\(\int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx\) [30]

Optimal result

Integrand size = 20, antiderivative size = 437 \[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=-\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac {3 \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac {\cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Chi}\left (\frac {6 c f}{d}+6 f x\right )}{8 a^3 d}+\frac {\log (c+d x)}{8 a^3 d}+\frac {\text {Chi}\left (\frac {6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{8 a^3 d}-\frac {3 \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{8 a^3 d}+\frac {3 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{8 a^3 d}+\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^3 d}+\frac {3 \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^3 d}+\frac {\cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{8 a^3 d}-\frac {\sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{8 a^3 d} \]

[Out]

-1/8*Chi(6*c*f/d+6*f*x)*cosh(-6*e+6*c*f/d)/a^3/d+3/8*Chi(4*c*f/d+4*f*x)*cosh(-4*e+4*c*f/d)/a^3/d-3/8*Chi(2*c*f
/d+2*f*x)*cosh(-2*e+2*c*f/d)/a^3/d+1/8*ln(d*x+c)/a^3/d+3/8*cosh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*f*x)/a^3/d-3/8*cos
h(-4*e+4*c*f/d)*Shi(4*c*f/d+4*f*x)/a^3/d+1/8*cosh(-6*e+6*c*f/d)*Shi(6*c*f/d+6*f*x)/a^3/d-1/8*Chi(6*c*f/d+6*f*x
)*sinh(-6*e+6*c*f/d)/a^3/d+1/8*Shi(6*c*f/d+6*f*x)*sinh(-6*e+6*c*f/d)/a^3/d+3/8*Chi(4*c*f/d+4*f*x)*sinh(-4*e+4*
c*f/d)/a^3/d-3/8*Shi(4*c*f/d+4*f*x)*sinh(-4*e+4*c*f/d)/a^3/d-3/8*Chi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/a^3/d+3
/8*Shi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/a^3/d

Rubi [A] (verified)

Time = 1.45 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.00, number of steps used = 53, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3809, 3384, 3379, 3382, 3393, 5556, 5578} \[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=\frac {3 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{8 a^3 d}+\frac {\text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{8 a^3 d}-\frac {3 \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{8 a^3 d}-\frac {3 \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{8 a^3 d}+\frac {3 \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{8 a^3 d}-\frac {\text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \cosh \left (6 e-\frac {6 c f}{d}\right )}{8 a^3 d}-\frac {3 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{8 a^3 d}+\frac {3 \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{8 a^3 d}-\frac {\sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{8 a^3 d}+\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{8 a^3 d}-\frac {3 \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{8 a^3 d}+\frac {\cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{8 a^3 d}+\frac {\log (c+d x)}{8 a^3 d} \]

[In]

Int[1/((c + d*x)*(a + a*Coth[e + f*x])^3),x]

[Out]

(-3*Cosh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(
4*c*f)/d + 4*f*x])/(8*a^3*d) - (Cosh[6*e - (6*c*f)/d]*CoshIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + Log[c + d*x
]/(8*a^3*d) + (CoshIntegral[(6*c*f)/d + 6*f*x]*Sinh[6*e - (6*c*f)/d])/(8*a^3*d) - (3*CoshIntegral[(4*c*f)/d +
4*f*x]*Sinh[4*e - (4*c*f)/d])/(8*a^3*d) + (3*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(8*a^3*d)
+ (3*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) - (3*Sinh[2*e - (2*c*f)/d]*SinhIntegral[
(2*c*f)/d + 2*f*x])/(8*a^3*d) - (3*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) + (3*Sinh[
4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) + (Cosh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d +
6*f*x])/(8*a^3*d) - (Sinh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d)

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3809

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c
+ d*x)^m, (1/(2*a) + Cos[2*e + 2*f*x]/(2*a) + Sin[2*e + 2*f*x]/(2*b))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f
}, x] && EqQ[a^2 + b^2, 0] && ILtQ[m, 0] && ILtQ[n, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5578

Int[((e_.) + (f_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(p_.)*Sinh[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigReduce[(e + f*x)^m, Sinh[a + b*x]^p*Sinh[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ
[p, 0] && IGtQ[q, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{8 a^3 (c+d x)}-\frac {3 \cosh (2 e+2 f x)}{8 a^3 (c+d x)}+\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 (c+d x)}-\frac {\cosh ^3(2 e+2 f x)}{8 a^3 (c+d x)}+\frac {3 \sinh (2 e+2 f x)}{8 a^3 (c+d x)}+\frac {3 \cosh ^2(2 e+2 f x) \sinh (2 e+2 f x)}{8 a^3 (c+d x)}+\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 (c+d x)}+\frac {\sinh ^3(2 e+2 f x)}{8 a^3 (c+d x)}-\frac {3 \sinh (4 e+4 f x)}{8 a^3 (c+d x)}-\frac {3 \sinh (2 e+2 f x) \sinh (4 e+4 f x)}{16 a^3 (c+d x)}\right ) \, dx \\ & = \frac {\log (c+d x)}{8 a^3 d}-\frac {\int \frac {\cosh ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac {\int \frac {\sinh ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac {3 \int \frac {\sinh (2 e+2 f x) \sinh (4 e+4 f x)}{c+d x} \, dx}{16 a^3}-\frac {3 \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac {3 \int \frac {\cosh ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac {3 \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac {3 \int \frac {\cosh ^2(2 e+2 f x) \sinh (2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac {3 \int \frac {\sinh ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac {3 \int \frac {\sinh (4 e+4 f x)}{c+d x} \, dx}{8 a^3} \\ & = \frac {\log (c+d x)}{8 a^3 d}+\frac {i \int \left (\frac {3 i \sinh (2 e+2 f x)}{4 (c+d x)}-\frac {i \sinh (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac {\int \left (\frac {3 \cosh (2 e+2 f x)}{4 (c+d x)}+\frac {\cosh (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac {3 \int \left (-\frac {\cosh (2 e+2 f x)}{2 (c+d x)}+\frac {\cosh (6 e+6 f x)}{2 (c+d x)}\right ) \, dx}{16 a^3}-\frac {3 \int \left (\frac {1}{2 (c+d x)}-\frac {\cosh (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}+\frac {3 \int \left (\frac {1}{2 (c+d x)}+\frac {\cosh (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}+\frac {3 \int \left (\frac {\sinh (2 e+2 f x)}{4 (c+d x)}+\frac {\sinh (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac {\left (3 \cosh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^3}-\frac {\left (3 \cosh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}+\frac {\left (3 \cosh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}-\frac {\left (3 \sinh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^3}+\frac {\left (3 \sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}-\frac {\left (3 \sinh \left (2 e-\frac {2 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3} \\ & = -\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac {\log (c+d x)}{8 a^3 d}-\frac {3 \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{8 a^3 d}+\frac {3 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{8 a^3 d}+\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac {\int \frac {\cosh (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+\frac {\int \frac {\sinh (6 e+6 f x)}{c+d x} \, dx}{32 a^3}-\frac {3 \int \frac {\cosh (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+\frac {3 \int \frac {\sinh (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+2 \frac {3 \int \frac {\cosh (4 e+4 f x)}{c+d x} \, dx}{16 a^3} \\ & = -\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac {\log (c+d x)}{8 a^3 d}-\frac {3 \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{8 a^3 d}+\frac {3 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{8 a^3 d}+\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac {\cosh \left (6 e-\frac {6 c f}{d}\right ) \int \frac {\cosh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac {\cosh \left (6 e-\frac {6 c f}{d}\right ) \int \frac {\sinh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac {\left (3 \cosh \left (6 e-\frac {6 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac {\left (3 \cosh \left (6 e-\frac {6 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac {\sinh \left (6 e-\frac {6 c f}{d}\right ) \int \frac {\cosh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac {\sinh \left (6 e-\frac {6 c f}{d}\right ) \int \frac {\sinh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac {\left (3 \sinh \left (6 e-\frac {6 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac {\left (3 \sinh \left (6 e-\frac {6 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+2 \left (\frac {\left (3 \cosh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{16 a^3}+\frac {\left (3 \sinh \left (4 e-\frac {4 c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {4 c f}{d}+4 f x\right )}{c+d x} \, dx}{16 a^3}\right ) \\ & = -\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {\cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Chi}\left (\frac {6 c f}{d}+6 f x\right )}{8 a^3 d}+\frac {\log (c+d x)}{8 a^3 d}+\frac {\text {Chi}\left (\frac {6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{8 a^3 d}-\frac {3 \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{8 a^3 d}+\frac {3 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{8 a^3 d}+\frac {3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac {3 \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{8 a^3 d}+2 \left (\frac {3 \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{16 a^3 d}+\frac {3 \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{16 a^3 d}\right )+\frac {\cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{8 a^3 d}-\frac {\sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{8 a^3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=\frac {\text {csch}^3(e+f x) (\cosh (f x)+\sinh (f x))^3 \left (\cosh (3 e) \log (f (c+d x))+\log (f (c+d x)) \sinh (3 e)+\left (-\cosh \left (e-\frac {4 c f}{d}\right )+\sinh \left (e-\frac {4 c f}{d}\right )\right ) \left (-3 \text {Chi}\left (\frac {4 f (c+d x)}{d}\right )+\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {6 f (c+d x)}{d}\right )-\text {Chi}\left (\frac {6 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )+3 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cosh \left (2 e-\frac {2 c f}{d}\right )+\sinh \left (2 e-\frac {2 c f}{d}\right )\right )-3 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-3 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+3 \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )-\cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {6 f (c+d x)}{d}\right )+\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {6 f (c+d x)}{d}\right )\right )\right )}{8 a^3 d (1+\coth (e+f x))^3} \]

[In]

Integrate[1/((c + d*x)*(a + a*Coth[e + f*x])^3),x]

[Out]

(Csch[e + f*x]^3*(Cosh[f*x] + Sinh[f*x])^3*(Cosh[3*e]*Log[f*(c + d*x)] + Log[f*(c + d*x)]*Sinh[3*e] + (-Cosh[e
- (4*c*f)/d] + Sinh[e - (4*c*f)/d])*(-3*CoshIntegral[(4*f*(c + d*x))/d] + Cosh[2*e - (2*c*f)/d]*CoshIntegral[
(6*f*(c + d*x))/d] - CoshIntegral[(6*f*(c + d*x))/d]*Sinh[2*e - (2*c*f)/d] + 3*CoshIntegral[(2*f*(c + d*x))/d]
*(Cosh[2*e - (2*c*f)/d] + Sinh[2*e - (2*c*f)/d]) - 3*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] - 3
*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] + 3*SinhIntegral[(4*f*(c + d*x))/d] - Cosh[2*e - (2*c*f
)/d]*SinhIntegral[(6*f*(c + d*x))/d] + Sinh[2*e - (2*c*f)/d]*SinhIntegral[(6*f*(c + d*x))/d])))/(8*a^3*d*(1 +
Coth[e + f*x])^3)

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.35

method	result	size
risch	\(\frac {\ln \left (d x +c \right )}{8 a^{3} d}+\frac {{\mathrm e}^{\frac {6 c f -6 d e}{d}} \operatorname {Ei}_{1}\left (6 f x +6 e +\frac {6 c f -6 d e}{d}\right )}{8 a^{3} d}-\frac {3 \,{\mathrm e}^{\frac {4 c f -4 d e}{d}} \operatorname {Ei}_{1}\left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{8 a^{3} d}+\frac {3 \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{8 a^{3} d}\)	\(151\)

[In]

int(1/(d*x+c)/(a+a*coth(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/8*ln(d*x+c)/a^3/d+1/8/a^3/d*exp(6*(c*f-d*e)/d)*Ei(1,6*f*x+6*e+6*(c*f-d*e)/d)-3/8/a^3/d*exp(4*(c*f-d*e)/d)*Ei
(1,4*f*x+4*e+4*(c*f-d*e)/d)+3/8/a^3/d*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=-\frac {3 \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 3 \, {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + {\rm Ei}\left (-\frac {6 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right ) + 3 \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 3 \, {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + {\rm Ei}\left (-\frac {6 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right ) - \log \left (d x + c\right )}{8 \, a^{3} d} \]

[In]

integrate(1/(d*x+c)/(a+a*coth(f*x+e))^3,x, algorithm="fricas")

[Out]

-1/8*(3*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) - 3*Ei(-4*(d*f*x + c*f)/d)*cosh(-4*(d*e - c*f)/d) + Ei(-
6*(d*f*x + c*f)/d)*cosh(-6*(d*e - c*f)/d) + 3*Ei(-2*(d*f*x + c*f)/d)*sinh(-2*(d*e - c*f)/d) - 3*Ei(-4*(d*f*x +
c*f)/d)*sinh(-4*(d*e - c*f)/d) + Ei(-6*(d*f*x + c*f)/d)*sinh(-6*(d*e - c*f)/d) - log(d*x + c))/(a^3*d)

Sympy [F]

\[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=\frac {\int \frac {1}{c \coth ^{3}{\left (e + f x \right )} + 3 c \coth ^{2}{\left (e + f x \right )} + 3 c \coth {\left (e + f x \right )} + c + d x \coth ^{3}{\left (e + f x \right )} + 3 d x \coth ^{2}{\left (e + f x \right )} + 3 d x \coth {\left (e + f x \right )} + d x}\, dx}{a^{3}} \]

[In]

integrate(1/(d*x+c)/(a+a*coth(f*x+e))**3,x)

[Out]

Integral(1/(c*coth(e + f*x)**3 + 3*c*coth(e + f*x)**2 + 3*c*coth(e + f*x) + c + d*x*coth(e + f*x)**3 + 3*d*x*c
oth(e + f*x)**2 + 3*d*x*coth(e + f*x) + d*x), x)/a**3

Maxima [A] (verification not implemented)

none

Time = 2.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.26 \[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=\frac {e^{\left (-6 \, e + \frac {6 \, c f}{d}\right )} E_{1}\left (\frac {6 \, {\left (d x + c\right )} f}{d}\right )}{8 \, a^{3} d} - \frac {3 \, e^{\left (-4 \, e + \frac {4 \, c f}{d}\right )} E_{1}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{8 \, a^{3} d} + \frac {3 \, e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{1}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{8 \, a^{3} d} + \frac {\log \left (d x + c\right )}{8 \, a^{3} d} \]

[In]

integrate(1/(d*x+c)/(a+a*coth(f*x+e))^3,x, algorithm="maxima")

[Out]

1/8*e^(-6*e + 6*c*f/d)*exp_integral_e(1, 6*(d*x + c)*f/d)/(a^3*d) - 3/8*e^(-4*e + 4*c*f/d)*exp_integral_e(1, 4
*(d*x + c)*f/d)/(a^3*d) + 3/8*e^(-2*e + 2*c*f/d)*exp_integral_e(1, 2*(d*x + c)*f/d)/(a^3*d) + 1/8*log(d*x + c)
/(a^3*d)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.24 \[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=-\frac {{\left (3 \, {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (4 \, e + \frac {2 \, c f}{d}\right )} - 3 \, {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (2 \, e + \frac {4 \, c f}{d}\right )} + {\rm Ei}\left (-\frac {6 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {6 \, c f}{d}\right )} - e^{\left (6 \, e\right )} \log \left (d x + c\right )\right )} e^{\left (-6 \, e\right )}}{8 \, a^{3} d} \]

[In]

integrate(1/(d*x+c)/(a+a*coth(f*x+e))^3,x, algorithm="giac")

[Out]

-1/8*(3*Ei(-2*(d*f*x + c*f)/d)*e^(4*e + 2*c*f/d) - 3*Ei(-4*(d*f*x + c*f)/d)*e^(2*e + 4*c*f/d) + Ei(-6*(d*f*x +
c*f)/d)*e^(6*c*f/d) - e^(6*e)*log(d*x + c))*e^(-6*e)/(a^3*d)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(c+d x) (a+a \coth (e+f x))^3} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^3\,\left (c+d\,x\right )} \,d x \]

[In]

int(1/((a + a*coth(e + f*x))^3*(c + d*x)),x)

[Out]

int(1/((a + a*coth(e + f*x))^3*(c + d*x)), x)