3.46 \(\int \frac{1}{(c+d x) (a+a \tanh (e+f x))^3} \, dx\)

Optimal. Leaf size=437 \[ -\frac{3 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{\text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \sinh \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \cosh \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \cosh \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{3 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\sinh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 1.82121, antiderivative size = 437, normalized size of antiderivative = 1., number of steps used = 53, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {3728, 3303, 3298, 3301, 3312, 5448, 5470} \[ -\frac{3 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{\text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \sinh \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \sinh \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{Chi}\left (2 x f+\frac{2 c f}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{Chi}\left (4 x f+\frac{4 c f}{d}\right ) \cosh \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\text{Chi}\left (6 x f+\frac{6 c f}{d}\right ) \cosh \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{3 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \sinh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\sinh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \cosh \left (4 e-\frac{4 c f}{d}\right ) \text{Shi}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\cosh \left (6 e-\frac{6 c f}{d}\right ) \text{Shi}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)*(a + a*Tanh[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 3728

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 3303

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 3298

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 3301

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 3312

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 5448

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 5470

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*} \int \frac{1}{(c+d x) (a+a \tanh (e+f x))^3} \, dx &=\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]  time = 0.672704, size = 312, normalized size = 0.71 \[ \frac{\text{sech}^3(e+f x) (\sinh (f x)+\cosh (f x))^3 \left (\left (\cosh \left (e-\frac{4 c f}{d}\right )-\sinh \left (e-\frac{4 c f}{d}\right )\right ) \left (-\text{Chi}\left (\frac{6 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac{2 c f}{d}\right )+\text{Chi}\left (\frac{6 f (c+d x)}{d}\right ) \cosh \left (2 e-\frac{2 c f}{d}\right )+3 \text{Chi}\left (\frac{2 f (c+d x)}{d}\right ) \left (\sinh \left (2 e-\frac{2 c f}{d}\right )+\cosh \left (2 e-\frac{2 c f}{d}\right )\right )+3 \text{Chi}\left (\frac{4 f (c+d x)}{d}\right )-3 \sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )+\sinh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{6 f (c+d x)}{d}\right )-3 \cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{2 f (c+d x)}{d}\right )-\cosh \left (2 e-\frac{2 c f}{d}\right ) \text{Shi}\left (\frac{6 f (c+d x)}{d}\right )-3 \text{Shi}\left (\frac{4 f (c+d x)}{d}\right )\right )+\sinh (3 e) \log (f (c+d x))+\cosh (3 e) \log (f (c+d x))\right )}{8 a^3 d (\tanh (e+f x)+1)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sech[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*S
inh[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 + Ta
nh[e + f*x])^3)

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Maple [A]  time = 0.395, size = 151, normalized size = 0.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{8\,{a}^{3}d}}-{\frac{1}{8\,{a}^{3}d}{{\rm e}^{6\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,6\,fx+6\,e+6\,{\frac{cf-de}{d}} \right ) }-{\frac{3}{8\,{a}^{3}d}{{\rm e}^{4\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,4\,fx+4\,e+4\,{\frac{cf-de}{d}} \right ) }-{\frac{3}{8\,{a}^{3}d}{{\rm e}^{2\,{\frac{cf-de}{d}}}}{\it Ei} \left ( 1,2\,fx+2\,e+2\,{\frac{cf-de}{d}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]  time = 11.1494, size = 154, normalized size = 0.35 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+a*tanh(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)

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Fricas [A]  time = 2.28808, size = 437, normalized size = 1. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)/(a+a*tanh(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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{c \tanh ^{3}{\left (e + f x \right )} + 3 c \tanh ^{2}{\left (e + f x \right )} + 3 c \tanh{\left (e + f x \right )} + c + d x \tanh ^{3}{\left (e + f x \right )} + 3 d x \tanh ^{2}{\left (e + f x \right )} + 3 d x \tanh{\left (e + f x \right )} + d x}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(c*tanh(e + f*x)**3 + 3*c*tanh(e + f*x)**2 + 3*c*tanh(e + f*x) + c + d*x*tanh(e + f*x)**3 + 3*d*x*t
anh(e + f*x)**2 + 3*d*x*tanh(e + f*x) + d*x), x)/a**3

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Giac [A]  time = 1.17891, size = 144, normalized size = 0.33 \begin{align*} \frac{{\left ({\rm Ei}\left (-\frac{6 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{6 \, c f}{d}\right )} + 3 \,{\rm Ei}\left (-\frac{4 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{4 \, c f}{d} + 2 \, e\right )} + 3 \,{\rm Ei}\left (-\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac{2 \, c f}{d} + 4 \, e\right )} + e^{\left (6 \, e\right )} \log \left (d x + c\right )\right )} e^{\left (-6 \, e\right )}}{8 \, a^{3} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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