrt(c - c/(a*x))) + atanh(sqrt(c - c/(a*x))/sqrt(c))/(a*c^(5/2)) + atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))/(sqrt(2)*a*c^(5/2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^(7/2)), x, 13, -(4/(3*a*c^2*(c - c/(a*x))^(3/2))) - 7/(2*a*c^3*sqrt(c - c/(a*x))) + x/(c^2*(c - c/(a*x))^(3/2)) + (3*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(a*c^(7/2)) + atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))/(2*sqrt(2)*a*c^(7/2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a*x))^(9/2)), x, 14, -(6/(5*a*c^2*(c - c/(a*x))^(5/2))) - 11/(6*a*c^3*(c - c/(a*x))^(3/2)) - 21/(4*a*c^4*sqrt(c - c/(a*x))) + x/(c^2*(c - c/(a*x))^(5/2)) + (5*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(a*c^(9/2)) + atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))/(4*sqrt(2)*a*c^(9/2))]

    @test_int [(c - c/(a*x))^(9/2)/ℯ^(3*acoth(a*x)), x, 9, (10*(a - 1/x)^4*(c - c/(a*x))^(9/2))/(a^5*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x))) + (5*(304*a - 65/x)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2))/(7*a^2*(1 - 1/(a*x))^(9/2)) + (135*(a - 1/x)^2*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2))/(7*a^3*(1 - 1/(a*x))^(9/2)) + (65*(a - 1/x)^3*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(9/2))/(7*a^4*(1 - 1/(a*x))^(9/2)) + ((a - 1/x)^5*(c - c/(a*x))^(9/2)*x)/(a^5*(1 - 1/(a*x))^(9/2)*sqrt(1 + 1/(a*x))) - (15*(c - c/(a*x))^(9/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(9/2))]
    @test_int [(c - c/(a*x))^(7/2)/ℯ^(3*acoth(a*x)), x, 8, (10*(a - 1/x)^3*(c - c/(a*x))^(7/2))/(a^4*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))) + ((1360*a - 311/x)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2))/(15*a^2*(1 - 1/(a*x))^(7/2)) + (47*(a - 1/x)^2*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2))/(5*a^3*(1 - 1/(a*x))^(7/2)) + ((a - 1/x)^4*(c - c/(a*x))^(7/2)*x)/(a^4*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))) - (13*(c - c/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(7/2))]
    @test_int [(c - c/(a*x))^(5/2)/ℯ^(3*acoth(a*x)), x, 7, (10*(a - 1/x)^2*(c - c/(a*x))^(5/2))/(a^3*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))) + ((112*a - 29/x)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(5/2))/(3*a^2*(1 - 1/(a*x))^(5/2)) + ((a - 1/x)^3*(c - c/(a*x))^(5/2)*x)/(a^3*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))) - (11*(c - c/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(5/2))]
    @test_int [(c - c/(a*x))^(3/2)/ℯ^(3*acoth(a*x)), x, 6, ((21*a + 1/x)*(c - c/(a*x))^(3/2))/(a^2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))) + ((a - 1/x)^2*(c - c/(a*x))^(3/2)*x)/(a^2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))) - (9*(c - c/(a*x))^(3/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(1 - 1/(a*x))^(3/2))]
    @test_int [sqrt(c - c/(a*x))/ℯ^(3*acoth(a*x)), x, 6, (9*sqrt(c - c/(a*x)))/(a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (sqrt(c - c/(a*x))*x)/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) - (7*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(a*sqrt(1 - 1/(a*x)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x))), x, 5, (5*sqrt(c - c/(a*x)))/(a*c*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a*x))*x)/(c*sqrt(1 - 1/(a^2*x^2))) - (5*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/(a*sqrt(c))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^(3/2)), x, 5, (3*sqrt(1 - 1/(a^2*x^2))*x)/(c*sqrt(c - c/(a*x))) - (2*sqrt(c - c/(a*x))*x)/(c^2*sqrt(1 - 1/(a^2*x^2))) - (3*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/(a*c^(3/2))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^(5/2)), x, 9, (2*(1 - 1/(a*x))^(5/2))/(a*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(5/2)) + ((1 - 1/(a*x))^(5/2)*x)/(sqrt(1 + 1/(a*x))*(c - c/(a*x))^(5/2)) - ((1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(5/2)) - ((1 - 1/(a*x))^(5/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(sqrt(2)*a*(c - c/(a*x))^(5/2))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a*x))^(7/2)), x, 10, (7*(1 - 1/(a*x))^(7/2))/(4*a*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2)) - (3*(1 - 1/(a*x))^(7/2))/(2*(a - x^(-1))*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2)) + (a*(1 - 1/(a*x))^(7/2)*x)/((a - x^(-1))*sqrt(1 + 1/(a*x))*(c - c/(a*x))^(7/2)) + ((1 - 1/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))))/(a*(c - c/(a*x))^(7/2)) - (11*(1 - 1/(a*x))^(7/2)*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(4*sqrt(2)*a*(c - c/(a*x))^(7/2))]

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

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

    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))*x^m, x, 3, (sqrt(c - c/(a*x))*x^(1 + m)*HypergeometricFunctions._₂F₁(-(1/2), -1 - m, -m, -(1/(a*x))))/((1 + m)*sqrt(1 - 1/(a*x)))]

    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))*x^2, x, 6, -((c*sqrt(1 - 1/(a^2*x^2))*x)/(8*a^2*sqrt(c - c/(a*x)))) + (c*sqrt(1 - 1/(a^2*x^2))*x^2)/(12*a*sqrt(c - c/(a*x))) + (c*sqrt(1 - 1/(a^2*x^2))*x^3)/(3*sqrt(c - c/(a*x))) + (sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/(8*a^3)]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))*x^1, x, 5, (c*sqrt(1 - 1/(a^2*x^2))*x)/(4*a*sqrt(c - c/(a*x))) + (c*sqrt(1 - 1/(a^2*x^2))*x^2)/(2*sqrt(c - c/(a*x))) - (sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/(4*a^2)]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))*x^0, x, 4, (c*sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a*x)) + (sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))/x^1, x, 4, -((2*c*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))) + 2*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))/x^2, x, 2, -((2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))/x^3, x, 3, -((2*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))/(15*(c - c/(a*x))^(3/2))) + (2*a^2*c*(1 - 1/(a^2*x^2))^(3/2))/(5*sqrt(c - c/(a*x)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))/x^4, x, 4, (8*a^3*c^2*(1 - 1/(a^2*x^2))^(3/2))/(105*(c - c/(a*x))^(3/2)) - (8*a^3*c*(1 - 1/(a^2*x^2))^(3/2))/(35*sqrt(c - c/(a*x))) - (2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(7*(c - c/(a*x))^(3/2)*x^2)]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a*x))/x^5, x, 5, -((16*a^4*c^2*(1 - 1/(a^2*x^2))^(3/2))/(315*(c - c/(a*x))^(3/2))) + (16*a^4*c*(1 - 1/(a^2*x^2))^(3/2))/(105*sqrt(c - c/(a*x))) - (2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(9*(c - c/(a*x))^(3/2)*x^3) + (4*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))/(21*(c - c/(a*x))^(3/2)*x^2)]

    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x))*x^3, x, 11, (75*sqrt(c - c/(a*x))*x)/(64*a^3) + (25*sqrt(c - c/(a*x))*x^2)/(32*a^2) + (5*sqrt(c - c/(a*x))*x^3)/(8*a) + (sqrt(c - c/(a*x))*x^4)/4 + (75*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(64*a^4)]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x))*x^2, x, 10, (11*sqrt(c - c/(a*x))*x)/(8*a^2) + (11*sqrt(c - c/(a*x))*x^2)/(12*a) + (sqrt(c - c/(a*x))*x^3)/3 + (11*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(8*a^3)]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x))*x, x, 9, (7*sqrt(c - c/(a*x))*x)/(4*a) + (sqrt(c - c/(a*x))*x^2)/2 + (7*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(4*a^2)]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x)), x, 8, sqrt(c - c/(a*x))*x + (3*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x)))/x, x, 8, 2*sqrt(c - c/(a*x)) + 2*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x)))/x^2, x, 7, 4*a*sqrt(c - c/(a*x)) - (2*a*(c - c/(a*x))^(3/2))/(3*c)]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x)))/x^3, x, 7, 4*a^2*sqrt(c - c/(a*x)) - (2*a^2*(c - c/(a*x))^(3/2))/c + (2*a^2*(c - c/(a*x))^(5/2))/(5*c^2)]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x)))/x^4, x, 7, 4*a^3*sqrt(c - c/(a*x)) - (10*a^3*(c - c/(a*x))^(3/2))/(3*c) + (8*a^3*(c - c/(a*x))^(5/2))/(5*c^2) - (2*a^3*(c - c/(a*x))^(7/2))/(7*c^3)]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a*x)))/x^5, x, 7, 4*a^4*sqrt(c - c/(a*x)) - (14*a^4*(c - c/(a*x))^(3/2))/(3*c) + (18*a^4*(c - c/(a*x))^(5/2))/(5*c^2) - (10*a^4*(c - c/(a*x))^(7/2))/(7*c^3) + (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4)]

    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x))*x^3, x, 11, (149*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x)/(64*a^3*sqrt(1 - 1/(a*x))) + (107*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x^2)/(96*a^2*sqrt(1 - 1/(a*x))) + (17*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x^3)/(24*a*sqrt(1 - 1/(a*x))) + (sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x^4)/(4*sqrt(1 - 1/(a*x))) + (363*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(64*a^4*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a^4*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x))*x^2, x, 10, (19*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x)/(8*a^2*sqrt(1 - 1/(a*x))) + (13*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x^2)/(12*a*sqrt(1 - 1/(a*x))) + (sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x^3)/(3*sqrt(1 - 1/(a*x))) + (45*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(8*a^3*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a^3*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x))*x, x, 9, (9*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x)/(4*a*sqrt(1 - 1/(a*x))) + (sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x^2)/(2*sqrt(1 - 1/(a*x))) + (23*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(4*a^2*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a^2*sqrt(1 - 1/(a*x)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x)), x, 8, (sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x)/sqrt(1 - 1/(a*x)) + (5*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(a*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/(a*sqrt(1 - 1/(a*x)))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x)))/x, x, 8, (2*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x)))/sqrt(1 - 1/(a*x)) + (2*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/sqrt(1 - 1/(a*x)) - (4*sqrt(2)*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/sqrt(1 - 1/(a*x))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x)))/x^2, x, 5, (2*a*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)) + (4*a*c*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x)) - 4*sqrt(2)*a*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/(sqrt(2)*sqrt(c - c/(a*x))))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x)))/x^3, x, 6, (2*a^2*c^3*(1 - 1/(a^2*x^2))^(5/2))/(5*(c - c/(a*x))^(5/2)) + (2*a^2*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)) + (4*a^2*c*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x)) - 4*sqrt(2)*a^2*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/(sqrt(2)*sqrt(c - c/(a*x))))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x)))/x^4, x, 7, (4*a^3*c^3*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(5/2)) + (2*a^3*c^2*(1 - 1/(a^2*x^2))^(3/2))/(3*(c - c/(a*x))^(3/2)) - (2*a^3*c^2*(1 - 1/(a^2*x^2))^(5/2))/(7*(c - c/(a*x))^(3/2)) + (4*a^3*c*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x)) - 4*sqrt(2)*a^3*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/(sqrt(2)*sqrt(c - c/(a*x))))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a*x)))/x^5, x, 8, (4*a^4*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x)))/sqrt(1 - 1/(a*x)) + (2*a^4*(1 + 1/(a*x))^(3/2)*sqrt(c - c/(a*x)))/(3*sqrt(1 - 1/(a*x))) + (2*a^4*(1 + 1/(a*x))^(5/2)*sqrt(c - c/(a*x)))/(5*sqrt(1 - 1/(a*x))) - (2*a^4*(1 + 1/(a*x))^(7/2)*sqrt(c - c/(a*x)))/(7*sqrt(1 - 1/(a*x))) + (2*a^4*(1 + 1/(a*x))^(9/2)*sqrt(c - c/(a*x)))/(9*sqrt(1 - 1/(a*x))) - (4*sqrt(2)*a^4*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))/sqrt(2)))/sqrt(1 - 1/(a*x))]

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

    @test_int [(sqrt(c - c/(a*x))*x^m)/ℯ^acoth(a*x), x, 4, (sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x))*x^(1 + m))/((1 + m)*sqrt(1 - 1/(a*x))) - ((3 + 4*m)*sqrt(c - c/(a*x))*x^m*HypergeometricFunctions._₂F₁(1/2, -m, 1 - m, -(1/(a*x))))/(2*a*m*(1 + m)*sqrt(1 - 1/(a*x)))]

    @test_int [(sqrt(c - c/(a*x))*x^2)/ℯ^acoth(a*x), x, 6, (11*c*sqrt(1 - 1/(a^2*x^2))*x)/(8*a^2*sqrt(c - c/(a*x))) - (11*c*sqrt(1 - 1/(a^2*x^2))*x^2)/(12*a*sqrt(c - c/(a*x))) + (c*sqrt(1 - 1/(a^2*x^2))*x^3)/(3*sqrt(c - c/(a*x))) - (11*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/(8*a^3)]
    @test_int [(sqrt(c - c/(a*x))*x)/ℯ^acoth(a*x), x, 5, -((7*c*sqrt(1 - 1/(a^2*x^2))*x)/(4*a*sqrt(c - c/(a*x)))) + (c*sqrt(1 - 1/(a^2*x^2))*x^2)/(2*sqrt(c - c/(a*x))) + (7*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/(4*a^2)]
    @test_int [sqrt(c - c/(a*x))/ℯ^acoth(a*x), x, 4, (c*sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a*x)) - (3*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x))))/a]
    @test_int [sqrt(c - c/(a*x))/(ℯ^acoth(a*x)*x), x, 4, (2*c*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x)) + 2*sqrt(c)*atanh((sqrt(c)*sqrt(1 - 1/(a^2*x^2)))/sqrt(c - c/(a*x)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^acoth(a*x)*x^2), x, 3, -((8*a*c*sqrt(1 - 1/(a^2*x^2)))/(3*sqrt(c - c/(a*x)))) - (2/3)*a*sqrt(1 - 1/(a^2*x^2))*sqrt(c - c/(a*x))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^acoth(a*x)*x^3), x, 4, (8*a^2*c*sqrt(1 - 1/(a^2*x^2)))/(5*sqrt(c - c/(a*x))) + (2/5)*a^2*sqrt(1 - 1/(a^2*x^2))*sqrt(c - c/(a*x)) + (2*a^2*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(3/2))/(5*c)]
    @test_int [sqrt(c - c/(a*x))/(ℯ^acoth(a*x)*x^4), x, 5, -((104*a^3*c*sqrt(1 - 1/(a^2*x^2)))/(105*sqrt(c - c/(a*x)))) - (104/105)*a^3*sqrt(1 - 1/(a^2*x^2))*sqrt(c - c/(a*x)) + (2*c*sqrt(1 - 1/(a^2*x^2)))/(7*sqrt(c - c/(a*x))*x^3) - (26*a*c*sqrt(1 - 1/(a^2*x^2)))/(35*sqrt(c - c/(a*x))*x^2)]

    @test_int [(sqrt(c - c/(a*x))*x^3)/ℯ^(2*acoth(a*x)), x, 14, -((149*sqrt(c - c/(a*x))*x)/(64*a^3)) + (107*sqrt(c - c/(a*x))*x^2)/(96*a^2) - (17*sqrt(c - c/(a*x))*x^3)/(24*a) + (1/4)*sqrt(c - c/(a*x))*x^4 + (363*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(64*a^4) - (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a^4]
    @test_int [(sqrt(c - c/(a*x))*x^2)/ℯ^(2*acoth(a*x)), x, 13, (19*sqrt(c - c/(a*x))*x)/(8*a^2) - (13*sqrt(c - c/(a*x))*x^2)/(12*a) + (1/3)*sqrt(c - c/(a*x))*x^3 - (45*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(8*a^3) + (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a^3]
    @test_int [(sqrt(c - c/(a*x))*x)/ℯ^(2*acoth(a*x)), x, 12, -((9*sqrt(c - c/(a*x))*x)/(4*a)) + (1/2)*sqrt(c - c/(a*x))*x^2 + (23*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/(4*a^2) - (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a^2]
    @test_int [sqrt(c - c/(a*x))/ℯ^(2*acoth(a*x)), x, 11, sqrt(c - c/(a*x))*x - (5*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)))/a + (4*sqrt(2)*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c))))/a]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(2*acoth(a*x))*x), x, 11, 2*sqrt(c - c/(a*x)) + 2*sqrt(c)*atanh(sqrt(c - c/(a*x))/sqrt(c)) - 4*sqrt(2)*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(2*acoth(a*x))*x^2), x, 9, -4*a*sqrt(c - c/(a*x)) - (2*a*(c - c/(a*x))^(3/2))/(3*c) + 4*sqrt(2)*a*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(2*acoth(a*x))*x^3), x, 10, 4*a^2*sqrt(c - c/(a*x)) + (2*a^2*(c - c/(a*x))^(3/2))/(3*c) + (2*a^2*(c - c/(a*x))^(5/2))/(5*c^2) - 4*sqrt(2)*a^2*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(2*acoth(a*x))*x^4), x, 11, -4*a^3*sqrt(c - c/(a*x)) - (2*a^3*(c - c/(a*x))^(3/2))/(3*c) - (2*a^3*(c - c/(a*x))^(7/2))/(7*c^3) + 4*sqrt(2)*a^3*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(2*acoth(a*x))*x^5), x, 11, 4*a^4*sqrt(c - c/(a*x)) + (2*a^4*(c - c/(a*x))^(3/2))/(3*c) + (2*a^4*(c - c/(a*x))^(5/2))/(5*c^2) - (2*a^4*(c - c/(a*x))^(7/2))/(7*c^3) + (2*a^4*(c - c/(a*x))^(9/2))/(9*c^4) - 4*sqrt(2)*a^4*sqrt(c)*atanh(sqrt(c - c/(a*x))/(sqrt(2)*sqrt(c)))]

    @test_int [(sqrt(c - c/(a*x))*x^3)/ℯ^(3*acoth(a*x)), x, 9, -((1115*sqrt(c - c/(a*x)))/(64*a^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))) - (1115*sqrt(c - c/(a*x))*x)/(192*a^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (223*sqrt(c - c/(a*x))*x^2)/(96*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) - (25*sqrt(c - c/(a*x))*x^3)/(24*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (sqrt(c - c/(a*x))*x^4)/(4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (1115*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(64*a^4*sqrt(1 - 1/(a*x)))]
    @test_int [(sqrt(c - c/(a*x))*x^2)/ℯ^(3*acoth(a*x)), x, 8, (119*sqrt(c - c/(a*x)))/(8*a^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (119*sqrt(c - c/(a*x))*x)/(24*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) - (19*sqrt(c - c/(a*x))*x^2)/(12*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (sqrt(c - c/(a*x))*x^3)/(3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) - (119*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(8*a^3*sqrt(1 - 1/(a*x)))]
    @test_int [(sqrt(c - c/(a*x))*x)/ℯ^(3*acoth(a*x)), x, 7, (-47*sqrt(c - c/(a*x)))/(4*a^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) - (13*sqrt(c - c/(a*x))*x)/(4*a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (sqrt(c - c/(a*x))*x^2)/(2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (47*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(4*a^2*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - c/(a*x))/ℯ^(3*acoth(a*x)), x, 6, (9*sqrt(c - c/(a*x)))/(a*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (sqrt(c - c/(a*x))*x)/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) - (7*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/(a*sqrt(1 - 1/(a*x)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(3*acoth(a*x))*x), x, 6, (-8*sqrt(c - c/(a*x)))/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) - (2*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x)))/sqrt(1 - 1/(a*x)) + (2*sqrt(c - c/(a*x))*atanh(sqrt(1 + 1/(a*x))))/sqrt(1 - 1/(a*x))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(3*acoth(a*x))*x^2), x, 4, (64*a*sqrt(c - c/(a*x)))/(3*sqrt(1 - 1/(a^2*x^2))) - (16*a*(c - c/(a*x))^(3/2))/(3*c*sqrt(1 - 1/(a^2*x^2))) - (2*a*(c - c/(a*x))^(5/2))/(3*c^2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(3*acoth(a*x))*x^3), x, 5, -((224*a^2*c*sqrt(1 - 1/(a^2*x^2)))/(15*sqrt(c - c/(a*x)))) - (56/15)*a^2*sqrt(1 - 1/(a^2*x^2))*sqrt(c - c/(a*x)) - (7*a^2*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(3/2))/(5*c) - (a^2*(c - c/(a*x))^(7/2))/(c^3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(3*acoth(a*x))*x^4), x, 6, (1888*a^3*c*sqrt(1 - 1/(a^2*x^2)))/(105*sqrt(c - c/(a*x))) + (472/105)*a^3*sqrt(1 - 1/(a^2*x^2))*sqrt(c - c/(a*x)) + (59*a^3*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(3/2))/(35*c) + (2*a^3*sqrt(1 - 1/(a^2*x^2))*(c - c/(a*x))^(5/2))/(7*c^2) + (a^3*(c - c/(a*x))^(7/2))/(c^3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a*x))/(ℯ^(3*acoth(a*x))*x^5), x, 4, -((8*a^4*sqrt(c - c/(a*x)))/(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))) - (32*a^4*sqrt(1 + 1/(a*x))*sqrt(c - c/(a*x)))/sqrt(1 - 1/(a*x)) + (50*a^4*(1 + 1/(a*x))^(3/2)*sqrt(c - c/(a*x)))/(3*sqrt(1 - 1/(a*x))) - (38*a^4*(1 + 1/(a*x))^(5/2)*sqrt(c - c/(a*x)))/(5*sqrt(1 - 1/(a*x))) + (2*a^4*(1 + 1/(a*x))^(7/2)*sqrt(c - c/(a*x)))/sqrt(1 - 1/(a*x)) - (2*a^4*(1 + 1/(a*x))^(9/2)*sqrt(c - c/(a*x)))/(9*sqrt(1 - 1/(a*x)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a*x))^p*with*n*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*(c - c/(a*x)), x, 5, c*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^(n/2)*x - (2*c*(1 - n)*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)))/((1 - 1/(a*x))^(n/2)*(a*n)) - (2^(n/2)*c*(1 - 1/(a*x))^(1 - n/2)*HypergeometricFunctions._₂F₁(1 - n/2, 1 - n/2, 2 - n/2, (a - 1/x)/(2*a)))/(a*(2 - n))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c/(a*x)), x, 3, ((1 + 1/(a*x))^((2 + n)/2)*x)/((1 - 1/(a*x))^(n/2)*c) - (2*(1 + n)*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)))/((1 - 1/(a*x))^(n/2)*(a*c*n))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c/(a*x))^2, x, 5, -(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2))/(a*c^2*(2 + n))) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x)/c^2 - (2*(2 + n)*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, -(n/2), 1 - n/2, (a - 1/x)/(a + 1/x)))/((1 - 1/(a*x))^(n/2)*(a*c^2*n))]

    @test_int [ℯ^(n*acoth(a*x))*(c - c/(a*x))^(3/2), x, 3, -((2^(5/2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - c/(a*x))^(3/2)*AppellF1((2 + n)/2, (1/2)*(-3 + n), 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)))/(a*(2 + n)*(1 - 1/(a*x))^(3/2)))]
    @test_int [ℯ^(n*acoth(a*x))*sqrt(c - c/(a*x)), x, 3, -((2^(3/2 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*sqrt(c - c/(a*x))*AppellF1((2 + n)/2, (1/2)*(-1 + n), 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)))/(a*(2 + n)*sqrt(1 - 1/(a*x))))]
    @test_int [ℯ^(n*acoth(a*x))/sqrt(c - c/(a*x)), x, 3, -((2^(1/2 - n/2)*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^((2 + n)/2)*AppellF1((2 + n)/2, (1 + n)/2, 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)))/(a*(2 + n)*sqrt(c - c/(a*x))))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c/(a*x))^(3/2), x, 3, -((2^(-(1/2) - n/2)*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^((2 + n)/2)*AppellF1((2 + n)/2, (3 + n)/2, 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)))/(a*(2 + n)*(c - c/(a*x))^(3/2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a*x))^p*with*p*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*(c - c/(a*x))^p, x, 3, -((2^(1 - n/2 + p)*(1 + 1/(a*x))^((2 + n)/2)*(c - c/(a*x))^p*AppellF1((2 + n)/2, (1/2)*(n - 2*p), 2, (4 + n)/2, (a + 1/x)/(2*a), 1 + 1/(a*x)))/((1 - 1/(a*x))^p*(a*(2 + n))))]

    @test_int [(c - c/(a*x))^p*ℯ^(2*p*acoth(a*x)), x, 3, -(((1 + 1/(a*x))^(1 + p)*(c - c/(a*x))^p*HypergeometricFunctions._₂F₁(2, 1 + p, 2 + p, 1 + 1/(a*x)))/((1 - 1/(a*x))^p*(a*(1 + p))))]
    @test_int [(c - c/(a*x))^p/ℯ^(2*p*acoth(a*x)), x, 3, -((4^p*(1 + 1/(a*x))^(1 - p)*(c - c/(a*x))^p*AppellF1(1 - p, -2*p, 2, 2 - p, (a + 1/x)/(2*a), 1 + 1/(a*x)))/((1 - 1/(a*x))^p*(a*(1 - p))))]

    @test_int [(c - c/(a*x))^p*ℯ^(2*acoth(a*x)), x, 7, (c - c/(a*x))^p*x + ((2 - p)*(c - c/(a*x))^p*HypergeometricFunctions._₂F₁(1, p, 1 + p, 1 - 1/(a*x)))/(a*p)]
    @test_int [(c - c/(a*x))^p*ℯ^acoth(a*x), x, 3, -((2^(1/2 + p)*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^p*AppellF1(3/2, 1/2 - p, 2, 5/2, (a + 1/x)/(2*a), 1 + 1/(a*x)))/((1 - 1/(a*x))^p*(3*a)))]
    @test_int [(c - c/(a*x))^p/ℯ^acoth(a*x), x, 3, -((2^(3/2 + p)*sqrt(1 + 1/(a*x))*(c - c/(a*x))^p*AppellF1(1/2, -(1/2) - p, 2, 3/2, (a + 1/x)/(2*a), 1 + 1/(a*x)))/((1 - 1/(a*x))^p*a))]
    @test_int [(c - c/(a*x))^p/ℯ^(2*acoth(a*x)), x, 9, ((c - c/(a*x))^(2 + p)*x)/c^2 + ((c - c/(a*x))^(2 + p)*HypergeometricFunctions._₂F₁(1, 2 + p, 3 + p, (a - 1/x)/(2*a)))/(2*a*c^2*(2 + p)) - ((c - c/(a*x))^(2 + p)*HypergeometricFunctions._₂F₁(1, 2 + p, 3 + p, 1 - 1/(a*x)))/(a*c^2)]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*ℯ^(n*acoth(a*x))*(c-c*a^2*x^2)^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c*a^2*x^2)^p=#

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

    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^4, x, 13, (35/128)*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (35/384)*a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 + (7/192)*a^2*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 + (1/64)*a^3*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 + (1/144)*a^4*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2)*x^5 - (5/144)*a^5*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(11/2)*x^6 + (5/72)*a^6*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2)*x^7 - (7/72)*a^7*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(11/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(11/2)*x^9 + (35*c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(128*a)]
    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^3, x, 11, (5/16)*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (5/48)*a*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 + (1/24)*a^2*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 + (1/56)*a^3*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 - (1/14)*a^4*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2)*x^5 + (5/42)*a^5*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7 + (5*c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(16*a)]
    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^2, x, 9, (3/8)*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (1/8)*a*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 + (1/20)*a^2*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (3/20)*a^3*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 + (1/5)*a^4*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5 + (3*c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(8*a)]
    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2), x, 7, (1/2)*c*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (1/6)*a*c*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (1/3)*a^2*c*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 + (c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(2*a)]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2), x, 1, ℯ^acoth(a*x)/(a*c)]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^2, x, 2, (2*ℯ^acoth(a*x))/(3*a*c^2) - (ℯ^acoth(a*x)*(1 - 2*a*x))/(3*a*c^2*(1 - a^2*x^2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^3, x, 3, (8*ℯ^acoth(a*x))/(15*a*c^3) - (ℯ^acoth(a*x)*(1 - 4*a*x))/(15*a*c^3*(1 - a^2*x^2)^2) - (4*ℯ^acoth(a*x)*(1 - 2*a*x))/(15*a*c^3*(1 - a^2*x^2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^4, x, 4, (16*ℯ^acoth(a*x))/(35*a*c^4) - (ℯ^acoth(a*x)*(1 - 6*a*x))/(35*a*c^4*(1 - a^2*x^2)^3) - (2*ℯ^acoth(a*x)*(1 - 4*a*x))/(35*a*c^4*(1 - a^2*x^2)^2) - (8*ℯ^acoth(a*x)*(1 - 2*a*x))/(35*a*c^4*(1 - a^2*x^2))]

    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^5, x, 4, -((16*c^5*(1 + a*x)^7)/(7*a)) + (4*c^5*(1 + a*x)^8)/a - (8*c^5*(1 + a*x)^9)/(3*a) + (4*c^5*(1 + a*x)^10)/(5*a) - (c^5*(1 + a*x)^11)/(11*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^4, x, 4, -((4*c^4*(1 + a*x)^6)/(3*a)) + (12*c^4*(1 + a*x)^7)/(7*a) - (3*c^4*(1 + a*x)^8)/(4*a) + (c^4*(1 + a*x)^9)/(9*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^3, x, 4, -((4*c^3*(1 + a*x)^5)/(5*a)) + (2*c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^2, x, 4, -((c^2*(1 + a*x)^4)/(2*a)) + (c^2*(1 + a*x)^5)/(5*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^1, x, 3, -((c*(1 + a*x)^3)/(3*a))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^1, x, 3, -(1/(a*c*(1 - a*x)))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^2, x, 5, -(1/(4*a*c^2*(1 - a*x)^2)) - 1/(4*a*c^2*(1 - a*x)) - atanh(a*x)/(4*a*c^2)]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^3, x, 5, -(1/(12*a*c^3*(1 - a*x)^3)) - 1/(8*a*c^3*(1 - a*x)^2) - 3/(16*a*c^3*(1 - a*x)) + 1/(16*a*c^3*(1 + a*x)) - atanh(a*x)/(4*a*c^3)]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^4, x, 5, -(1/(32*a*c^4*(1 - a*x)^4)) - 1/(16*a*c^4*(1 - a*x)^3) - 3/(32*a*c^4*(1 - a*x)^2) - 5/(32*a*c^4*(1 - a*x)) + 1/(64*a*c^4*(1 + a*x)^2) + 5/(64*a*c^4*(1 + a*x)) - (15*atanh(a*x))/(64*a*c^4)]

    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^4, x, 13, (-(55/128))*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (55/384)*a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (11/192)*a^2*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (11/448)*a^3*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 - (11*a^4*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2)*x^5)/1008 - (5*a^5*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(11/2)*x^6)/1008 + (5/168)*a^6*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(13/2)*x^7 - (5/72)*a^7*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(13/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(13/2)*x^9 - (55*c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(128*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^3, x, 11, (-(9/16))*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (3/16)*a*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (3/40)*a^2*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (9/280)*a^3*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 - (1/70)*a^4*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2)*x^5 + (1/14)*a^5*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(11/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2)*x^7 - (9*c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(16*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^2, x, 9, (-(7/8))*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (7/24)*a*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (7/60)*a^2*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (1/20)*a^3*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 + (1/5)*a^4*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2)*x^5 - (7*c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(8*a)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2), x, 7, (-(5/2))*c*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (5/6)*a*c*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (1/3)*a^2*c*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (5*c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(2*a)]
    @test_int [ℯ^(3*acoth(a*x))/(c - a^2*c*x^2), x, 1, ℯ^(3*acoth(a*x))/(3*a*c)]
    @test_int [ℯ^(3*acoth(a*x))/(c - a^2*c*x^2)^2, x, 2, -((2*ℯ^(3*acoth(a*x)))/(15*a*c^2)) + (ℯ^(3*acoth(a*x))*(3 - 2*a*x))/(5*a*c^2*(1 - a^2*x^2))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a^2*c*x^2)^3, x, 3, -((8*ℯ^(3*acoth(a*x)))/(35*a*c^3)) - (ℯ^(3*acoth(a*x))*(3 - 4*a*x))/(7*a*c^3*(1 - a^2*x^2)^2) + (12*ℯ^(3*acoth(a*x))*(3 - 2*a*x))/(35*a*c^3*(1 - a^2*x^2))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a^2*c*x^2)^4, x, 4, -((16*ℯ^(3*acoth(a*x)))/(63*a*c^4)) - (ℯ^(3*acoth(a*x))*(1 - 2*a*x))/(9*a*c^4*(1 - a^2*x^2)^3) - (10*ℯ^(3*acoth(a*x))*(3 - 4*a*x))/(63*a*c^4*(1 - a^2*x^2)^2) + (8*ℯ^(3*acoth(a*x))*(3 - 2*a*x))/(21*a*c^4*(1 - a^2*x^2))]

    @test_int [ℯ^(4*acoth(a*x))*(c - a^2*c*x^2)^5, x, 4, (c^5*(1 + a*x)^8)/a - (4*c^5*(1 + a*x)^9)/(3*a) + (3*c^5*(1 + a*x)^10)/(5*a) - (c^5*(1 + a*x)^11)/(11*a)]
    @test_int [ℯ^(4*acoth(a*x))*(c - a^2*c*x^2)^4, x, 4, (4*c^4*(1 + a*x)^7)/(7*a) - (c^4*(1 + a*x)^8)/(2*a) + (c^4*(1 + a*x)^9)/(9*a)]
    @test_int [ℯ^(4*acoth(a*x))*(c - a^2*c*x^2)^3, x, 4, (c^3*(1 + a*x)^6)/(3*a) - (c^3*(1 + a*x)^7)/(7*a)]
    @test_int [ℯ^(4*acoth(a*x))*(c - a^2*c*x^2)^2, x, 3, (c^2*(1 + a*x)^5)/(5*a)]
    @test_int [ℯ^(4*acoth(a*x))*(c - a^2*c*x^2)^1, x, 4, -4*c*x - (c*(1 + a*x)^2)/a - (c*(1 + a*x)^3)/(3*a) - (8*c*log(1 - a*x))/a]
    @test_int [ℯ^(4*acoth(a*x))/(c - a^2*c*x^2)^1, x, 3, x/(c*(1 - a*x)^2)]
    @test_int [ℯ^(4*acoth(a*x))/(c - a^2*c*x^2)^2, x, 3, 1/(3*a*c^2*(1 - a*x)^3)]
    @test_int [ℯ^(4*acoth(a*x))/(c - a^2*c*x^2)^3, x, 5, 1/(8*a*c^3*(1 - a*x)^4) + 1/(12*a*c^3*(1 - a*x)^3) + 1/(16*a*c^3*(1 - a*x)^2) + 1/(16*a*c^3*(1 - a*x)) + atanh(a*x)/(16*a*c^3)]
    @test_int [ℯ^(4*acoth(a*x))/(c - a^2*c*x^2)^4, x, 5, 1/(20*a*c^4*(1 - a*x)^5) + 1/(16*a*c^4*(1 - a*x)^4) + 1/(16*a*c^4*(1 - a*x)^3) + 1/(16*a*c^4*(1 - a*x)^2) + 5/(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (3*atanh(a*x))/(32*a*c^4)]

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

    @test_int [(c - a^2*c*x^2)^4/ℯ^acoth(a*x), x, 13, (-(35/128))*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (35/384)*a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (7/192)*a^2*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (1/64)*a^3*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 + (1/16)*a^4*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2)*x^5 - (5/48)*a^5*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*x^6 + (1/8)*a^6*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2)*x^7 - (1/8)*a^7*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(9/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(9/2)*x^9 - (35*c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(128*a)]
    @test_int [(c - a^2*c*x^2)^3/ℯ^acoth(a*x), x, 11, (-(5/16))*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (5/48)*a*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (1/24)*a^2*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 + (1/8)*a^3*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 - (1/6)*a^4*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5 + (1/6)*a^5*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2)*x^7 - (5*c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(16*a)]
    @test_int [(c - a^2*c*x^2)^2/ℯ^acoth(a*x), x, 9, (-(3/8))*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (1/8)*a*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 + (1/4)*a^2*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (1/4)*a^3*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4 + (1/5)*a^4*c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5 - (3*c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(8*a)]
    @test_int [(c - a^2*c*x^2)/ℯ^acoth(a*x), x, 7, (-(1/2))*c*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (1/2)*a*c*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (1/3)*a^2*c*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)*x^3 - (c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(2*a)]
    @test_int [1/(ℯ^acoth(a*x)*(c - a^2*c*x^2)), x, 1, -(1/(a*c*ℯ^acoth(a*x)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a^2*c*x^2)^2), x, 2, -(2/(ℯ^acoth(a*x)*(3*a*c^2))) + (1 + 2*a*x)/(ℯ^acoth(a*x)*(3*a*c^2*(1 - a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a^2*c*x^2)^3), x, 3, -(8/(ℯ^acoth(a*x)*(15*a*c^3))) + (1 + 4*a*x)/(ℯ^acoth(a*x)*(15*a*c^3*(1 - a^2*x^2)^2)) + (4*(1 + 2*a*x))/(ℯ^acoth(a*x)*(15*a*c^3*(1 - a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a^2*c*x^2)^4), x, 4, -(16/(ℯ^acoth(a*x)*(35*a*c^4))) + (1 + 6*a*x)/(ℯ^acoth(a*x)*(35*a*c^4*(1 - a^2*x^2)^3)) + (2*(1 + 4*a*x))/(ℯ^acoth(a*x)*(35*a*c^4*(1 - a^2*x^2)^2)) + (8*(1 + 2*a*x))/(ℯ^acoth(a*x)*(35*a*c^4*(1 - a^2*x^2)))]

    @test_int [(c - a^2*c*x^2)^4/ℯ^(2*acoth(a*x)), x, 4, (4*c^4*(1 - a*x)^6)/(3*a) - (12*c^4*(1 - a*x)^7)/(7*a) + (3*c^4*(1 - a*x)^8)/(4*a) - (c^4*(1 - a*x)^9)/(9*a)]
    @test_int [(c - a^2*c*x^2)^3/ℯ^(2*acoth(a*x)), x, 4, (4*c^3*(1 - a*x)^5)/(5*a) - (2*c^3*(1 - a*x)^6)/(3*a) + (c^3*(1 - a*x)^7)/(7*a)]
    @test_int [(c - a^2*c*x^2)^2/ℯ^(2*acoth(a*x)), x, 4, (c^2*(1 - a*x)^4)/(2*a) - (c^2*(1 - a*x)^5)/(5*a)]
    @test_int [(c - a^2*c*x^2)/ℯ^(2*acoth(a*x)), x, 3, (c*(1 - a*x)^3)/(3*a)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)), x, 3, 1/(a*c*(1 + a*x))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^2), x, 5, 1/(4*a*c^2*(1 + a*x)^2) + 1/(4*a*c^2*(1 + a*x)) - atanh(a*x)/(4*a*c^2)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^3), x, 5, -(1/(16*a*c^3*(1 - a*x))) + 1/(12*a*c^3*(1 + a*x)^3) + 1/(8*a*c^3*(1 + a*x)^2) + 3/(16*a*c^3*(1 + a*x)) - atanh(a*x)/(4*a*c^3)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^4), x, 5, -(1/(64*a*c^4*(1 - a*x)^2)) - 5/(64*a*c^4*(1 - a*x)) + 1/(32*a*c^4*(1 + a*x)^4) + 1/(16*a*c^4*(1 + a*x)^3) + 3/(32*a*c^4*(1 + a*x)^2) + 5/(32*a*c^4*(1 + a*x)) - (15*atanh(a*x))/(64*a*c^4)]

    @test_int [(c - a^2*c*x^2)^4/ℯ^(3*acoth(a*x)), x, 13, (55/128)*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (55/384)*a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 + (11/192)*a^2*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 - (11/64)*a^3*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x^4 + (11/48)*a^4*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)*x^5 - (11/48)*a^5*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x^6 + (11/56)*a^6*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2)*x^7 - (11/72)*a^7*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(7/2)*x^8 + (1/9)*a^8*c^4*(1 - 1/(a*x))^(11/2)*(1 + 1/(a*x))^(7/2)*x^9 + (55*c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(128*a)]
    @test_int [(c - a^2*c*x^2)^3/ℯ^(3*acoth(a*x)), x, 11, (9/16)*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (3/16)*a*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 - (3/8)*a^2*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)*x^3 + (3/8)*a^3*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x^4 - (3/10)*a^4*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)*x^5 + (3/14)*a^5*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x^6 - (1/7)*a^6*c^3*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(5/2)*x^7 + (9*c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(16*a)]
    @test_int [(c - a^2*c*x^2)^2/ℯ^(3*acoth(a*x)), x, 9, (7/8)*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x - (7/8)*a*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x^2 + (7/12)*a^2*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)*x^3 - (7/20)*a^3*c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x^4 + (1/5)*a^4*c^2*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)*x^5 + (7*c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(8*a)]
    @test_int [(c - a^2*c*x^2)/ℯ^(3*acoth(a*x)), x, 7, (-(5/2))*c*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))*x + (5/6)*a*c*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x^2 - (1/3)*a^2*c*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))*x^3 + (5*c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(2*a)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)), x, 1, -1/(3*a*c*ℯ^(3*acoth(a*x)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^2), x, 2, 2/(ℯ^(3*acoth(a*x))*(15*a*c^2)) - (3 + 2*a*x)/(ℯ^(3*acoth(a*x))*(5*a*c^2*(1 - a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^3), x, 3, 8/(ℯ^(3*acoth(a*x))*(35*a*c^3)) + (3 + 4*a*x)/(ℯ^(3*acoth(a*x))*(7*a*c^3*(1 - a^2*x^2)^2)) - (12*(3 + 2*a*x))/(ℯ^(3*acoth(a*x))*(35*a*c^3*(1 - a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^4), x, 4, 16/(ℯ^(3*acoth(a*x))*(63*a*c^4)) + (1 + 2*a*x)/(ℯ^(3*acoth(a*x))*(9*a*c^4*(1 - a^2*x^2)^3)) + (10*(3 + 4*a*x))/(ℯ^(3*acoth(a*x))*(63*a*c^4*(1 - a^2*x^2)^2)) - (8*(3 + 2*a*x))/(ℯ^(3*acoth(a*x))*(21*a*c^4*(1 - a^2*x^2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))  (c-c*a^2*x^2)^(p/2)=#

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

    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^(9/2), x, 4, (8*(1 + a*x)^6*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (32*(1 + a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + (3*(1 + a*x)^8*(c - a^2*c*x^2)^(9/2))/(a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (8*(1 + a*x)^9*(c - a^2*c*x^2)^(9/2))/(9*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 + a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)]
    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^(7/2), x, 4, -((8*(1 + a*x)^5*(c - a^2*c*x^2)^(7/2))/(5*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)) + (2*(1 + a*x)^6*(c - a^2*c*x^2)^(7/2))/(a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (6*(1 + a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 + a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)]
    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^(5/2), x, 4, ((1 + a*x)^4*(c - a^2*c*x^2)^(5/2))/(a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) - (4*(1 + a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) + ((1 + a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)]
    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^(3/2), x, 4, -((2*(1 + a*x)^3*(c - a^2*c*x^2)^(3/2))/(3*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)) + ((1 + a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)]
    @test_int [ℯ^acoth(a*x)*sqrt(c - a^2*c*x^2), x, 3, sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)/sqrt(c - a^2*c*x^2), x, 3, (sqrt(1 - 1/(a^2*x^2))*x*log(1 - a*x))/sqrt(c - a^2*c*x^2)]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2), x, 5, (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*atanh(a*x))/(2*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2), x, 5, -((a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (3*a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(7/2), x, 5, (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(24*(1 - a*x)^3*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) + (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*atanh(a*x))/(16*(c - a^2*c*x^2)^(7/2))]

    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(9/2), x, 10, (-(77/256))*c^4*x*sqrt(c - a^2*c*x^2) - (77/384)*c^3*x*(c - a^2*c*x^2)^(3/2) - (77/480)*c^2*x*(c - a^2*c*x^2)^(5/2) - (11/80)*c*x*(c - a^2*c*x^2)^(7/2) + (11*(c - a^2*c*x^2)^(9/2))/(90*a) + ((1 + a*x)*(c - a^2*c*x^2)^(9/2))/(10*a) - (77*c^(9/2)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(256*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(7/2), x, 9, (-(45/128))*c^3*x*sqrt(c - a^2*c*x^2) - (15/64)*c^2*x*(c - a^2*c*x^2)^(3/2) - (3/16)*c*x*(c - a^2*c*x^2)^(5/2) + (9*(c - a^2*c*x^2)^(7/2))/(56*a) + ((1 + a*x)*(c - a^2*c*x^2)^(7/2))/(8*a) - (45*c^(7/2)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(128*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(5/2), x, 8, (-(7/16))*c^2*x*sqrt(c - a^2*c*x^2) - (7/24)*c*x*(c - a^2*c*x^2)^(3/2) + (7*(c - a^2*c*x^2)^(5/2))/(30*a) + ((1 + a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) - (7*c^(5/2)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(16*a)]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(3/2), x, 7, (-(5/8))*c*x*sqrt(c - a^2*c*x^2) + (5*(c - a^2*c*x^2)^(3/2))/(12*a) + ((1 + a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) - (5*c^(3/2)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(8*a)]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2), x, 6, (3*sqrt(c - a^2*c*x^2))/(2*a) + ((1 + a*x)*sqrt(c - a^2*c*x^2))/(2*a) - (3*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(2*a)]
    @test_int [ℯ^(2*acoth(a*x))/sqrt(c - a^2*c*x^2), x, 5, -((2*(1 + a*x))/(a*sqrt(c - a^2*c*x^2))) + atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2))/(a*sqrt(c))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^(3/2), x, 4, -((2*(1 + a*x))/(3*a*(c - a^2*c*x^2)^(3/2))) - x/(3*c*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^(5/2), x, 5, -((2*(1 + a*x))/(5*a*(c - a^2*c*x^2)^(5/2))) - x/(5*c*(c - a^2*c*x^2)^(3/2)) - (2*x)/(5*c^2*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^(7/2), x, 6, -((2*(1 + a*x))/(7*a*(c - a^2*c*x^2)^(7/2))) - x/(7*c*(c - a^2*c*x^2)^(5/2)) - (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2)) - (8*x)/(21*c^3*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(2*acoth(a*x))/(c - a^2*c*x^2)^(9/2), x, 7, -((2*(1 + a*x))/(9*a*(c - a^2*c*x^2)^(9/2))) - x/(9*c*(c - a^2*c*x^2)^(7/2)) - (2*x)/(15*c^2*(c - a^2*c*x^2)^(5/2)) - (8*x)/(45*c^3*(c - a^2*c*x^2)^(3/2)) - (16*x)/(45*c^4*sqrt(c - a^2*c*x^2))]

    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^(9/2), x, 4, -((8*(1 + a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)) + (3*(1 + a*x)^8*(c - a^2*c*x^2)^(9/2))/(2*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (2*(1 + a*x)^9*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 + a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^(7/2), x, 4, (2*(1 + a*x)^6*(c - a^2*c*x^2)^(7/2))/(3*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (4*(1 + a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 + a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^(5/2), x, 4, -((2*(1 + a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)) + ((1 + a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)]
    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^(3/2), x, 3, ((1 + a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2), x, 4, (3*sqrt(c - a^2*c*x^2))/(a*sqrt(1 - 1/(a^2*x^2))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(a^2*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [ℯ^(3*acoth(a*x))/sqrt(c - a^2*c*x^2), x, 4, (2*sqrt(1 - 1/(a^2*x^2))*x)/((1 - a*x)*sqrt(c - a^2*c*x^2)) + (sqrt(1 - 1/(a^2*x^2))*x*log(1 - a*x))/sqrt(c - a^2*c*x^2)]
    @test_int [ℯ^(3*acoth(a*x))/(c - a^2*c*x^2)^(3/2), x, 3, -((a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)^2*(c - a^2*c*x^2)^(3/2)))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a^2*c*x^2)^(5/2), x, 5, (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(6*(1 - a*x)^3*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^(3*acoth(a*x))/(c - a^2*c*x^2)^(7/2), x, 5, -((a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^4*(c - a^2*c*x^2)^(7/2))) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(12*(1 - a*x)^3*(c - a^2*c*x^2)^(7/2)) - (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) + (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) - (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*atanh(a*x))/(32*(c - a^2*c*x^2)^(7/2))]

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

    @test_int [(c - a^2*c*x^2)^(9/2)/ℯ^acoth(a*x), x, 4, (8*(1 - a*x)^6*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (32*(1 - a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + (3*(1 - a*x)^8*(c - a^2*c*x^2)^(9/2))/(a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (8*(1 - a*x)^9*(c - a^2*c*x^2)^(9/2))/(9*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 - a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)]
    @test_int [(c - a^2*c*x^2)^(7/2)/ℯ^acoth(a*x), x, 4, -((8*(1 - a*x)^5*(c - a^2*c*x^2)^(7/2))/(5*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)) + (2*(1 - a*x)^6*(c - a^2*c*x^2)^(7/2))/(a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (6*(1 - a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 - a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)]
    @test_int [(c - a^2*c*x^2)^(5/2)/ℯ^acoth(a*x), x, 4, ((1 - a*x)^4*(c - a^2*c*x^2)^(5/2))/(a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) - (4*(1 - a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5) + ((1 - a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)]
    @test_int [(c - a^2*c*x^2)^(3/2)/ℯ^acoth(a*x), x, 4, -((2*(1 - a*x)^3*(c - a^2*c*x^2)^(3/2))/(3*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)) + ((1 - a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)]
    @test_int [sqrt(c - a^2*c*x^2)/ℯ^acoth(a*x), x, 3, -(sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2)))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*sqrt(c - a^2*c*x^2)), x, 3, (sqrt(1 - 1/(a^2*x^2))*x*log(1 + a*x))/sqrt(c - a^2*c*x^2)]
    @test_int [1/(ℯ^acoth(a*x)*(c - a^2*c*x^2)^(3/2)), x, 5, (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 + a*x)*(c - a^2*c*x^2)^(3/2)) - (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*atanh(a*x))/(2*(c - a^2*c*x^2)^(3/2))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a^2*c*x^2)^(5/2)), x, 5, (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)^2*(c - a^2*c*x^2)^(5/2)) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (3*a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [1/(ℯ^acoth(a*x)*(c - a^2*c*x^2)^(7/2)), x, 5, -((a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)^2*(c - a^2*c*x^2)^(7/2))) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) + (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(24*(1 + a*x)^3*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) + (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) - (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*atanh(a*x))/(16*(c - a^2*c*x^2)^(7/2))]

    @test_int [(c - a^2*c*x^2)^(5/2)/ℯ^(2*acoth(a*x)), x, 8, (-(7/16))*c^2*x*sqrt(c - a^2*c*x^2) - (7/24)*c*x*(c - a^2*c*x^2)^(3/2) - (7*(c - a^2*c*x^2)^(5/2))/(30*a) - ((1 - a*x)*(c - a^2*c*x^2)^(5/2))/(6*a) - (7*c^(5/2)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(16*a)]
    @test_int [(c - a^2*c*x^2)^(3/2)/ℯ^(2*acoth(a*x)), x, 7, (-(5/8))*c*x*sqrt(c - a^2*c*x^2) - (5*(c - a^2*c*x^2)^(3/2))/(12*a) - ((1 - a*x)*(c - a^2*c*x^2)^(3/2))/(4*a) - (5*c^(3/2)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(8*a)]
    @test_int [sqrt(c - a^2*c*x^2)/ℯ^(2*acoth(a*x)), x, 6, -((3*sqrt(c - a^2*c*x^2))/(2*a)) - ((1 - a*x)*sqrt(c - a^2*c*x^2))/(2*a) - (3*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(2*a)]
    @test_int [1/(ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2)), x, 5, (2*(1 - a*x))/(a*sqrt(c - a^2*c*x^2)) + atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2))/(a*sqrt(c))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(3/2)), x, 4, (2*(1 - a*x))/(3*a*(c - a^2*c*x^2)^(3/2)) - x/(3*c*sqrt(c - a^2*c*x^2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(5/2)), x, 5, (2*(1 - a*x))/(5*a*(c - a^2*c*x^2)^(5/2)) - x/(5*c*(c - a^2*c*x^2)^(3/2)) - (2*x)/(5*c^2*sqrt(c - a^2*c*x^2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(7/2)), x, 6, (2*(1 - a*x))/(7*a*(c - a^2*c*x^2)^(7/2)) - x/(7*c*(c - a^2*c*x^2)^(5/2)) - (4*x)/(21*c^2*(c - a^2*c*x^2)^(3/2)) - (8*x)/(21*c^3*sqrt(c - a^2*c*x^2))]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^(9/2)), x, 7, (2*(1 - a*x))/(9*a*(c - a^2*c*x^2)^(9/2)) - x/(9*c*(c - a^2*c*x^2)^(7/2)) - (2*x)/(15*c^2*(c - a^2*c*x^2)^(5/2)) - (8*x)/(45*c^3*(c - a^2*c*x^2)^(3/2)) - (16*x)/(45*c^4*sqrt(c - a^2*c*x^2))]

    @test_int [(c - a^2*c*x^2)^(9/2)/ℯ^(3*acoth(a*x)), x, 4, -((8*(1 - a*x)^7*(c - a^2*c*x^2)^(9/2))/(7*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)) + (3*(1 - a*x)^8*(c - a^2*c*x^2)^(9/2))/(2*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) - (2*(1 - a*x)^9*(c - a^2*c*x^2)^(9/2))/(3*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9) + ((1 - a*x)^10*(c - a^2*c*x^2)^(9/2))/(10*a^10*(1 - 1/(a^2*x^2))^(9/2)*x^9)]
    @test_int [(c - a^2*c*x^2)^(7/2)/ℯ^(3*acoth(a*x)), x, 4, (2*(1 - a*x)^6*(c - a^2*c*x^2)^(7/2))/(3*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) - (4*(1 - a*x)^7*(c - a^2*c*x^2)^(7/2))/(7*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7) + ((1 - a*x)^8*(c - a^2*c*x^2)^(7/2))/(8*a^8*(1 - 1/(a^2*x^2))^(7/2)*x^7)]
    @test_int [(c - a^2*c*x^2)^(5/2)/ℯ^(3*acoth(a*x)), x, 4, -((2*(1 - a*x)^5*(c - a^2*c*x^2)^(5/2))/(5*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)) + ((1 - a*x)^6*(c - a^2*c*x^2)^(5/2))/(6*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)]
    @test_int [(c - a^2*c*x^2)^(3/2)/ℯ^(3*acoth(a*x)), x, 3, ((1 - a*x)^4*(c - a^2*c*x^2)^(3/2))/(4*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)]
    @test_int [sqrt(c - a^2*c*x^2)/ℯ^(3*acoth(a*x)), x, 4, -((3*sqrt(c - a^2*c*x^2))/(a*sqrt(1 - 1/(a^2*x^2)))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(a^2*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [1/(ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2)), x, 4, (2*sqrt(1 - 1/(a^2*x^2))*x)/((1 + a*x)*sqrt(c - a^2*c*x^2)) + (sqrt(1 - 1/(a^2*x^2))*x*log(1 + a*x))/sqrt(c - a^2*c*x^2)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^(3/2)), x, 3, -((a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 + a*x)^2*(c - a^2*c*x^2)^(3/2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^(5/2)), x, 5, (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(6*(1 + a*x)^3*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)^2*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^(7/2)), x, 5, (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 - a*x)*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(16*(1 + a*x)^4*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(12*(1 + a*x)^3*(c - a^2*c*x^2)^(7/2)) - (3*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(32*(1 + a*x)^2*(c - a^2*c*x^2)^(7/2)) - (a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7)/(8*(1 + a*x)*(c - a^2*c*x^2)^(7/2)) + (5*a^6*(1 - 1/(a^2*x^2))^(7/2)*x^7*atanh(a*x))/(32*(c - a^2*c*x^2)^(7/2))]

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

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

    @test_int [ℯ^acoth(a*x)*x^2*sqrt(c - a^2*c*x^2), x, 4, (x^2*sqrt(c - a^2*c*x^2))/(3*a*sqrt(1 - 1/(a^2*x^2))) + (x^3*sqrt(c - a^2*c*x^2))/(4*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)*x*sqrt(c - a^2*c*x^2), x, 4, (x*sqrt(c - a^2*c*x^2))/(2*a*sqrt(1 - 1/(a^2*x^2))) + (x^2*sqrt(c - a^2*c*x^2))/(3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - a^2*c*x^2), x, 3, sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(ℯ^acoth(a*x)*sqrt(c - a^2*c*x^2))/x, x, 4, sqrt(c - a^2*c*x^2)/sqrt(1 - 1/(a^2*x^2)) + (sqrt(c - a^2*c*x^2)*log(x))/(a*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(ℯ^acoth(a*x)*sqrt(c - a^2*c*x^2))/x^2, x, 4, -(sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2))*x^2)) + (sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x)]

    @test_int [ℯ^(2*acoth(a*x))*x^3*sqrt(c - a^2*c*x^2), x, 8, (3*x^2*sqrt(c - a^2*c*x^2))/(5*a^2) + (x^3*sqrt(c - a^2*c*x^2))/(2*a) + (1/5)*x^4*sqrt(c - a^2*c*x^2) + (3*(8 + 5*a*x)*sqrt(c - a^2*c*x^2))/(20*a^4) - (3*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(4*a^4)]
    @test_int [ℯ^(2*acoth(a*x))*x^2*sqrt(c - a^2*c*x^2), x, 7, (2*x^2*sqrt(c - a^2*c*x^2))/(3*a) + (1/4)*x^3*sqrt(c - a^2*c*x^2) + ((32 + 21*a*x)*sqrt(c - a^2*c*x^2))/(24*a^3) - (7*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(8*a^3)]
    @test_int [ℯ^(2*acoth(a*x))*x*sqrt(c - a^2*c*x^2), x, 6, (1/3)*x^2*sqrt(c - a^2*c*x^2) + ((5 + 3*a*x)*sqrt(c - a^2*c*x^2))/(3*a^2) - (sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/a^2]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2), x, 6, (3*sqrt(c - a^2*c*x^2))/(2*a) + ((1 + a*x)*sqrt(c - a^2*c*x^2))/(2*a) - (3*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(2*a)]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2))/x, x, 9, sqrt(c - a^2*c*x^2) - 2*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)) + sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^2, x, 9, sqrt(c - a^2*c*x^2)/x - a*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)) + 2*a*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^3, x, 7, sqrt(c - a^2*c*x^2)/(2*x^2) + (2*a*sqrt(c - a^2*c*x^2))/x + (3/2)*a^2*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^4, x, 8, sqrt(c - a^2*c*x^2)/(3*x^3) + (a*sqrt(c - a^2*c*x^2))/x^2 + (5*a^2*sqrt(c - a^2*c*x^2))/(3*x) + a^3*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^5, x, 9, sqrt(c - a^2*c*x^2)/(4*x^4) + (2*a*sqrt(c - a^2*c*x^2))/(3*x^3) + (7*a^2*sqrt(c - a^2*c*x^2))/(8*x^2) + (4*a^3*sqrt(c - a^2*c*x^2))/(3*x) + (7/8)*a^4*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]

    @test_int [ℯ^(3*acoth(a*x))*x^3*sqrt(c - a^2*c*x^2), x, 4, (4*sqrt(c - a^2*c*x^2))/(a^4*sqrt(1 - 1/(a^2*x^2))) + (2*x*sqrt(c - a^2*c*x^2))/(a^3*sqrt(1 - 1/(a^2*x^2))) + (4*x^2*sqrt(c - a^2*c*x^2))/(3*a^2*sqrt(1 - 1/(a^2*x^2))) + (3*x^3*sqrt(c - a^2*c*x^2))/(4*a*sqrt(1 - 1/(a^2*x^2))) + (x^4*sqrt(c - a^2*c*x^2))/(5*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(a^5*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [ℯ^(3*acoth(a*x))*x^2*sqrt(c - a^2*c*x^2), x, 4, (4*sqrt(c - a^2*c*x^2))/(a^3*sqrt(1 - 1/(a^2*x^2))) + (2*x*sqrt(c - a^2*c*x^2))/(a^2*sqrt(1 - 1/(a^2*x^2))) + (x^2*sqrt(c - a^2*c*x^2))/(a*sqrt(1 - 1/(a^2*x^2))) + (x^3*sqrt(c - a^2*c*x^2))/(4*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(a^4*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [ℯ^(3*acoth(a*x))*x*sqrt(c - a^2*c*x^2), x, 4, (4*sqrt(c - a^2*c*x^2))/(a^2*sqrt(1 - 1/(a^2*x^2))) + (3*x*sqrt(c - a^2*c*x^2))/(2*a*sqrt(1 - 1/(a^2*x^2))) + (x^2*sqrt(c - a^2*c*x^2))/(3*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(a^3*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2), x, 4, (3*sqrt(c - a^2*c*x^2))/(a*sqrt(1 - 1/(a^2*x^2))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(a^2*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2))/x, x, 4, sqrt(c - a^2*c*x^2)/sqrt(1 - 1/(a^2*x^2)) - (sqrt(c - a^2*c*x^2)*log(x))/(a*sqrt(1 - 1/(a^2*x^2))*x) + (4*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(a*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^2, x, 4, sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2))*x^2) - (3*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^3, x, 4, sqrt(c - a^2*c*x^2)/(2*a*sqrt(1 - 1/(a^2*x^2))*x^3) + (3*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^2) - (4*a*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*a*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^4, x, 4, sqrt(c - a^2*c*x^2)/(3*a*sqrt(1 - 1/(a^2*x^2))*x^4) + (3*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2))*x^3) + (4*a*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^2) - (4*a^2*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*a^2*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - a^2*c*x^2))/x^5, x, 4, sqrt(c - a^2*c*x^2)/(4*a*sqrt(1 - 1/(a^2*x^2))*x^5) + sqrt(c - a^2*c*x^2)/(sqrt(1 - 1/(a^2*x^2))*x^4) + (2*a*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^3) + (4*a^2*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^2) - (4*a^3*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*a^3*sqrt(c - a^2*c*x^2)*log(1 - a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]

    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)*x^4, x, 4, ((1 - 1/(a^2*x^2))^(3/2)*x^4)/(a*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^5)/(2*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*a^2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (7*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 - a*x))/(4*a^2*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 + a*x))/(4*a^2*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)*x^3, x, 4, ((1 - 1/(a^2*x^2))^(3/2)*x^4)/(c - a^2*c*x^2)^(3/2) + ((1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*a*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (5*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 - a*x))/(4*a*(c - a^2*c*x^2)^(3/2)) - ((1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 + a*x))/(4*a*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)*x^2, x, 4, ((1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (3*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 - a*x))/(4*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 + a*x))/(4*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)*x^1, x, 5, (a*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) - (a*(1 - 1/(a^2*x^2))^(3/2)*x^3*atanh(a*x))/(2*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)*x^0, x, 5, (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^2*(1 - 1/(a^2*x^2))^(3/2)*x^3*atanh(a*x))/(2*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)/x^1, x, 4, (a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(x))/(c - a^2*c*x^2)^(3/2) - (3*a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 - a*x))/(4*(c - a^2*c*x^2)^(3/2)) - (a^3*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 + a*x))/(4*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)/x^2, x, 4, -((a^3*(1 - 1/(a^2*x^2))^(3/2)*x^2)/(c - a^2*c*x^2)^(3/2)) + (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(x))/(c - a^2*c*x^2)^(3/2) - (5*a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 - a*x))/(4*(c - a^2*c*x^2)^(3/2)) + (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 + a*x))/(4*(c - a^2*c*x^2)^(3/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(3/2)/x^3, x, 4, -((a^3*(1 - 1/(a^2*x^2))^(3/2)*x)/(2*(c - a^2*c*x^2)^(3/2))) - (a^4*(1 - 1/(a^2*x^2))^(3/2)*x^2)/(c - a^2*c*x^2)^(3/2) + (a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(c - a^2*c*x^2)^(3/2)) + (2*a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(x))/(c - a^2*c*x^2)^(3/2) - (7*a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 - a*x))/(4*(c - a^2*c*x^2)^(3/2)) - (a^5*(1 - 1/(a^2*x^2))^(3/2)*x^3*log(1 + a*x))/(4*(c - a^2*c*x^2)^(3/2))]

    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)*x^5, x, 4, ((1 - 1/(a^2*x^2))^(5/2)*x^6)/(c - a^2*c*x^2)^(5/2) - ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*a*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2)) + ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(a*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*a*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (23*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 - a*x))/(16*a*(c - a^2*c*x^2)^(5/2)) - (7*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 + a*x))/(16*a*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)*x^4, x, 4, -(((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) + (3*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + ((1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (11*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 - a*x))/(16*(c - a^2*c*x^2)^(5/2)) + (5*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 + a*x))/(16*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)*x^3, x, 5, -((a*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) + (a*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(2*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - (a*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (3*a*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)*x^2, x, 5, -((a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) + (a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (a^2*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)*x^1, x, 5, -((a^3*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^3*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) + (a^3*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)*x^0, x, 5, -((a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (3*a^4*(1 - 1/(a^2*x^2))^(5/2)*x^5*atanh(a*x))/(8*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)/x^1, x, 4, -((a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2))) - (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(2*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) - (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(x))/(c - a^2*c*x^2)^(5/2) + (11*a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 - a*x))/(16*(c - a^2*c*x^2)^(5/2)) + (5*a^5*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 + a*x))/(16*(c - a^2*c*x^2)^(5/2))]
    @test_int [ℯ^acoth(a*x)/(c - a^2*c*x^2)^(5/2)/x^2, x, 4, (a^5*(1 - 1/(a^2*x^2))^(5/2)*x^4)/(c - a^2*c*x^2)^(5/2) - (a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(c - a^2*c*x^2)^(5/2)) - (3*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(4*(1 - a*x)*(c - a^2*c*x^2)^(5/2)) + (a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5)/(8*(1 + a*x)*(c - a^2*c*x^2)^(5/2)) - (a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(x))/(c - a^2*c*x^2)^(5/2) + (23*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 - a*x))/(16*(c - a^2*c*x^2)^(5/2)) - (7*a^6*(1 - 1/(a^2*x^2))^(5/2)*x^5*log(1 + a*x))/(16*(c - a^2*c*x^2)^(5/2))]

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

    @test_int [(x^2*sqrt(c - a^2*c*x^2))/ℯ^acoth(a*x), x, 4, -(x^2*sqrt(c - a^2*c*x^2))/(3*a*sqrt(1 - 1/(a^2*x^2))) + (x^3*sqrt(c - a^2*c*x^2))/(4*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(x*sqrt(c - a^2*c*x^2))/ℯ^acoth(a*x), x, 4, -(x*sqrt(c - a^2*c*x^2))/(2*a*sqrt(1 - 1/(a^2*x^2))) + (x^2*sqrt(c - a^2*c*x^2))/(3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - a^2*c*x^2)/ℯ^acoth(a*x), x, 3, -(sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2)))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^acoth(a*x)*x), x, 4, sqrt(c - a^2*c*x^2)/sqrt(1 - 1/(a^2*x^2)) - (sqrt(c - a^2*c*x^2)*log(x))/(a*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^acoth(a*x)*x^2), x, 4, sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2))*x^2) + (sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x)]

    @test_int [(x^3*sqrt(c - a^2*c*x^2))/ℯ^(2*acoth(a*x)), x, 8, (3*x^2*sqrt(c - a^2*c*x^2))/(5*a^2) - (x^3*sqrt(c - a^2*c*x^2))/(2*a) + (1/5)*x^4*sqrt(c - a^2*c*x^2) + (3*(8 - 5*a*x)*sqrt(c - a^2*c*x^2))/(20*a^4) + (3*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(4*a^4)]
    @test_int [(x^2*sqrt(c - a^2*c*x^2))/ℯ^(2*acoth(a*x)), x, 7, -((2*x^2*sqrt(c - a^2*c*x^2))/(3*a)) + (1/4)*x^3*sqrt(c - a^2*c*x^2) - ((32 - 21*a*x)*sqrt(c - a^2*c*x^2))/(24*a^3) - (7*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(8*a^3)]
    @test_int [(x^1*sqrt(c - a^2*c*x^2))/ℯ^(2*acoth(a*x)), x, 6, (1/3)*x^2*sqrt(c - a^2*c*x^2) + ((5 - 3*a*x)*sqrt(c - a^2*c*x^2))/(3*a^2) + (sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/a^2]
    @test_int [sqrt(c - a^2*c*x^2)/ℯ^(2*acoth(a*x)), x, 6, -((3*sqrt(c - a^2*c*x^2))/(2*a)) - ((1 - a*x)*sqrt(c - a^2*c*x^2))/(2*a) - (3*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)))/(2*a)]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(2*acoth(a*x))*x^1), x, 9, sqrt(c - a^2*c*x^2) + 2*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)) + sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(2*acoth(a*x))*x^2), x, 9, sqrt(c - a^2*c*x^2)/x - a*sqrt(c)*atan((a*sqrt(c)*x)/sqrt(c - a^2*c*x^2)) - 2*a*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(2*acoth(a*x))*x^3), x, 7, sqrt(c - a^2*c*x^2)/(2*x^2) - (2*a*sqrt(c - a^2*c*x^2))/x + (3/2)*a^2*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(2*acoth(a*x))*x^4), x, 8, sqrt(c - a^2*c*x^2)/(3*x^3) - (a*sqrt(c - a^2*c*x^2))/x^2 + (5*a^2*sqrt(c - a^2*c*x^2))/(3*x) - a^3*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(2*acoth(a*x))*x^5), x, 9, sqrt(c - a^2*c*x^2)/(4*x^4) - (2*a*sqrt(c - a^2*c*x^2))/(3*x^3) + (7*a^2*sqrt(c - a^2*c*x^2))/(8*x^2) - (4*a^3*sqrt(c - a^2*c*x^2))/(3*x) + (7/8)*a^4*sqrt(c)*atanh(sqrt(c - a^2*c*x^2)/sqrt(c))]

    @test_int [(x^3*sqrt(c - a^2*c*x^2))/ℯ^(3*acoth(a*x)), x, 4, (4*sqrt(c - a^2*c*x^2))/(a^4*sqrt(1 - 1/(a^2*x^2))) - (2*x*sqrt(c - a^2*c*x^2))/(a^3*sqrt(1 - 1/(a^2*x^2))) + (4*x^2*sqrt(c - a^2*c*x^2))/(3*a^2*sqrt(1 - 1/(a^2*x^2))) - (3*x^3*sqrt(c - a^2*c*x^2))/(4*a*sqrt(1 - 1/(a^2*x^2))) + (x^4*sqrt(c - a^2*c*x^2))/(5*sqrt(1 - 1/(a^2*x^2))) - (4*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(a^5*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(x^2*sqrt(c - a^2*c*x^2))/ℯ^(3*acoth(a*x)), x, 4, (-4*sqrt(c - a^2*c*x^2))/(a^3*sqrt(1 - 1/(a^2*x^2))) + (2*x*sqrt(c - a^2*c*x^2))/(a^2*sqrt(1 - 1/(a^2*x^2))) - (x^2*sqrt(c - a^2*c*x^2))/(a*sqrt(1 - 1/(a^2*x^2))) + (x^3*sqrt(c - a^2*c*x^2))/(4*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(a^4*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [(x*sqrt(c - a^2*c*x^2))/ℯ^(3*acoth(a*x)), x, 4, (4*sqrt(c - a^2*c*x^2))/(a^2*sqrt(1 - 1/(a^2*x^2))) - (3*x*sqrt(c - a^2*c*x^2))/(2*a*sqrt(1 - 1/(a^2*x^2))) + (x^2*sqrt(c - a^2*c*x^2))/(3*sqrt(1 - 1/(a^2*x^2))) - (4*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(a^3*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [sqrt(c - a^2*c*x^2)/ℯ^(3*acoth(a*x)), x, 4, -((3*sqrt(c - a^2*c*x^2))/(a*sqrt(1 - 1/(a^2*x^2)))) + (x*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(a^2*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(3*acoth(a*x))*x), x, 4, sqrt(c - a^2*c*x^2)/sqrt(1 - 1/(a^2*x^2)) + (sqrt(c - a^2*c*x^2)*log(x))/(a*sqrt(1 - 1/(a^2*x^2))*x) - (4*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(a*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(3*acoth(a*x))*x^2), x, 4, -(sqrt(c - a^2*c*x^2)/(a*sqrt(1 - 1/(a^2*x^2))*x^2)) - (3*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(3*acoth(a*x))*x^3), x, 4, -(sqrt(c - a^2*c*x^2)/(2*a*sqrt(1 - 1/(a^2*x^2))*x^3)) + (3*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^2) + (4*a*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(3*acoth(a*x))*x^4), x, 4, -(sqrt(c - a^2*c*x^2)/(3*a*sqrt(1 - 1/(a^2*x^2))*x^4)) + (3*sqrt(c - a^2*c*x^2))/(2*sqrt(1 - 1/(a^2*x^2))*x^3) - (4*a*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^2) - (4*a^2*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*a^2*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [sqrt(c - a^2*c*x^2)/(ℯ^(3*acoth(a*x))*x^5), x, 4, -(sqrt(c - a^2*c*x^2)/(4*a*sqrt(1 - 1/(a^2*x^2))*x^5)) + sqrt(c - a^2*c*x^2)/(sqrt(1 - 1/(a^2*x^2))*x^4) - (2*a*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^3) + (4*a^2*sqrt(c - a^2*c*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^2) + (4*a^3*sqrt(c - a^2*c*x^2)*log(x))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a^3*sqrt(c - a^2*c*x^2)*log(1 + a*x))/(sqrt(1 - 1/(a^2*x^2))*x)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*ℯ^(n*acoth(a*x))*(c-c*a^2*x^2)^p*with*m*symbolic=#

    @test_int [ℯ^(3*acoth(a*x))*x^m*sqrt(c - a^2*c*x^2), x, 5, (3*x^m*sqrt(c - a^2*c*x^2))/(a*(1 + m)*sqrt(1 - 1/(a^2*x^2))) + (x^(1 + m)*sqrt(c - a^2*c*x^2))/((2 + m)*sqrt(1 - 1/(a^2*x^2))) - (4*x^m*sqrt(c - a^2*c*x^2)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, a*x))/(a*(1 + m)*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(2*acoth(a*x))*x^m*sqrt(c - a^2*c*x^2), x, 8, If(13>=8, (x^(1 + m)*sqrt(c - a^2*c*x^2))/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, a^2*x^2))/((1 + m)*(2 + m)*sqrt(c - a^2*c*x^2)) - (2*a*c*x^(2 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, a^2*x^2))/((2 + m)*sqrt(c - a^2*c*x^2)), (x^(1 + m)*sqrt(c - a^2*c*x^2))/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, a^2*x^2))/((2 + 3*m + m^2)*sqrt(c - a^2*c*x^2)) - (2*a*c*x^(2 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, a^2*x^2))/((2 + m)*sqrt(c - a^2*c*x^2)))]
    @test_int [ℯ^(1*acoth(a*x))*x^m*sqrt(c - a^2*c*x^2), x, 4, (x^m*sqrt(c - a^2*c*x^2))/(a*(1 + m)*sqrt(1 - 1/(a^2*x^2))) + (x^(1 + m)*sqrt(c - a^2*c*x^2))/((2 + m)*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(x^m*sqrt(c - a^2*c*x^2))/ℯ^(1*acoth(a*x)), x, 4, -((x^m*sqrt(c - a^2*c*x^2))/(a*(1 + m)*sqrt(1 - 1/(a^2*x^2)))) + (x^(1 + m)*sqrt(c - a^2*c*x^2))/((2 + m)*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(x^m*sqrt(c - a^2*c*x^2))/ℯ^(2*acoth(a*x)), x, 8, If(13>=8, (x^(1 + m)*sqrt(c - a^2*c*x^2))/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, a^2*x^2))/((1 + m)*(2 + m)*sqrt(c - a^2*c*x^2)) + (2*a*c*x^(2 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, a^2*x^2))/((2 + m)*sqrt(c - a^2*c*x^2)), (x^(1 + m)*sqrt(c - a^2*c*x^2))/(2 + m) - (c*(3 + 2*m)*x^(1 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (1 + m)/2, (3 + m)/2, a^2*x^2))/((2 + 3*m + m^2)*sqrt(c - a^2*c*x^2)) + (2*a*c*x^(2 + m)*sqrt(1 - a^2*x^2)*HypergeometricFunctions._₂F₁(1/2, (2 + m)/2, (4 + m)/2, a^2*x^2))/((2 + m)*sqrt(c - a^2*c*x^2)))]
    @test_int [(x^m*sqrt(c - a^2*c*x^2))/ℯ^(3*acoth(a*x)), x, 5, -((3*x^m*sqrt(c - a^2*c*x^2))/(a*(1 + m)*sqrt(1 - 1/(a^2*x^2)))) + (x^(1 + m)*sqrt(c - a^2*c*x^2))/((2 + m)*sqrt(1 - 1/(a^2*x^2))) + (4*x^m*sqrt(c - a^2*c*x^2)*HypergeometricFunctions._₂F₁(1, 1 + m, 2 + m, (-a)*x))/(a*(1 + m)*sqrt(1 - 1/(a^2*x^2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*x^m*ℯ^(n*acoth(a*x))*(c-c*a^2*x^2)^p*with*n*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*(c - c*a^2*x^2)^3, x, 3, -((256*c^3*(1 - 1/(a*x))^(4 - n/2)*(1 + 1/(a*x))^((1/2)*(-8 + n))*HypergeometricFunctions._₂F₁(8, 4 - n/2, 5 - n/2, (a - 1/x)/(a + 1/x)))/(a*(8 - n)))]
    @test_int [ℯ^(n*acoth(a*x))*(c - c*a^2*x^2)^2, x, 3, (64*c^2*(1 - 1/(a*x))^(3 - n/2)*(1 + 1/(a*x))^((1/2)*(-6 + n))*HypergeometricFunctions._₂F₁(6, 3 - n/2, 4 - n/2, (a - 1/x)/(a + 1/x)))/(a*(6 - n))]
    @test_int [ℯ^(n*acoth(a*x))*(c - c*a^2*x^2)^1, x, 3, -((16*c*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((1/2)*(-4 + n))*HypergeometricFunctions._₂F₁(4, 2 - n/2, 3 - n/2, (a - 1/x)/(a + 1/x)))/(a*(4 - n)))]
    @test_int [ℯ^(n*acoth(a*x))*(c - c*a^2*x^2)^0, x, 2, (4*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*HypergeometricFunctions._₂F₁(2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)))/(a*(2 - n))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c*a^2*x^2)^1, x, 1, ℯ^(n*acoth(a*x))/(a*c*n)]
    @test_int [ℯ^(n*acoth(a*x))/(c - c*a^2*x^2)^2, x, 2, (2*ℯ^(n*acoth(a*x)))/(a*c^2*n*(4 - n^2)) - (ℯ^(n*acoth(a*x))*(n - 2*a*x))/(a*c^2*(4 - n^2)*(1 - a^2*x^2))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c*a^2*x^2)^3, x, 3, If(13>=8, (24*ℯ^(n*acoth(a*x)))/(a*c^3*n*(64 - 20*n^2 + n^4)) - (ℯ^(n*acoth(a*x))*(n - 4*a*x))/(a*c^3*(16 - n^2)*(1 - a^2*x^2)^2) - (12*ℯ^(n*acoth(a*x))*(n - 2*a*x))/(a*c^3*(4 - n^2)*(16 - n^2)*(1 - a^2*x^2)), (24*ℯ^(n*acoth(a*x)))/(a*c^3*n*(64 - 20*n^2 + n^4)) - (ℯ^(n*acoth(a*x))*(n - 4*a*x))/(a*c^3*(16 - n^2)*(1 - a^2*x^2)^2) - (12*ℯ^(n*acoth(a*x))*(n - 2*a*x))/(a*c^3*(64 - 20*n^2 + n^4)*(1 - a^2*x^2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c*a^2*x^2)^4, x, 4, If(13>=8, (720*ℯ^(n*acoth(a*x)))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (ℯ^(n*acoth(a*x))*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3) - (30*ℯ^(n*acoth(a*x))*(n - 4*a*x))/(a*c^4*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)^2) - (360*ℯ^(n*acoth(a*x))*(n - 2*a*x))/(a*c^4*(4 - n^2)*(16 - n^2)*(36 - n^2)*(1 - a^2*x^2)), (720*ℯ^(n*acoth(a*x)))/(a*c^4*n*(36 - n^2)*(64 - 20*n^2 + n^4)) - (ℯ^(n*acoth(a*x))*(n - 6*a*x))/(a*c^4*(36 - n^2)*(1 - a^2*x^2)^3) - (30*ℯ^(n*acoth(a*x))*(n - 4*a*x))/(a*c^4*(576 - 52*n^2 + n^4)*(1 - a^2*x^2)^2) - (360*ℯ^(n*acoth(a*x))*(n - 2*a*x))/(a*c^4*(36 - n^2)*(64 - 20*n^2 + n^4)*(1 - a^2*x^2)))]

    @test_int [ℯ^(n*acoth(a*x))*(c - a^2*c*x^2)^(3/2), x, 3, (32*(1 - 1/(a*x))^((5 - n)/2)*(1 + 1/(a*x))^((1/2)*(-5 + n))*(c - a^2*c*x^2)^(3/2)*HypergeometricFunctions._₂F₁(5, (5 - n)/2, (7 - n)/2, (a - 1/x)/(a + 1/x)))/(a^4*(5 - n)*(1 - 1/(a^2*x^2))^(3/2)*x^3)]
    @test_int [ℯ^(n*acoth(a*x))*(c - a^2*c*x^2)^(1/2), x, 3, (8*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*sqrt(c - a^2*c*x^2)*HypergeometricFunctions._₂F₁(3, (3 - n)/2, (5 - n)/2, (a - 1/x)/(a + 1/x)))/(a^2*(3 - n)*sqrt(1 - 1/(a^2*x^2))*x)]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(1/2), x, 3, (2*sqrt(1 - 1/(a^2*x^2))*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x*HypergeometricFunctions._₂F₁(1, (1 - n)/2, (3 - n)/2, (a - 1/x)/(a + 1/x)))/((1 - n)*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(3/2), x, 1, -((ℯ^(n*acoth(a*x))*(n - a*x))/(a*c*(1 - n^2)*sqrt(c - a^2*c*x^2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(5/2), x, 2, If(13>=8, -((ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^2*(1 - n^2)*(9 - n^2)*sqrt(c - a^2*c*x^2)), -((ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^2*(9 - 10*n^2 + n^4)*sqrt(c - a^2*c*x^2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(7/2), x, 3, If(13>=8, -((ℯ^(n*acoth(a*x))*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c^2*(9 - n^2)*(25 - n^2)*(c - a^2*c*x^2)^(3/2)) - (120*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^3*(1 - n^2)*(9 - n^2)*(25 - n^2)*sqrt(c - a^2*c*x^2)), -((ℯ^(n*acoth(a*x))*(n - 5*a*x))/(a*c*(25 - n^2)*(c - a^2*c*x^2)^(5/2))) - (20*ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c^2*(225 - 34*n^2 + n^4)*(c - a^2*c*x^2)^(3/2)) - (120*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^3*(25 - n^2)*(9 - 10*n^2 + n^4)*sqrt(c - a^2*c*x^2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(9/2), x, 4, If(13>=8, -((ℯ^(n*acoth(a*x))*(n - 7*a*x))/(a*c*(49 - n^2)*(c - a^2*c*x^2)^(7/2))) - (42*ℯ^(n*acoth(a*x))*(n - 5*a*x))/(a*c^2*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(5/2)) - (840*ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c^3*(9 - n^2)*(25 - n^2)*(49 - n^2)*(c - a^2*c*x^2)^(3/2)) - (5040*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^4*(1 - n^2)*(9 - n^2)*(25 - n^2)*(49 - n^2)*sqrt(c - a^2*c*x^2)),-((ℯ^(n*acoth(a*x))*(n - 7*a*x))/(a*c*(49 - n^2)*(c - a^2*c*x^2)^(7/2))) - (42*ℯ^(n*acoth(a*x))*(n - 5*a*x))/(a*c^2*(1225 - 74*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) - (840*ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c^3*(49 - n^2)*(225 - 34*n^2 + n^4)*(c - a^2*c*x^2)^(3/2)) - (5040*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^4*(1225 - 74*n^2 + n^4)*(9 - 10*n^2 + n^4)*sqrt(c - a^2*c*x^2)))]

    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(3/2)*x^3, x, 7, If(13>=8, -(((2 + n)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 + n)*(c - a^2*c*x^2)^(3/2))) + ((2 + 2*n + n^2)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 - n)*(1 + n)*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^4)/(c - a^2*c*x^2)^(3/2) - (2*n*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)))/(a*(1 - n)*(c - a^2*c*x^2)^(3/2)), -(((2 + n)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 + n)*(c - a^2*c*x^2)^(3/2))) + ((2 + 2*n + n^2)*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/(a*(1 - n^2)*(c - a^2*c*x^2)^(3/2)) + ((1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^4)/(c - a^2*c*x^2)^(3/2) - (2*n*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)))/(a*(1 - n)*(c - a^2*c*x^2)^(3/2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(3/2)*x^2, x, 4, -((ℯ^(n*acoth(a*x))*(n - a*x))/(a^3*c*(1 - n^2)*sqrt(c - a^2*c*x^2))) - (2*sqrt(1 - 1/(a^2*x^2))*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x*HypergeometricFunctions._₂F₁(1, (1 - n)/2, (3 - n)/2, (a - 1/x)/(a + 1/x)))/(a^2*c*(1 - n)*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(3/2)*x^1, x, 1, (ℯ^(n*acoth(a*x))*(1 - a*n*x))/(a^2*c*(1 - n^2)*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(3/2)*x^0, x, 1, -((ℯ^(n*acoth(a*x))*(n - a*x))/(a*c*(1 - n^2)*sqrt(c - a^2*c*x^2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(3/2)/x^1, x, 5, -((a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/((1 + n)*(c - a^2*c*x^2)^(3/2))) + (a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^3)/((1 - n^2)*(c - a^2*c*x^2)^(3/2)) - (2^((1 + n)/2)*a^3*(1 - 1/(a^2*x^2))^(3/2)*(1 - 1/(a*x))^((1 - n)/2)*x^3*HypergeometricFunctions._₂F₁((1 - n)/2, (1 - n)/2, (3 - n)/2, (a - 1/x)/(2*a)))/((1 - n)*(c - a^2*c*x^2)^(3/2))]

    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(5/2)*x^4, x, 8, If(13>=8, -(((1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - ((6 + n)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((1 + n)*(3 + n)*(c - a^2*c*x^2)^(5/2)) + ((15 + 6*n + n^2)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((1 - n)*(1 + n)*(3 + n)*(c - a^2*c*x^2)^(5/2)) - ((18 + 7*n - 2*n^2 - n^3)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) - (2*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)))/((1 - n)*(c - a^2*c*x^2)^(5/2)), -(((1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - ((6 + n)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + ((15 + 6*n + n^2)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - ((18 + 7*n - 2*n^2 - n^3)*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) - (2*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5*HypergeometricFunctions._₂F₁(1, (1/2)*(-1 + n), (1 + n)/2, (a + 1/x)/(a - 1/x)))/((1 - n)*(c - a^2*c*x^2)^(5/2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(5/2)*x^3, x, 6, If(13>=8, -((a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)), -((a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - (6*a*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(5/2)*x^2, x, 2, -((ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a^3*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) + (ℯ^(n*acoth(a*x))*(3 - n^2)*(n - a*x))/(a^3*c^2*(9 - 10*n^2 + n^4)*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(5/2)*x^1, x, 2, (ℯ^(n*acoth(a*x))*(3 - a*n*x))/(a^2*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2)) + (2*ℯ^(n*acoth(a*x))*n*(n - a*x))/(a^2*c^2*(9 - 10*n^2 + n^4)*sqrt(c - a^2*c*x^2))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(5/2)*x^0, x, 2, If(13>=8, -((ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^2*(1 - n^2)*(9 - n^2)*sqrt(c - a^2*c*x^2)), -((ℯ^(n*acoth(a*x))*(n - 3*a*x))/(a*c*(9 - n^2)*(c - a^2*c*x^2)^(3/2))) - (6*ℯ^(n*acoth(a*x))*(n - a*x))/(a*c^2*(9 - 10*n^2 + n^4)*sqrt(c - a^2*c*x^2)))]
    @test_int [ℯ^(n*acoth(a*x))/(c - a^2*c*x^2)^(5/2)/x^1, x, 15, If(13>=8, -((a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) + (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) - (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n)*(1 - n^2)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (2^((5 + n)/2)*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*x^5*HypergeometricFunctions._₂F₁((1/2)*(-3 - n), (1/2)*(-3 - n), (1/2)*(-1 - n), (a - 1/x)/(2*a)))/((3 + n)*(c - a^2*c*x^2)^(5/2)), -((a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2))) - (3*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((3 - n)/2)*(1 + 1/(a*x))^((1/2)*(-3 + n))*x^5)/((9 - 10*n^2 + n^4)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) + (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) - (8*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1 - n)/2)*(1 + 1/(a*x))^((1/2)*(-1 + n))*x^5)/((3 + n - 3*n^2 - n^3)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (6*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-1 - n))*(1 + 1/(a*x))^((1 + n)/2)*x^5)/((3 + 4*n + n^2)*(c - a^2*c*x^2)^(5/2)) + (4*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*(1 + 1/(a*x))^((3 + n)/2)*x^5)/((3 + n)*(c - a^2*c*x^2)^(5/2)) - (2^((5 + n)/2)*a^5*(1 - 1/(a^2*x^2))^(5/2)*(1 - 1/(a*x))^((1/2)*(-3 - n))*x^5*HypergeometricFunctions._₂F₁((1/2)*(-3 - n), (1/2)*(-3 - n), (1/2)*(-1 - n), (a - 1/x)/(2*a)))/((3 + n)*(c - a^2*c*x^2)^(5/2)))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c*a^2*x^2)^p*with*p*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*(c - c*a^2*x^2)^p, x, 3, (((a - 1/x)/(a + 1/x))^((1/2)*(n - 2*p))*(1 - 1/(a*x))^(-(n/2) + p)*(1 + 1/(a*x))^(1 + n/2 + p)*x*(c - a^2*c*x^2)^p*HypergeometricFunctions._₂F₁(-1 - 2*p, (1/2)*(n - 2*p), -2*p, 2/((a + 1/x)*x)))/((1 - 1/(a^2*x^2))^p*(1 + 2*p))]

    @test_int [(c - a^2*c*x^2)^p*ℯ^(2*p*acoth(a*x)), x, 3, ((1 + 1/(a*x))^(1 + 2*p)*x*(c - a^2*c*x^2)^p)/((1 + 2*p)*(1 - 1/(a^2*x^2))^p)]
    @test_int [(c - a^2*c*x^2)^p/ℯ^(2*p*acoth(a*x)), x, 3, ((1 - 1/(a*x))^(1 + 2*p)*x*(c - a^2*c*x^2)^p)/((1 + 2*p)*(1 - 1/(a^2*x^2))^p)]

    @test_int [ℯ^(4*acoth(a*x))*(c - a^2*c*x^2)^p, x, 4, (2^(2 + p)*c*(1 + a*x)^(1 - p)*(c - a^2*c*x^2)^(-1 + p)*HypergeometricFunctions._₂F₁(-2 - p, -1 + p, p, (1/2)*(1 - a*x)))/(a*(1 - p))]
    @test_int [ℯ^(3*acoth(a*x))*(c - a^2*c*x^2)^p, x, 3, (((a - 1/x)/(a + 1/x))^(3/2 - p)*(1 - 1/(a*x))^(-(3/2) + p)*(1 + 1/(a*x))^(5/2 + p)*x*(c - a^2*c*x^2)^p*HypergeometricFunctions._₂F₁(-1 - 2*p, 3/2 - p, -2*p, 2/((a + 1/x)*x)))/((1 - 1/(a^2*x^2))^p*(1 + 2*p))]
    @test_int [ℯ^(2*acoth(a*x))*(c - a^2*c*x^2)^p, x, 4, (2^(1 + p)*(c - a^2*c*x^2)^p*HypergeometricFunctions._₂F₁(-1 - p, p, 1 + p, (1/2)*(1 - a*x)))/((1 + a*x)^p*(a*p))]
    @test_int [ℯ^acoth(a*x)*(c - a^2*c*x^2)^p, x, 3, (((a - 1/x)/(a + 1/x))^(1/2 - p)*(1 - 1/(a*x))^(-(1/2) + p)*(1 + 1/(a*x))^(3/2 + p)*x*(c - a^2*c*x^2)^p*HypergeometricFunctions._₂F₁(-1 - 2*p, 1/2 - p, -2*p, 2/((a + 1/x)*x)))/((1 - 1/(a^2*x^2))^p*(1 + 2*p))]
    @test_int [(c - a^2*c*x^2)^p/ℯ^acoth(a*x), x, 3, (((a - 1/x)/(a + 1/x))^(-(1/2) - p)*(1 - 1/(a*x))^(1/2 + p)*(1 + 1/(a*x))^(1/2 + p)*x*(c - a^2*c*x^2)^p*HypergeometricFunctions._₂F₁(-1 - 2*p, -(1/2) - p, -2*p, 2/((a + 1/x)*x)))/((1 - 1/(a^2*x^2))^p*(1 + 2*p))]
    @test_int [(c - a^2*c*x^2)^p/ℯ^(2*acoth(a*x)), x, 4, -((2^(1 + p)*(c - a^2*c*x^2)^p*HypergeometricFunctions._₂F₁(-1 - p, p, 1 + p, (1/2)*(1 + a*x)))/((1 - a*x)^p*(a*p)))]
    @test_int [(c - a^2*c*x^2)^p/ℯ^(3*acoth(a*x)), x, 3, (((a - 1/x)/(a + 1/x))^(-(3/2) - p)*(1 - 1/(a*x))^(3/2 + p)*(1 + 1/(a*x))^(-(1/2) + p)*x*(c - a^2*c*x^2)^p*HypergeometricFunctions._₂F₁(-1 - 2*p, -(3/2) - p, -2*p, 2/((a + 1/x)*x)))/((1 - 1/(a^2*x^2))^p*(1 + 2*p))]

    #= ::Section::Closed:: =#
    #=Integrands*of*the*form*u*ℯ^(n*acoth(a*x))*(c-c/(a^2*x^2))^p=#

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a^2*x^2))^p=#

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

    @test_int [ℯ^acoth(a*x)*(c - c/(a^2*x^2))^4, x, 14, -((51*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(16*a)) - (67*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(48*a) - (91*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(120*a) - (131*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2))/(280*a) + (61*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2))/(70*a) + (47*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))/(42*a) + (8*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(9/2))/(7*a) + c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(9/2)*x + (35*c^4*acsc(a*x))/(16*a) + (c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^acoth(a*x)*(c - c/(a^2*x^2))^3, x, 12, -((23*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(8*a)) - (31*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(24*a) - (43*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(60*a) + (23*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2))/(20*a) + (6*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2))/(5*a) + c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)*x + (15*c^3*acsc(a*x))/(8*a) + (c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^acoth(a*x)*(c - c/(a^2*x^2))^2, x, 10, -((5*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(2*a)) - (7*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(6*a) + (4*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(3*a) + c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)*x + (3*c^2*acsc(a*x))/(2*a) + (c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^acoth(a*x)*(c - c/(a^2*x^2)), x, 9, -((2*c*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/a) + c*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x + (c*acsc(a*x))/a + (c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^acoth(a*x)/(c - c/(a^2*x^2)), x, 6, -((2*sqrt(1 + 1/(a*x)))/(a*c*sqrt(1 - 1/(a*x)))) + (sqrt(1 + 1/(a*x))*x)/(c*sqrt(1 - 1/(a*x))) + atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c)]
    @test_int [ℯ^acoth(a*x)/(c - c/(a^2*x^2))^2, x, 8, -(4/(3*a*c^2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x)))) - 11/(3*a*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (8*sqrt(1 - 1/(a*x)))/(3*a*c^2*sqrt(1 + 1/(a*x))) + x/(c^2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))) + atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c^2)]
    @test_int [ℯ^acoth(a*x)/(c - c/(a^2*x^2))^3, x, 10, -(6/(5*a*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2))) - 29/(15*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) - 34/(5*a*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)) + (21*sqrt(1 - 1/(a*x)))/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (16*sqrt(1 - 1/(a*x)))/(5*a*c^3*sqrt(1 + 1/(a*x))) + x/(c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) + atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c^3)]
    @test_int [ℯ^acoth(a*x)/(c - c/(a^2*x^2))^4, x, 12, -(8/(7*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2))) - 11/(7*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2)) - 62/(21*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) - 269/(21*a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)) + (262*sqrt(1 - 1/(a*x)))/(35*a*c^4*(1 + 1/(a*x))^(5/2)) + (163*sqrt(1 - 1/(a*x)))/(35*a*c^4*(1 + 1/(a*x))^(3/2)) + (128*sqrt(1 - 1/(a*x)))/(35*a*c^4*sqrt(1 + 1/(a*x))) + x/(c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)) + atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c^4)]

    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^5, x, 5, -(c^5/(9*a^10*x^9)) - c^5/(4*a^9*x^8) + (3*c^5)/(7*a^8*x^7) + (4*c^5)/(3*a^7*x^6) - (2*c^5)/(5*a^6*x^5) - (3*c^5)/(a^5*x^4) - (2*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) + (3*c^5)/(a^2*x) + c^5*x + (2*c^5*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^4, x, 5, c^4/(7*a^8*x^7) + c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) - (3*c^4)/(2*a^5*x^4) + (3*c^4)/(a^3*x^2) + (2*c^4)/(a^2*x) + c^4*x + (2*c^4*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^3, x, 5, -(c^3/(5*a^6*x^5)) - c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) + (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x + (2*c^3*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^2, x, 5, c^2/(3*a^4*x^3) + c^2/(a^3*x^2) + c^2*x + (2*c^2*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2)), x, 5, -(c/(a^2*x)) + c*x + (2*c*log(x))/a]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a^2*x^2)), x, 5, x/c + 1/(a*c*(1 - a*x)) + (2*log(1 - a*x))/(a*c)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a^2*x^2))^2, x, 5, x/c^2 - 1/(4*a*c^2*(1 - a*x)^2) + 7/(4*a*c^2*(1 - a*x)) + (17*log(1 - a*x))/(8*a*c^2) - log(1 + a*x)/(8*a*c^2)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a^2*x^2))^3, x, 5, x/c^3 + 1/(12*a*c^3*(1 - a*x)^3) - 5/(8*a*c^3*(1 - a*x)^2) + 39/(16*a*c^3*(1 - a*x)) - 1/(16*a*c^3*(1 + a*x)) + (9*log(1 - a*x))/(4*a*c^3) - log(1 + a*x)/(4*a*c^3)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a^2*x^2))^4, x, 5, x/c^4 - 1/(32*a*c^4*(1 - a*x)^4) + 13/(48*a*c^4*(1 - a*x)^3) - 35/(32*a*c^4*(1 - a*x)^2) + 99/(32*a*c^4*(1 - a*x)) + 1/(64*a*c^4*(1 + a*x)^2) - 11/(64*a*c^4*(1 + a*x)) + (303*log(1 - a*x))/(128*a*c^4) - (47*log(1 + a*x))/(128*a*c^4)]

    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^4, x, 14, -((63*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(16*a)) - (37*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(16*a) - (61*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(40*a) - (303*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2))/(280*a) - (57*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2))/(70*a) + (15*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(11/2))/(14*a) + (8*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(11/2))/(7*a) + c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(11/2)*x + (15*c^4*acsc(a*x))/(16*a) + (3*c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^3, x, 12, -((27*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(8*a)) - (17*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(8*a) - (29*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(20*a) - (21*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2))/(20*a) + (6*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2))/(5*a) + c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)*x + (3*c^3*acsc(a*x))/(8*a) + (3*c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^2, x, 10, -((5*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(2*a)) - (11*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(6*a) - (4*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(3*a) + c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)*x - (c^2*acsc(a*x))/(2*a) + (3*c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2)), x, 8, c*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)*x - (3*c*acsc(a*x))/a + (3*c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a^2*x^2)), x, 7, -((5*sqrt(1 + 1/(a*x)))/(3*a*c*(1 - 1/(a*x))^(3/2))) - (14*sqrt(1 + 1/(a*x)))/(3*a*c*sqrt(1 - 1/(a*x))) + (sqrt(1 + 1/(a*x))*x)/(c*(1 - 1/(a*x))^(3/2)) + (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c)]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a^2*x^2))^2, x, 9, -((6*sqrt(1 + 1/(a*x)))/(5*a*c^2*(1 - 1/(a*x))^(5/2))) - (9*sqrt(1 + 1/(a*x)))/(5*a*c^2*(1 - 1/(a*x))^(3/2)) - (24*sqrt(1 + 1/(a*x)))/(5*a*c^2*sqrt(1 - 1/(a*x))) + (sqrt(1 + 1/(a*x))*x)/(c^2*(1 - 1/(a*x))^(5/2)) + (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c^2)]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a^2*x^2))^3, x, 10, -(8/(7*a*c^3*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x)))) - 53/(35*a*c^3*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x))) - 88/(35*a*c^3*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))) - 281/(35*a*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))) + (176*sqrt(1 - 1/(a*x)))/(35*a*c^3*sqrt(1 + 1/(a*x))) + x/(c^3*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))) + (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c^3)]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a^2*x^2))^4, x, 12, -(10/(9*a*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2))) - 29/(21*a*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2)) - 208/(105*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)) - 1147/(315*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2)) - 1462/(105*a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)) + (2609*sqrt(1 - 1/(a*x)))/(315*a*c^4*(1 + 1/(a*x))^(3/2)) + (1664*sqrt(1 - 1/(a*x)))/(315*a*c^4*sqrt(1 + 1/(a*x))) + x/(c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)) + (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c^4)]

    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a^2*x^2))^5, x, 5, c^5/(9*a^10*x^9) + c^5/(2*a^9*x^8) + (3*c^5)/(7*a^8*x^7) - (4*c^5)/(3*a^7*x^6) - (14*c^5)/(5*a^6*x^5) + (14*c^5)/(3*a^4*x^3) + (4*c^5)/(a^3*x^2) - (3*c^5)/(a^2*x) + c^5*x + (4*c^5*log(x))/a]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a^2*x^2))^4, x, 5, -(c^4/(7*a^8*x^7)) - (2*c^4)/(3*a^7*x^6) - (4*c^4)/(5*a^6*x^5) + c^4/(a^5*x^4) + (10*c^4)/(3*a^4*x^3) + (2*c^4)/(a^3*x^2) - (4*c^4)/(a^2*x) + c^4*x + (4*c^4*log(x))/a]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a^2*x^2))^3, x, 5, c^3/(5*a^6*x^5) + c^3/(a^5*x^4) + (5*c^3)/(3*a^4*x^3) - (5*c^3)/(a^2*x) + c^3*x + (4*c^3*log(x))/a]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a^2*x^2))^2, x, 5, -(c^2/(3*a^4*x^3)) - (2*c^2)/(a^3*x^2) - (6*c^2)/(a^2*x) + c^2*x + (4*c^2*log(x))/a]
    @test_int [ℯ^(4*acoth(a*x))*(c - c/(a^2*x^2)), x, 5, c/(a^2*x) + c*x - (4*c*log(x))/a + (8*c*log(1 - a*x))/a]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a^2*x^2)), x, 5, x/c - 1/(a*c*(1 - a*x)^2) + 5/(a*c*(1 - a*x)) + (4*log(1 - a*x))/(a*c)]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a^2*x^2))^2, x, 5, x/c^2 + 1/(3*a*c^2*(1 - a*x)^3) - 2/(a*c^2*(1 - a*x)^2) + 6/(a*c^2*(1 - a*x)) + (4*log(1 - a*x))/(a*c^2)]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a^2*x^2))^3, x, 5, x/c^3 - 1/(8*a*c^3*(1 - a*x)^4) + 11/(12*a*c^3*(1 - a*x)^3) - 49/(16*a*c^3*(1 - a*x)^2) + 111/(16*a*c^3*(1 - a*x)) + (129*log(1 - a*x))/(32*a*c^3) - log(1 + a*x)/(32*a*c^3)]
    @test_int [ℯ^(4*acoth(a*x))/(c - c/(a^2*x^2))^4, x, 5, x/c^4 + 1/(20*a*c^4*(1 - a*x)^5) - 7/(16*a*c^4*(1 - a*x)^4) + 83/(48*a*c^4*(1 - a*x)^3) - 67/(16*a*c^4*(1 - a*x)^2) + 501/(64*a*c^4*(1 - a*x)) - 1/(64*a*c^4*(1 + a*x)) + (261*log(1 - a*x))/(64*a*c^4) - (5*log(1 + a*x))/(64*a*c^4)]

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

    @test_int [(c - c/(a^2*x^2))^4/ℯ^acoth(a*x), x, 14, -((19*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(16*a)) - (c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(16*a) + (7*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(40*a) + (19*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2))/(40*a) + (29*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2))/(30*a) + (7*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2))/(6*a) + (8*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(7/2))/(7*a) + c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(7/2)*x + (35*c^4*acsc(a*x))/(16*a) - (c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [(c - c/(a^2*x^2))^3/ℯ^acoth(a*x), x, 12, -((7*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(8*a)) + (c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(24*a) + (11*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(12*a) + (5*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2))/(4*a) + (6*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2))/(5*a) + c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2)*x + (15*c^3*acsc(a*x))/(8*a) - (c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [(c - c/(a^2*x^2))^2/ℯ^acoth(a*x), x, 10, -((c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(2*a)) + (3*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(2*a) + (4*c^2*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2))/(3*a) + c^2*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2)*x + (3*c^2*acsc(a*x))/(2*a) - (c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [(c - c/(a^2*x^2))/ℯ^acoth(a*x), x, 9, (2*c*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/a + c*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x + (c*acsc(a*x))/a - (c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a^2*x^2))), x, 6, (2*sqrt(1 - 1/(a*x)))/(a*c*sqrt(1 + 1/(a*x))) + (sqrt(1 - 1/(a*x))*x)/(c*sqrt(1 + 1/(a*x))) - atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c)]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a^2*x^2))^2), x, 8, -(2/(a*c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))) + (5*sqrt(1 - 1/(a*x)))/(3*a*c^2*(1 + 1/(a*x))^(3/2)) + (8*sqrt(1 - 1/(a*x)))/(3*a*c^2*sqrt(1 + 1/(a*x))) + x/(c^2*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2)) - atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c^2)]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a^2*x^2))^3), x, 10, -(4/(3*a*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2))) - 13/(3*a*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2)) + (14*sqrt(1 - 1/(a*x)))/(5*a*c^3*(1 + 1/(a*x))^(5/2)) + (11*sqrt(1 - 1/(a*x)))/(5*a*c^3*(1 + 1/(a*x))^(3/2)) + (16*sqrt(1 - 1/(a*x)))/(5*a*c^3*sqrt(1 + 1/(a*x))) + x/(c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2)) - atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c^3)]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a^2*x^2))^4), x, 12, -(6/(5*a*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2))) - 31/(15*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(7/2)) - 28/(3*a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)) + (115*sqrt(1 - 1/(a*x)))/(21*a*c^4*(1 + 1/(a*x))^(7/2)) + (122*sqrt(1 - 1/(a*x)))/(35*a*c^4*(1 + 1/(a*x))^(5/2)) + (93*sqrt(1 - 1/(a*x)))/(35*a*c^4*(1 + 1/(a*x))^(3/2)) + (128*sqrt(1 - 1/(a*x)))/(35*a*c^4*sqrt(1 + 1/(a*x))) + x/(c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(7/2)) - atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(a*c^4)]

    @test_int [(c - c/(a^2*x^2))^4/ℯ^(2*acoth(a*x)), x, 5, c^4/(7*a^8*x^7) - c^4/(3*a^7*x^6) - (2*c^4)/(5*a^6*x^5) + (3*c^4)/(2*a^5*x^4) - (3*c^4)/(a^3*x^2) + (2*c^4)/(a^2*x) + c^4*x - (2*c^4*log(x))/a]
    @test_int [(c - c/(a^2*x^2))^3/ℯ^(2*acoth(a*x)), x, 5, -(c^3/(5*a^6*x^5)) + c^3/(2*a^5*x^4) + c^3/(3*a^4*x^3) - (2*c^3)/(a^3*x^2) + c^3/(a^2*x) + c^3*x - (2*c^3*log(x))/a]
    @test_int [(c - c/(a^2*x^2))^2/ℯ^(2*acoth(a*x)), x, 5, c^2/(3*a^4*x^3) - c^2/(a^3*x^2) + c^2*x - (2*c^2*log(x))/a]
    @test_int [(c - c/(a^2*x^2))/ℯ^(2*acoth(a*x)), x, 5, -(c/(a^2*x)) + c*x - (2*c*log(x))/a]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))), x, 5, x/c - 1/(a*c*(1 + a*x)) - (2*log(1 + a*x))/(a*c)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^2), x, 5, x/c^2 + 1/(4*a*c^2*(1 + a*x)^2) - 7/(4*a*c^2*(1 + a*x)) + log(1 - a*x)/(8*a*c^2) - (17*log(1 + a*x))/(8*a*c^2)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^3), x, 5, x/c^3 + 1/(16*a*c^3*(1 - a*x)) - 1/(12*a*c^3*(1 + a*x)^3) + 5/(8*a*c^3*(1 + a*x)^2) - 39/(16*a*c^3*(1 + a*x)) + log(1 - a*x)/(4*a*c^3) - (9*log(1 + a*x))/(4*a*c^3)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^4), x, 5, x/c^4 - 1/(64*a*c^4*(1 - a*x)^2) + 11/(64*a*c^4*(1 - a*x)) + 1/(32*a*c^4*(1 + a*x)^4) - 13/(48*a*c^4*(1 + a*x)^3) + 35/(32*a*c^4*(1 + a*x)^2) - 99/(32*a*c^4*(1 + a*x)) + (47*log(1 - a*x))/(128*a*c^4) - (303*log(1 + a*x))/(128*a*c^4)]

    @test_int [(c - c/(a^2*x^2))^4/ℯ^(3*acoth(a*x)), x, 14, (33*c^4*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(16*a) + (27*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(16*a) - (3*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(5/2))/(8*a) + (5*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(5/2))/(8*a) + (11*c^4*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(5/2))/(10*a) + (17*c^4*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(5/2))/(14*a) + (8*c^4*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(5/2))/(7*a) + c^4*(1 - 1/(a*x))^(11/2)*(1 + 1/(a*x))^(5/2)*x + (15*c^4*acsc(a*x))/(16*a) - (3*c^4*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [(c - c/(a^2*x^2))^3/ℯ^(3*acoth(a*x)), x, 12, (21*c^3*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(8*a) + (3*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(3/2))/(8*a) + (5*c^3*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(3/2))/(4*a) + (27*c^3*(1 - 1/(a*x))^(5/2)*(1 + 1/(a*x))^(3/2))/(20*a) + (6*c^3*(1 - 1/(a*x))^(7/2)*(1 + 1/(a*x))^(3/2))/(5*a) + c^3*(1 - 1/(a*x))^(9/2)*(1 + 1/(a*x))^(3/2)*x + (3*c^3*acsc(a*x))/(8*a) - (3*c^3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [(c - c/(a^2*x^2))^2/ℯ^(3*acoth(a*x)), x, 10, (5*c^2*sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x)))/(2*a) + (11*c^2*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x)))/(6*a) + (4*c^2*(1 - 1/(a*x))^(5/2)*sqrt(1 + 1/(a*x)))/(3*a) + c^2*(1 - 1/(a*x))^(7/2)*sqrt(1 + 1/(a*x))*x - (c^2*acsc(a*x))/(2*a) - (3*c^2*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [(c - c/(a^2*x^2))/ℯ^(3*acoth(a*x)), x, 8, c*(1 - 1/(a*x))^(3/2)*sqrt(1 + 1/(a*x))*x - (3*c*acsc(a*x))/a - (3*c*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/a]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))), x, 7, (5*sqrt(1 - 1/(a*x)))/(3*a*c*(1 + 1/(a*x))^(3/2)) + (14*sqrt(1 - 1/(a*x)))/(3*a*c*sqrt(1 + 1/(a*x))) + (sqrt(1 - 1/(a*x))*x)/(c*(1 + 1/(a*x))^(3/2)) - (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^2), x, 9, (6*sqrt(1 - 1/(a*x)))/(5*a*c^2*(1 + 1/(a*x))^(5/2)) + (9*sqrt(1 - 1/(a*x)))/(5*a*c^2*(1 + 1/(a*x))^(3/2)) + (24*sqrt(1 - 1/(a*x)))/(5*a*c^2*sqrt(1 + 1/(a*x))) + (sqrt(1 - 1/(a*x))*x)/(c^2*(1 + 1/(a*x))^(5/2)) - (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c^2)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^3), x, 10, -(2/(a*c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2))) + (11*sqrt(1 - 1/(a*x)))/(7*a*c^3*(1 + 1/(a*x))^(7/2)) + (54*sqrt(1 - 1/(a*x)))/(35*a*c^3*(1 + 1/(a*x))^(5/2)) + (71*sqrt(1 - 1/(a*x)))/(35*a*c^3*(1 + 1/(a*x))^(3/2)) + (176*sqrt(1 - 1/(a*x)))/(35*a*c^3*sqrt(1 + 1/(a*x))) + x/(c^3*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(7/2)) - (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c^3)]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^4), x, 12, -(4/(3*a*c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2))) - 5/(a*c^4*sqrt(1 - 1/(a*x))*(1 + 1/(a*x))^(9/2)) + (28*sqrt(1 - 1/(a*x)))/(9*a*c^4*(1 + 1/(a*x))^(9/2)) + (139*sqrt(1 - 1/(a*x)))/(63*a*c^4*(1 + 1/(a*x))^(7/2)) + (202*sqrt(1 - 1/(a*x)))/(105*a*c^4*(1 + 1/(a*x))^(5/2)) + (719*sqrt(1 - 1/(a*x)))/(315*a*c^4*(1 + 1/(a*x))^(3/2)) + (1664*sqrt(1 - 1/(a*x)))/(315*a*c^4*sqrt(1 + 1/(a*x))) + x/(c^4*(1 - 1/(a*x))^(3/2)*(1 + 1/(a*x))^(9/2)) - (3*atanh(sqrt(1 - 1/(a*x))*sqrt(1 + 1/(a*x))))/(a*c^4)]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a^2*x^2))^(p/2)=#

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

    @test_int [ℯ^acoth(a*x)*(c - c/(a^2*x^2))^(7/2), x, 4, (c^3*sqrt(c - c/(a^2*x^2)))/(6*a^7*sqrt(1 - 1/(a^2*x^2))*x^6) + (c^3*sqrt(c - c/(a^2*x^2)))/(5*a^6*sqrt(1 - 1/(a^2*x^2))*x^5) - (3*c^3*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) - (c^3*sqrt(c - c/(a^2*x^2)))/(a^4*sqrt(1 - 1/(a^2*x^2))*x^3) + (3*c^3*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2) + (3*c^3*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^3*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (c^3*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)*(c - c/(a^2*x^2))^(5/2), x, 4, -((c^2*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4)) - (c^2*sqrt(c - c/(a^2*x^2)))/(3*a^4*sqrt(1 - 1/(a^2*x^2))*x^3) + (c^2*sqrt(c - c/(a^2*x^2)))/(a^3*sqrt(1 - 1/(a^2*x^2))*x^2) + (2*c^2*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^2*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (c^2*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)*(c - c/(a^2*x^2))^(3/2), x, 4, (c*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2) + (c*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (c*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2)), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)/sqrt(c - c/(a^2*x^2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a^2*x^2)) + (sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(a*sqrt(c - c/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)/(c - c/(a^2*x^2))^(3/2), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c*sqrt(c - c/(a^2*x^2))) + sqrt(1 - 1/(a^2*x^2))/(2*a*c*sqrt(c - c/(a^2*x^2))*(1 - a*x)) + (5*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(4*a*c*sqrt(c - c/(a^2*x^2))) - (sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(4*a*c*sqrt(c - c/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)/(c - c/(a^2*x^2))^(5/2), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^2*sqrt(c - c/(a^2*x^2))) - sqrt(1 - 1/(a^2*x^2))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 - a*x)^2) + sqrt(1 - 1/(a^2*x^2))/(a*c^2*sqrt(c - c/(a^2*x^2))*(1 - a*x)) - sqrt(1 - 1/(a^2*x^2))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (23*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2))) - (7*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)/(c - c/(a^2*x^2))^(7/2), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^3*sqrt(c - c/(a^2*x^2))) + sqrt(1 - 1/(a^2*x^2))/(24*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)^3) - (11*sqrt(1 - 1/(a^2*x^2)))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)^2) + (3*sqrt(1 - 1/(a^2*x^2)))/(2*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)) + sqrt(1 - 1/(a^2*x^2))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)^2) - (5*sqrt(1 - 1/(a^2*x^2)))/(16*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (51*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(32*a*c^3*sqrt(c - c/(a^2*x^2))) - (19*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(32*a*c^3*sqrt(c - c/(a^2*x^2)))]

    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^(7/2), x, 15, (11*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(30*(1 - a*x)^3) - (57*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3) + (41*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(24*(1 - a*x)^3*(1 + a*x)^2) + (57*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(80*(1 - a*x)^3*(1 + a*x)) - (13*a^2*(c - c/(a^2*x^2))^(7/2)*x^3*(1 + a*x))/(40*(1 - a*x)^3) + (a*(c - c/(a^2*x^2))^(7/2)*x^2*(1 + a*x))/(15*(1 - a*x)^2) + ((c - c/(a^2*x^2))^(7/2)*x*(1 + a*x))/(6*(1 - a*x)) + (2*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*asin(a*x))/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) + (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^(5/2), x, 13, -((5*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(8*(1 - a*x)^2)) + (25*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1 + a*x)^2) - (17*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/(12*(1 - a*x)^2*(1 + a*x)) + (a*(c - c/(a^2*x^2))^(5/2)*x^2*(1 + a*x))/(6*(1 - a*x)^2) + ((c - c/(a^2*x^2))^(5/2)*x*(1 + a*x))/(4*(1 - a*x)) - (2*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*asin(a*x))/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) - (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))]
    @test_int [ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^(3/2), x, 11, (a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 - a*x) - (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(3/2)*x*(1 + a*x))/(2*(1 - a*x)) + (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*asin(a*x))/((1 - a*x)^(3/2)*(1 + a*x)^(3/2)) + (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2)), x, 9, sqrt(c - c/(a^2*x^2))*x - (2*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(sqrt(1 - a*x)*sqrt(1 + a*x)) + (sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [ℯ^(2*acoth(a*x))/sqrt(c - c/(a^2*x^2)), x, 7, -((2*(1 - a*x)*(1 + a*x))/(a^2*sqrt(c - c/(a^2*x^2))*x)) - (1 + a*x)^2/(a^2*sqrt(c - c/(a^2*x^2))*x) + (2*sqrt(1 - a*x)*sqrt(1 + a*x)*asin(a*x))/(a^2*sqrt(c - c/(a^2*x^2))*x)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a^2*x^2))^(3/2), x, 7, -((1 + a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(3/2)*x)) + (2*(5 - 2*a*x)*(1 - a*x)*(1 + a*x)^2)/(3*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) - (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*asin(a*x))/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a^2*x^2))^(5/2), x, 9, -((1 + a*x)^2/(5*a^2*(c - c/(a^2*x^2))^(5/2)*x)) + (2*(1 - a*x)*(1 + a*x)^2)/(3*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) - (58*(1 - a*x)^2*(1 + a*x)^2)/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) - (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 43*a*x))/(15*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) + (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*asin(a*x))/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)]
    @test_int [ℯ^(2*acoth(a*x))/(c - c/(a^2*x^2))^(7/2), x, 11, -((1 + a*x)^2/(7*a^2*(c - c/(a^2*x^2))^(7/2)*x)) + (2*(1 - a*x)*(1 + a*x)^2)/(5*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) - (124*(1 - a*x)^2*(1 + a*x)^2)/(105*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) + (782*(1 - a*x)^3*(1 + a*x)^2)/(105*a^5*(c - c/(a^2*x^2))^(7/2)*x^4) + (142*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) + (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 107*a*x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) - (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*asin(a*x))/(a^8*(c - c/(a^2*x^2))^(7/2)*x^7)]

    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^(9/2), x, 4, (c^4*sqrt(c - c/(a^2*x^2)))/(8*a^9*sqrt(1 - 1/(a^2*x^2))*x^8) + (3*c^4*sqrt(c - c/(a^2*x^2)))/(7*a^8*sqrt(1 - 1/(a^2*x^2))*x^7) - (8*c^4*sqrt(c - c/(a^2*x^2)))/(5*a^6*sqrt(1 - 1/(a^2*x^2))*x^5) - (3*c^4*sqrt(c - c/(a^2*x^2)))/(2*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) + (2*c^4*sqrt(c - c/(a^2*x^2)))/(a^4*sqrt(1 - 1/(a^2*x^2))*x^3) + (4*c^4*sqrt(c - c/(a^2*x^2)))/(a^3*sqrt(1 - 1/(a^2*x^2))*x^2) + (c^4*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (3*c^4*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^(7/2), x, 4, -((c^3*sqrt(c - c/(a^2*x^2)))/(6*a^7*sqrt(1 - 1/(a^2*x^2))*x^6)) - (3*c^3*sqrt(c - c/(a^2*x^2)))/(5*a^6*sqrt(1 - 1/(a^2*x^2))*x^5) - (c^3*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) + (5*c^3*sqrt(c - c/(a^2*x^2)))/(3*a^4*sqrt(1 - 1/(a^2*x^2))*x^3) + (5*c^3*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2) - (c^3*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^3*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (3*c^3*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^(5/2), x, 4, (c^2*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) + (c^2*sqrt(c - c/(a^2*x^2)))/(a^4*sqrt(1 - 1/(a^2*x^2))*x^3) + (c^2*sqrt(c - c/(a^2*x^2)))/(a^3*sqrt(1 - 1/(a^2*x^2))*x^2) - (2*c^2*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^2*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (3*c^2*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^(3/2), x, 4, -((c*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2)) - (3*c*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (3*c*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2)), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))/sqrt(c - c/(a^2*x^2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a^2*x^2)) + (2*sqrt(1 - 1/(a^2*x^2)))/(a*sqrt(c - c/(a^2*x^2))*(1 - a*x)) + (3*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(a*sqrt(c - c/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a^2*x^2))^(3/2), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c*sqrt(c - c/(a^2*x^2))) - sqrt(1 - 1/(a^2*x^2))/(2*a*c*sqrt(c - c/(a^2*x^2))*(1 - a*x)^2) + (3*sqrt(1 - 1/(a^2*x^2)))/(a*c*sqrt(c - c/(a^2*x^2))*(1 - a*x)) + (3*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(a*c*sqrt(c - c/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a^2*x^2))^(5/2), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^2*sqrt(c - c/(a^2*x^2))) + sqrt(1 - 1/(a^2*x^2))/(6*a*c^2*sqrt(c - c/(a^2*x^2))*(1 - a*x)^3) - (9*sqrt(1 - 1/(a^2*x^2)))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 - a*x)^2) + (31*sqrt(1 - 1/(a^2*x^2)))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 - a*x)) + (49*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2))) - (sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))/(c - c/(a^2*x^2))^(7/2), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^3*sqrt(c - c/(a^2*x^2))) - sqrt(1 - 1/(a^2*x^2))/(16*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)^4) + sqrt(1 - 1/(a^2*x^2))/(2*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)^3) - (59*sqrt(1 - 1/(a^2*x^2)))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)^2) + (75*sqrt(1 - 1/(a^2*x^2)))/(16*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)) - sqrt(1 - 1/(a^2*x^2))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (201*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(64*a*c^3*sqrt(c - c/(a^2*x^2))) - (9*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(64*a*c^3*sqrt(c - c/(a^2*x^2)))]

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

    @test_int [(c - c/(a^2*x^2))^(7/2)/ℯ^acoth(a*x), x, 4, -((c^3*sqrt(c - c/(a^2*x^2)))/(6*a^7*sqrt(1 - 1/(a^2*x^2))*x^6)) + (c^3*sqrt(c - c/(a^2*x^2)))/(5*a^6*sqrt(1 - 1/(a^2*x^2))*x^5) + (3*c^3*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) - (c^3*sqrt(c - c/(a^2*x^2)))/(a^4*sqrt(1 - 1/(a^2*x^2))*x^3) - (3*c^3*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2) + (3*c^3*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^3*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (c^3*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(c - c/(a^2*x^2))^(5/2)/ℯ^acoth(a*x), x, 4, (c^2*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) - (c^2*sqrt(c - c/(a^2*x^2)))/(3*a^4*sqrt(1 - 1/(a^2*x^2))*x^3) - (c^2*sqrt(c - c/(a^2*x^2)))/(a^3*sqrt(1 - 1/(a^2*x^2))*x^2) + (2*c^2*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^2*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (c^2*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(c - c/(a^2*x^2))^(3/2)/ℯ^acoth(a*x), x, 4, -((c*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2)) + (c*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (c*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a^2*x^2))/ℯ^acoth(a*x), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2))), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a^2*x^2)) - (sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(a*sqrt(c - c/(a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a^2*x^2))^(3/2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c*sqrt(c - c/(a^2*x^2))) - sqrt(1 - 1/(a^2*x^2))/(2*a*c*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(4*a*c*sqrt(c - c/(a^2*x^2))) - (5*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(4*a*c*sqrt(c - c/(a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a^2*x^2))^(5/2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^2*sqrt(c - c/(a^2*x^2))) + sqrt(1 - 1/(a^2*x^2))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 - a*x)) + sqrt(1 - 1/(a^2*x^2))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 + a*x)^2) - sqrt(1 - 1/(a^2*x^2))/(a*c^2*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (7*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2))) - (23*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2)))]
    @test_int [1/(ℯ^acoth(a*x)*(c - c/(a^2*x^2))^(7/2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^3*sqrt(c - c/(a^2*x^2))) - sqrt(1 - 1/(a^2*x^2))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)^2) + (5*sqrt(1 - 1/(a^2*x^2)))/(16*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)) - sqrt(1 - 1/(a^2*x^2))/(24*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)^3) + (11*sqrt(1 - 1/(a^2*x^2)))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)^2) - (3*sqrt(1 - 1/(a^2*x^2)))/(2*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (19*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(32*a*c^3*sqrt(c - c/(a^2*x^2))) - (51*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(32*a*c^3*sqrt(c - c/(a^2*x^2)))]

    @test_int [(c - c/(a^2*x^2))^(7/2)/ℯ^(2*acoth(a*x)), x, 15, (7*a^6*(c - c/(a^2*x^2))^(7/2)*x^7)/(16*(1 - a*x)^3*(1 + a*x)^3) + (3*a^5*(c - c/(a^2*x^2))^(7/2)*x^6)/(8*(1 - a*x)^3*(1 + a*x)^2) - (a*(c - c/(a^2*x^2))^(7/2)*x^2)/(15*(1 + a*x)) - (19*a^4*(c - c/(a^2*x^2))^(7/2)*x^5)/(16*(1 - a*x)^3*(1 + a*x)) + (2*a^3*(c - c/(a^2*x^2))^(7/2)*x^4)/(3*(1 - a*x)^2*(1 + a*x)) - (23*a^2*(c - c/(a^2*x^2))^(7/2)*x^3)/(120*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(7/2)*x*(1 - a*x))/(6*(1 + a*x)) - (2*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*asin(a*x))/((1 - a*x)^(7/2)*(1 + a*x)^(7/2)) + (25*a^6*(c - c/(a^2*x^2))^(7/2)*x^7*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(16*(1 - a*x)^(7/2)*(1 + a*x)^(7/2))]
    @test_int [(c - c/(a^2*x^2))^(5/2)/ℯ^(2*acoth(a*x)), x, 13, -((7*a^4*(c - c/(a^2*x^2))^(5/2)*x^5)/(8*(1 - a*x)^2*(1 + a*x)^2)) - (a*(c - c/(a^2*x^2))^(5/2)*x^2)/(6*(1 + a*x)) + (2*a^3*(c - c/(a^2*x^2))^(5/2)*x^4)/((1 - a*x)^2*(1 + a*x)) - (7*a^2*(c - c/(a^2*x^2))^(5/2)*x^3)/(24*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(5/2)*x*(1 - a*x))/(4*(1 + a*x)) + (2*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*asin(a*x))/((1 - a*x)^(5/2)*(1 + a*x)^(5/2)) - (9*a^4*(c - c/(a^2*x^2))^(5/2)*x^5*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(8*(1 - a*x)^(5/2)*(1 + a*x)^(5/2))]
    @test_int [(c - c/(a^2*x^2))^(3/2)/ℯ^(2*acoth(a*x)), x, 11, -((a*(c - c/(a^2*x^2))^(3/2)*x^2)/(1 + a*x)) - (5*a^2*(c - c/(a^2*x^2))^(3/2)*x^3)/(2*(1 - a*x)*(1 + a*x)) + ((c - c/(a^2*x^2))^(3/2)*x*(1 - a*x))/(2*(1 + a*x)) - (2*a^2*(c - c/(a^2*x^2))^(3/2)*x^3*asin(a*x))/((1 - a*x)^(3/2)*(1 + a*x)^(3/2)) + (a^2*(c - c/(a^2*x^2))^(3/2)*x^3*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2))]
    @test_int [sqrt(c - c/(a^2*x^2))/ℯ^(2*acoth(a*x)), x, 9, sqrt(c - c/(a^2*x^2))*x + (2*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(sqrt(1 - a*x)*sqrt(1 + a*x)) + (sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [1/(ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2))), x, 7, -((1 - a*x)^2/(a^2*sqrt(c - c/(a^2*x^2))*x)) - (2*(1 - a*x)*(1 + a*x))/(a^2*sqrt(c - c/(a^2*x^2))*x) - (2*sqrt(1 - a*x)*sqrt(1 + a*x)*asin(a*x))/(a^2*sqrt(c - c/(a^2*x^2))*x)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^(3/2)), x, 7, -((1 - a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(3/2)*x)) + (2*(1 - a*x)^2*(1 + a*x)*(5 + 2*a*x))/(3*a^4*(c - c/(a^2*x^2))^(3/2)*x^3) + (2*(1 - a*x)^(3/2)*(1 + a*x)^(3/2)*asin(a*x))/(a^4*(c - c/(a^2*x^2))^(3/2)*x^3)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^(5/2)), x, 9, -((1 - a*x)^2/(a^2*(c - c/(a^2*x^2))^(5/2)*x)) - (2*(1 - a*x)^3)/(5*a^3*(c - c/(a^2*x^2))^(5/2)*x^2) + (2*(1 - a*x)^3*(1 + a*x))/(15*a^4*(c - c/(a^2*x^2))^(5/2)*x^3) - (2*(1 - a*x)^3*(1 + a*x)^2*(28 + 13*a*x))/(15*a^6*(c - c/(a^2*x^2))^(5/2)*x^5) - (2*(1 - a*x)^(5/2)*(1 + a*x)^(5/2)*asin(a*x))/(a^6*(c - c/(a^2*x^2))^(5/2)*x^5)]
    @test_int [1/(ℯ^(2*acoth(a*x))*(c - c/(a^2*x^2))^(7/2)), x, 11, -((1 - a*x)^2/(3*a^2*(c - c/(a^2*x^2))^(7/2)*x)) + (10*(1 - a*x)^3)/(3*a^3*(c - c/(a^2*x^2))^(7/2)*x^2) + (12*(1 - a*x)^4)/(7*a^4*(c - c/(a^2*x^2))^(7/2)*x^3) + (82*(1 - a*x)^4*(1 + a*x))/(105*a^5*(c - c/(a^2*x^2))^(7/2)*x^4) + (2*(1 - a*x)^4*(1 + a*x)^2)/(35*a^6*(c - c/(a^2*x^2))^(7/2)*x^5) + (2*(1 - a*x)^4*(1 + a*x)^3*(72 + 37*a*x))/(35*a^8*(c - c/(a^2*x^2))^(7/2)*x^7) + (2*(1 - a*x)^(7/2)*(1 + a*x)^(7/2)*asin(a*x))/(a^8*(c - c/(a^2*x^2))^(7/2)*x^7)]

    @test_int [(c - c/(a^2*x^2))^(9/2)/ℯ^(3*acoth(a*x)), x, 4, -((c^4*sqrt(c - c/(a^2*x^2)))/(8*a^9*sqrt(1 - 1/(a^2*x^2))*x^8)) + (3*c^4*sqrt(c - c/(a^2*x^2)))/(7*a^8*sqrt(1 - 1/(a^2*x^2))*x^7) - (8*c^4*sqrt(c - c/(a^2*x^2)))/(5*a^6*sqrt(1 - 1/(a^2*x^2))*x^5) + (3*c^4*sqrt(c - c/(a^2*x^2)))/(2*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) + (2*c^4*sqrt(c - c/(a^2*x^2)))/(a^4*sqrt(1 - 1/(a^2*x^2))*x^3) - (4*c^4*sqrt(c - c/(a^2*x^2)))/(a^3*sqrt(1 - 1/(a^2*x^2))*x^2) + (c^4*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (3*c^4*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(c - c/(a^2*x^2))^(7/2)/ℯ^(3*acoth(a*x)), x, 4, (c^3*sqrt(c - c/(a^2*x^2)))/(6*a^7*sqrt(1 - 1/(a^2*x^2))*x^6) - (3*c^3*sqrt(c - c/(a^2*x^2)))/(5*a^6*sqrt(1 - 1/(a^2*x^2))*x^5) + (c^3*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4) + (5*c^3*sqrt(c - c/(a^2*x^2)))/(3*a^4*sqrt(1 - 1/(a^2*x^2))*x^3) - (5*c^3*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2) - (c^3*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^3*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (3*c^3*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(c - c/(a^2*x^2))^(5/2)/ℯ^(3*acoth(a*x)), x, 4, -((c^2*sqrt(c - c/(a^2*x^2)))/(4*a^5*sqrt(1 - 1/(a^2*x^2))*x^4)) + (c^2*sqrt(c - c/(a^2*x^2)))/(a^4*sqrt(1 - 1/(a^2*x^2))*x^3) - (c^2*sqrt(c - c/(a^2*x^2)))/(a^3*sqrt(1 - 1/(a^2*x^2))*x^2) - (2*c^2*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c^2*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (3*c^2*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(c - c/(a^2*x^2))^(3/2)/ℯ^(3*acoth(a*x)), x, 4, (c*sqrt(c - c/(a^2*x^2)))/(2*a^3*sqrt(1 - 1/(a^2*x^2))*x^2) - (3*c*sqrt(c - c/(a^2*x^2)))/(a^2*sqrt(1 - 1/(a^2*x^2))*x) + (c*sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (3*c*sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a^2*x^2))/ℯ^(3*acoth(a*x)), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2))) - (4*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2))), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/sqrt(c - c/(a^2*x^2)) - (2*sqrt(1 - 1/(a^2*x^2)))/(a*sqrt(c - c/(a^2*x^2))*(1 + a*x)) - (3*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(a*sqrt(c - c/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^(3/2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c*sqrt(c - c/(a^2*x^2))) + sqrt(1 - 1/(a^2*x^2))/(2*a*c*sqrt(c - c/(a^2*x^2))*(1 + a*x)^2) - (3*sqrt(1 - 1/(a^2*x^2)))/(a*c*sqrt(c - c/(a^2*x^2))*(1 + a*x)) - (3*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(a*c*sqrt(c - c/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^(5/2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^2*sqrt(c - c/(a^2*x^2))) - sqrt(1 - 1/(a^2*x^2))/(6*a*c^2*sqrt(c - c/(a^2*x^2))*(1 + a*x)^3) + (9*sqrt(1 - 1/(a^2*x^2)))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 + a*x)^2) - (31*sqrt(1 - 1/(a^2*x^2)))/(8*a*c^2*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2))) - (49*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(16*a*c^2*sqrt(c - c/(a^2*x^2)))]
    @test_int [1/(ℯ^(3*acoth(a*x))*(c - c/(a^2*x^2))^(7/2)), x, 4, (sqrt(1 - 1/(a^2*x^2))*x)/(c^3*sqrt(c - c/(a^2*x^2))) + sqrt(1 - 1/(a^2*x^2))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 - a*x)) + sqrt(1 - 1/(a^2*x^2))/(16*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)^4) - sqrt(1 - 1/(a^2*x^2))/(2*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)^3) + (59*sqrt(1 - 1/(a^2*x^2)))/(32*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)^2) - (75*sqrt(1 - 1/(a^2*x^2)))/(16*a*c^3*sqrt(c - c/(a^2*x^2))*(1 + a*x)) + (9*sqrt(1 - 1/(a^2*x^2))*log(1 - a*x))/(64*a*c^3*sqrt(c - c/(a^2*x^2))) - (201*sqrt(1 - 1/(a^2*x^2))*log(1 + a*x))/(64*a*c^3*sqrt(c - c/(a^2*x^2)))]

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

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

    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2))*x^m, x, 4, (sqrt(c - c/(a^2*x^2))*x^m)/(a*m*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^(1 + m))/((1 + m)*sqrt(1 - 1/(a^2*x^2)))]

    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2))*x^2, x, 4, (sqrt(c - c/(a^2*x^2))*x^2)/(2*a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^3)/(3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2))*x, x, 3, (sqrt(c - c/(a^2*x^2))*x)/(a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^2)/(2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2)), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2)))/x, x, 4, -(sqrt(c - c/(a^2*x^2))/(a*sqrt(1 - 1/(a^2*x^2))*x)) + (sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [(ℯ^acoth(a*x)*sqrt(c - c/(a^2*x^2)))/x^2, x, 3, -((sqrt(c - c/(a^2*x^2))*(1 + a*x)^2)/(2*a*sqrt(1 - 1/(a^2*x^2))*x^2))]

    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2))*x^3, x, 9, (7*sqrt(c - c/(a^2*x^2))*x)/(8*a^3) + (7*sqrt(c - c/(a^2*x^2))*x*(1 + a*x))/(24*a^3) + (sqrt(c - c/(a^2*x^2))*x*(1 + a*x)^2)/(6*a^3) + (sqrt(c - c/(a^2*x^2))*x^2*(1 + a*x)^2)/(4*a^2) - (7*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(8*a^3*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2))*x^2, x, 8, (sqrt(c - c/(a^2*x^2))*x)/a^2 + (sqrt(c - c/(a^2*x^2))*x*(1 + a*x))/(3*a^2) + (sqrt(c - c/(a^2*x^2))*x*(1 + a*x)^2)/(3*a^2) - (sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(a^2*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2))*x, x, 7, (3*sqrt(c - c/(a^2*x^2))*x)/(2*a) + (sqrt(c - c/(a^2*x^2))*x*(1 + a*x))/(2*a) - (3*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(2*a*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2)), x, 9, sqrt(c - c/(a^2*x^2))*x - (2*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(sqrt(1 - a*x)*sqrt(1 + a*x)) + (sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x, x, 9, sqrt(c - c/(a^2*x^2)) - (a*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(sqrt(1 - a*x)*sqrt(1 + a*x)) + (2*a*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^2, x, 7, (3/2)*a*sqrt(c - c/(a^2*x^2)) + (sqrt(c - c/(a^2*x^2))*(1 + a*x))/(2*x) + (3*a^2*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(2*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^3, x, 8, a^2*sqrt(c - c/(a^2*x^2)) + (a*sqrt(c - c/(a^2*x^2))*(1 + a*x))/(3*x) + (sqrt(c - c/(a^2*x^2))*(1 + a*x)^2)/(3*x^2) + (a^3*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^4, x, 10, (4/3)*a^3*sqrt(c - c/(a^2*x^2)) + sqrt(c - c/(a^2*x^2))/(4*x^3) + (2*a*sqrt(c - c/(a^2*x^2)))/(3*x^2) + (7*a^2*sqrt(c - c/(a^2*x^2)))/(8*x) + (7*a^4*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(8*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(ℯ^(2*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^5, x, 11, (6/5)*a^4*sqrt(c - c/(a^2*x^2)) + sqrt(c - c/(a^2*x^2))/(5*x^4) + (a*sqrt(c - c/(a^2*x^2)))/(2*x^3) + (3*a^2*sqrt(c - c/(a^2*x^2)))/(5*x^2) + (3*a^3*sqrt(c - c/(a^2*x^2)))/(4*x) + (3*a^5*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(4*sqrt(1 - a*x)*sqrt(1 + a*x))]

    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2))*x^3, x, 4, (4*sqrt(c - c/(a^2*x^2))*x)/(a^3*sqrt(1 - 1/(a^2*x^2))) + (2*sqrt(c - c/(a^2*x^2))*x^2)/(a^2*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^3)/(a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^4)/(4*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/(a^4*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2))*x^2, x, 4, (4*sqrt(c - c/(a^2*x^2))*x)/(a^2*sqrt(1 - 1/(a^2*x^2))) + (3*sqrt(c - c/(a^2*x^2))*x^2)/(2*a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^3)/(3*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/(a^3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2))*x, x, 4, (3*sqrt(c - c/(a^2*x^2))*x)/(a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^2)/(2*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/(a^2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2)), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x, x, 4, sqrt(c - c/(a^2*x^2))/(a*sqrt(1 - 1/(a^2*x^2))*x) - (3*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^2, x, 4, sqrt(c - c/(a^2*x^2))/(2*a*sqrt(1 - 1/(a^2*x^2))*x^2) + (3*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*a*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^3, x, 4, sqrt(c - c/(a^2*x^2))/(3*a*sqrt(1 - 1/(a^2*x^2))*x^3) + (3*sqrt(c - c/(a^2*x^2)))/(2*sqrt(1 - 1/(a^2*x^2))*x^2) + (4*a*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a^2*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*a^2*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^4, x, 4, sqrt(c - c/(a^2*x^2))/(4*a*sqrt(1 - 1/(a^2*x^2))*x^4) + sqrt(c - c/(a^2*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^3) + (2*a*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x^2) + (4*a^2*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a^3*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*a^3*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [(ℯ^(3*acoth(a*x))*sqrt(c - c/(a^2*x^2)))/x^5, x, 4, sqrt(c - c/(a^2*x^2))/(5*a*sqrt(1 - 1/(a^2*x^2))*x^5) + (3*sqrt(c - c/(a^2*x^2)))/(4*sqrt(1 - 1/(a^2*x^2))*x^4) + (4*a*sqrt(c - c/(a^2*x^2)))/(3*sqrt(1 - 1/(a^2*x^2))*x^3) + (2*a^2*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x^2) + (4*a^3*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a^4*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*a^4*sqrt(c - c/(a^2*x^2))*log(1 - a*x))/sqrt(1 - 1/(a^2*x^2))]

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

    @test_int [(sqrt(c - c/(a^2*x^2))*x^m)/ℯ^acoth(a*x), x, 4, -((sqrt(c - c/(a^2*x^2))*x^m)/(a*m*sqrt(1 - 1/(a^2*x^2)))) + (sqrt(c - c/(a^2*x^2))*x^(1 + m))/((1 + m)*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(sqrt(c - c/(a^2*x^2))*x^2)/ℯ^acoth(a*x), x, 4, -(sqrt(c - c/(a^2*x^2))*x^2)/(2*a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^3)/(3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(sqrt(c - c/(a^2*x^2))*x)/ℯ^acoth(a*x), x, 3, -((sqrt(c - c/(a^2*x^2))*x)/(a*sqrt(1 - 1/(a^2*x^2)))) + (sqrt(c - c/(a^2*x^2))*x^2)/(2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a^2*x^2))/ℯ^acoth(a*x), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) - (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^acoth(a*x)*x), x, 4, sqrt(c - c/(a^2*x^2))/(a*sqrt(1 - 1/(a^2*x^2))*x) + (sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^acoth(a*x)*x^2), x, 3, (sqrt(c - c/(a^2*x^2))*(1 - a*x)^2)/(2*a*sqrt(1 - 1/(a^2*x^2))*x^2)]

    @test_int [(sqrt(c - c/(a^2*x^2))*x^3)/ℯ^(2*acoth(a*x)), x, 9, -((7*sqrt(c - c/(a^2*x^2))*x)/(8*a^3)) - (7*sqrt(c - c/(a^2*x^2))*x*(1 - a*x))/(24*a^3) - (sqrt(c - c/(a^2*x^2))*x*(1 - a*x)^2)/(6*a^3) + (sqrt(c - c/(a^2*x^2))*x^2*(1 - a*x)^2)/(4*a^2) - (7*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(8*a^3*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(sqrt(c - c/(a^2*x^2))*x^2)/ℯ^(2*acoth(a*x)), x, 8, (sqrt(c - c/(a^2*x^2))*x)/a^2 + (sqrt(c - c/(a^2*x^2))*x*(1 - a*x))/(3*a^2) + (sqrt(c - c/(a^2*x^2))*x*(1 - a*x)^2)/(3*a^2) + (sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(a^2*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [(sqrt(c - c/(a^2*x^2))*x)/ℯ^(2*acoth(a*x)), x, 7, -((3*sqrt(c - c/(a^2*x^2))*x)/(2*a)) - (sqrt(c - c/(a^2*x^2))*x*(1 - a*x))/(2*a) - (3*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(2*a*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [sqrt(c - c/(a^2*x^2))/ℯ^(2*acoth(a*x)), x, 9, sqrt(c - c/(a^2*x^2))*x + (2*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(sqrt(1 - a*x)*sqrt(1 + a*x)) + (sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(2*acoth(a*x))*x), x, 9, sqrt(c - c/(a^2*x^2)) - (a*sqrt(c - c/(a^2*x^2))*x*asin(a*x))/(sqrt(1 - a*x)*sqrt(1 + a*x)) - (2*a*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(2*acoth(a*x))*x^2), x, 7, (-(3/2))*a*sqrt(c - c/(a^2*x^2)) + (sqrt(c - c/(a^2*x^2))*(1 - a*x))/(2*x) + (3*a^2*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(2*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(2*acoth(a*x))*x^3), x, 8, a^2*sqrt(c - c/(a^2*x^2)) - (a*sqrt(c - c/(a^2*x^2))*(1 - a*x))/(3*x) + (sqrt(c - c/(a^2*x^2))*(1 - a*x)^2)/(3*x^2) - (a^3*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(2*acoth(a*x))*x^4), x, 10, (-(4/3))*a^3*sqrt(c - c/(a^2*x^2)) + sqrt(c - c/(a^2*x^2))/(4*x^3) - (2*a*sqrt(c - c/(a^2*x^2)))/(3*x^2) + (7*a^2*sqrt(c - c/(a^2*x^2)))/(8*x) + (7*a^4*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(8*sqrt(1 - a*x)*sqrt(1 + a*x))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(2*acoth(a*x))*x^5), x, 11, (6/5)*a^4*sqrt(c - c/(a^2*x^2)) + sqrt(c - c/(a^2*x^2))/(5*x^4) - (a*sqrt(c - c/(a^2*x^2)))/(2*x^3) + (3*a^2*sqrt(c - c/(a^2*x^2)))/(5*x^2) - (3*a^3*sqrt(c - c/(a^2*x^2)))/(4*x) - (3*a^5*sqrt(c - c/(a^2*x^2))*x*atanh(sqrt(1 - a*x)*sqrt(1 + a*x)))/(4*sqrt(1 - a*x)*sqrt(1 + a*x))]

    @test_int [(sqrt(c - c/(a^2*x^2))*x^3)/ℯ^(3*acoth(a*x)), x, 4, (-4*sqrt(c - c/(a^2*x^2))*x)/(a^3*sqrt(1 - 1/(a^2*x^2))) + (2*sqrt(c - c/(a^2*x^2))*x^2)/(a^2*sqrt(1 - 1/(a^2*x^2))) - (sqrt(c - c/(a^2*x^2))*x^3)/(a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^4)/(4*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/(a^4*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(sqrt(c - c/(a^2*x^2))*x^2)/ℯ^(3*acoth(a*x)), x, 4, (4*sqrt(c - c/(a^2*x^2))*x)/(a^2*sqrt(1 - 1/(a^2*x^2))) - (3*sqrt(c - c/(a^2*x^2))*x^2)/(2*a*sqrt(1 - 1/(a^2*x^2))) + (sqrt(c - c/(a^2*x^2))*x^3)/(3*sqrt(1 - 1/(a^2*x^2))) - (4*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/(a^3*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [(sqrt(c - c/(a^2*x^2))*x)/ℯ^(3*acoth(a*x)), x, 4, -((3*sqrt(c - c/(a^2*x^2))*x)/(a*sqrt(1 - 1/(a^2*x^2)))) + (sqrt(c - c/(a^2*x^2))*x^2)/(2*sqrt(1 - 1/(a^2*x^2))) + (4*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/(a^2*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a^2*x^2))/ℯ^(3*acoth(a*x)), x, 4, (sqrt(c - c/(a^2*x^2))*x)/sqrt(1 - 1/(a^2*x^2)) + (sqrt(c - c/(a^2*x^2))*log(x))/(a*sqrt(1 - 1/(a^2*x^2))) - (4*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/(a*sqrt(1 - 1/(a^2*x^2)))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(3*acoth(a*x))*x), x, 4, -(sqrt(c - c/(a^2*x^2))/(a*sqrt(1 - 1/(a^2*x^2))*x)) - (3*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(3*acoth(a*x))*x^2), x, 4, -(sqrt(c - c/(a^2*x^2))/(2*a*sqrt(1 - 1/(a^2*x^2))*x^2)) + (3*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*a*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) - (4*a*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(3*acoth(a*x))*x^3), x, 4, -(sqrt(c - c/(a^2*x^2))/(3*a*sqrt(1 - 1/(a^2*x^2))*x^3)) + (3*sqrt(c - c/(a^2*x^2)))/(2*sqrt(1 - 1/(a^2*x^2))*x^2) - (4*a*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a^2*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*a^2*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(3*acoth(a*x))*x^4), x, 4, -(sqrt(c - c/(a^2*x^2))/(4*a*sqrt(1 - 1/(a^2*x^2))*x^4)) + sqrt(c - c/(a^2*x^2))/(sqrt(1 - 1/(a^2*x^2))*x^3) - (2*a*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x^2) + (4*a^2*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) + (4*a^3*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) - (4*a^3*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/sqrt(1 - 1/(a^2*x^2))]
    @test_int [sqrt(c - c/(a^2*x^2))/(ℯ^(3*acoth(a*x))*x^5), x, 4, -(sqrt(c - c/(a^2*x^2))/(5*a*sqrt(1 - 1/(a^2*x^2))*x^5)) + (3*sqrt(c - c/(a^2*x^2)))/(4*sqrt(1 - 1/(a^2*x^2))*x^4) - (4*a*sqrt(c - c/(a^2*x^2)))/(3*sqrt(1 - 1/(a^2*x^2))*x^3) + (2*a^2*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x^2) - (4*a^3*sqrt(c - c/(a^2*x^2)))/(sqrt(1 - 1/(a^2*x^2))*x) - (4*a^4*sqrt(c - c/(a^2*x^2))*log(x))/sqrt(1 - 1/(a^2*x^2)) + (4*a^4*sqrt(c - c/(a^2*x^2))*log(1 + a*x))/sqrt(1 - 1/(a^2*x^2))]

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a^2*x^2))^p*with*n*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*(c - c/(a^2*x^2)), x, 4, (4*c*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*HypergeometricFunctions._₂F₁(2, 1 - n/2, 2 - n/2, (a - 1/x)/(a + 1/x)))/(a*(2 - n)) - (2^(1 + n/2)*c*(1 - 1/(a*x))^(1 - n/2)*HypergeometricFunctions._₂F₁(1 - n/2, -(n/2), 2 - n/2, (a - 1/x)/(2*a)))/(a*(2 - n))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c/(a^2*x^2)), x, 5, -(((1 + n)*(1 + 1/(a*x))^(n/2))/((1 - 1/(a*x))^(n/2)*(a*c*n))) + ((1 + 1/(a*x))^(n/2)*x)/((1 - 1/(a*x))^(n/2)*c) + (2*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)))/((1 - 1/(a*x))^(n/2)*(a*c))]
    @test_int [ℯ^(n*acoth(a*x))/(c - c/(a^2*x^2))^2, x, 7, If(13>=8, -(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*(2 + n))) + ((6 + 4*n - n^2 - n^3)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*(2 - n)*n*(2 + n)) - ((6 + 4*n + n^2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/((1 - 1/(a*x))^(n/2)*(a*c^2*n*(2 + n))) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*x)/c^2 + (2*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)))/((1 - 1/(a*x))^(n/2)*(a*c^2)), -(((3 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*(2 + n))) + ((6 + 4*n - n^2 - n^3)*(1 - 1/(a*x))^(1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/(a*c^2*n*(4 - n^2)) - ((6 + 4*n + n^2)*(1 + 1/(a*x))^((1/2)*(-2 + n)))/((1 - 1/(a*x))^(n/2)*(a*c^2*n*(2 + n))) + ((1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((1/2)*(-2 + n))*x)/c^2 + (2*(1 + 1/(a*x))^(n/2)*HypergeometricFunctions._₂F₁(1, n/2, (2 + n)/2, (a + 1/x)/(a - 1/x)))/((1 - 1/(a*x))^(n/2)*(a*c^2)))]

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

    #= ::Subsection::Closed:: =#
    #=Integrands*of*the*form*ℯ^(n*acoth(a*x))*(c-c/(a^2*x^2))^p*with*p*symbolic=#

    @test_int [ℯ^(n*acoth(a*x))*(c - c/(a^2*x^2))^p, x, 3, -((2^(1 - n/2 + p)*(c - c/(a^2*x^2))^p*(1 + 1/(a*x))^(1 + n/2 + p)*AppellF1(1 + n/2 + p, (1/2)*(n - 2*p), 2, 2 + n/2 + p, (a + 1/x)/(2*a), 1 + 1/(a*x)))/((1 - 1/(a^2*x^2))^p*(a*(2 + n + 2*p))))]

    @test_int [(c - c/(a^2*x^2))^p/ℯ^(2*p*acoth(a*x)), x, 3, ((c - c/(a^2*x^2))^p*(1 - 1/(a*x))^(1 + 2*p)*HypergeometricFunctions._₂F₁(2, 1 + 2*p, 2*(1 + p), 1 - 1/(a*x)))/((1 - 1/(a^2*x^2))^p*(a*(1 + 2*p)))]
    @test_int [ℯ^(2*p*acoth(a*x))*(c - c/(a^2*x^2))^p, x, 3, -(((c - c/(a^2*x^2))^p*(1 + 1/(a*x))^(1 + 2*p)*HypergeometricFunctions._₂F₁(2, 1 + 2*p, 2*(1 + p), 1 + 1/(a*x)))/((1 - 1/(a^2*x^2))^p*(a*(1 + 2*p))))]
    end

max_stars_repo_path: test/7 Inverse hyperbolic functions/7.4 Inverse hyperbolic cotangent/7.4.2 Exponentials of inverse hyperbolic cotangent functions.jl