3.31 \(\int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx\)

Optimal. Leaf size=692 \[ -\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \cosh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {1}{8 a^3 d (c+d x)} \]

[Out]

-1/8/a^3/d/(d*x+c)+3/4*f*Chi(6*c*f/d+6*f*x)*cosh(-6*e+6*c*f/d)/a^3/d^2-3/2*f*Chi(4*c*f/d+4*f*x)*cosh(-4*e+4*c*
f/d)/a^3/d^2+3/4*f*Chi(2*c*f/d+2*f*x)*cosh(-2*e+2*c*f/d)/a^3/d^2+9/32*cosh(2*f*x+2*e)/a^3/d/(d*x+c)-3/8*cosh(2
*f*x+2*e)^2/a^3/d/(d*x+c)+1/8*cosh(2*f*x+2*e)^3/a^3/d/(d*x+c)+3/32*cosh(6*f*x+6*e)/a^3/d/(d*x+c)-3/4*f*cosh(-2
*e+2*c*f/d)*Shi(2*c*f/d+2*f*x)/a^3/d^2+3/2*f*cosh(-4*e+4*c*f/d)*Shi(4*c*f/d+4*f*x)/a^3/d^2-3/4*f*cosh(-6*e+6*c
*f/d)*Shi(6*c*f/d+6*f*x)/a^3/d^2+3/4*f*Chi(6*c*f/d+6*f*x)*sinh(-6*e+6*c*f/d)/a^3/d^2-3/4*f*Shi(6*c*f/d+6*f*x)*
sinh(-6*e+6*c*f/d)/a^3/d^2-3/2*f*Chi(4*c*f/d+4*f*x)*sinh(-4*e+4*c*f/d)/a^3/d^2+3/2*f*Shi(4*c*f/d+4*f*x)*sinh(-
4*e+4*c*f/d)/a^3/d^2+3/4*f*Chi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/a^3/d^2-3/4*f*Shi(2*c*f/d+2*f*x)*sinh(-2*e+2*
c*f/d)/a^3/d^2-15/32*sinh(2*f*x+2*e)/a^3/d/(d*x+c)-3/8*sinh(2*f*x+2*e)^2/a^3/d/(d*x+c)-1/8*sinh(2*f*x+2*e)^3/a
^3/d/(d*x+c)+3/8*sinh(4*f*x+4*e)/a^3/d/(d*x+c)-3/32*sinh(6*f*x+6*e)/a^3/d/(d*x+c)

________________________________________________________________________________________

Rubi [A]  time = 1.76, antiderivative size = 692, normalized size of antiderivative = 1.00, number of steps used = 60, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3728, 3297, 3303, 3298, 3301, 3313, 12, 5448, 5470} \[ -\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \cosh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {1}{8 a^3 d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^2*(a + a*Coth[e + f*x])^3),x]

[Out]

-1/(8*a^3*d*(c + d*x)) + (9*Cosh[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Cosh[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*
x)) + Cosh[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) + (3*Cosh[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) + (3*f*Cosh[2*e - (
2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) - (3*f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c*f)/d + 4
*f*x])/(2*a^3*d^2) + (3*f*Cosh[6*e - (6*c*f)/d]*CoshIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) - (3*f*CoshIntegr
al[(6*c*f)/d + 6*f*x]*Sinh[6*e - (6*c*f)/d])/(4*a^3*d^2) + (3*f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - (4*
c*f)/d])/(2*a^3*d^2) - (3*f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/(4*a^3*d^2) - (15*Sinh[2*e
+ 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Sinh[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) - Sinh[2*e + 2*f*x]^3/(8*a^3*d*(c
+ d*x)) + (3*Sinh[4*e + 4*f*x])/(8*a^3*d*(c + d*x)) - (3*Sinh[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) - (3*f*Cosh[
2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (3*f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f
)/d + 2*f*x])/(4*a^3*d^2) + (3*f*Cosh[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*f*Sin
h[4*e - (4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(2*a^3*d^2) - (3*f*Cosh[6*e - (6*c*f)/d]*SinhIntegral[(6*c
*f)/d + 6*f*x])/(4*a^3*d^2) + (3*f*Sinh[6*e - (6*c*f)/d]*SinhIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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 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 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

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 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)^2 (a+a \coth (e+f x))^3} \, dx &=\int \left (\frac {1}{8 a^3 (c+d x)^2}-\frac {3 \cosh (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {\cosh ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \sinh (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \cosh ^2(2 e+2 f x) \sinh (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {\sinh ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 \sinh (4 e+4 f x)}{8 a^3 (c+d x)^2}-\frac {3 \sinh (2 e+2 f x) \sinh (4 e+4 f x)}{16 a^3 (c+d x)^2}\right ) \, dx\\ &=-\frac {1}{8 a^3 d (c+d x)}-\frac {\int \frac {\cosh ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac {\int \frac {\sinh ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac {3 \int \frac {\sinh (2 e+2 f x) \sinh (4 e+4 f x)}{(c+d x)^2} \, dx}{16 a^3}-\frac {3 \int \frac {\cosh (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac {3 \int \frac {\cosh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac {3 \int \frac {\sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac {3 \int \frac {\cosh ^2(2 e+2 f x) \sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac {3 \int \frac {\sinh ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac {3 \int \frac {\sinh (4 e+4 f x)}{(c+d x)^2} \, dx}{8 a^3}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {3 \cosh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \int \left (-\frac {\cosh (2 e+2 f x)}{2 (c+d x)^2}+\frac {\cosh (6 e+6 f x)}{2 (c+d x)^2}\right ) \, dx}{16 a^3}+\frac {3 \int \left (\frac {\sinh (2 e+2 f x)}{4 (c+d x)^2}+\frac {\sinh (6 e+6 f x)}{4 (c+d x)^2}\right ) \, dx}{8 a^3}-\frac {(3 i f) \int \left (-\frac {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}{4 a^3 d}+\frac {(3 i f) \int -\frac {i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}-\frac {(3 i f) \int \frac {i \sinh (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}+\frac {(3 f) \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}-\frac {(3 f) \int \left (\frac {\cosh (2 e+2 f x)}{4 (c+d x)}-\frac {\cosh (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{4 a^3 d}-\frac {(3 f) \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}-\frac {(3 f) \int \frac {\cosh (4 e+4 f x)}{c+d x} \, dx}{2 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {3 \cosh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 \int \frac {\cosh (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac {3 \int \frac {\cosh (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac {3 \int \frac {\sinh (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac {3 \int \frac {\sinh (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac {(3 f) \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac {(3 f) \int \frac {\cosh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac {(3 f) \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac {(3 f) \int \frac {\sinh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}+2 \frac {(3 f) \int \frac {\sinh (4 e+4 f x)}{c+d x} \, dx}{4 a^3 d}-\frac {\left (3 f \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}{2 a^3 d}+\frac {\left (3 f \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}{4 a^3 d}-\frac {\left (3 f \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}{4 a^3 d}-\frac {\left (3 f \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}{2 a^3 d}-\frac {\left (3 f \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}{4 a^3 d}+\frac {\left (3 f \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}{4 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac {(3 f) \int \frac {\cosh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac {(3 f) \int \frac {\sinh (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac {(9 f) \int \frac {\cosh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac {(9 f) \int \frac {\sinh (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}-\frac {\left (3 f \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}{16 a^3 d}-\frac {\left (3 f \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}{16 a^3 d}-\frac {\left (3 f \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}{16 a^3 d}-\frac {\left (3 f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}+2 \left (\frac {\left (3 f \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}{4 a^3 d}+\frac {\left (3 f \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}{4 a^3 d}\right )-\frac {\left (3 f \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}{16 a^3 d}-\frac {\left (3 f \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}{16 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {9 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{16 a^3 d^2}-\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Chi}\left (\frac {6 c f}{d}+6 f x\right )}{16 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{16 a^3 d^2}-\frac {15 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{16 a^3 d^2}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {15 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{16 a^3 d^2}+\frac {9 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{16 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+2 \left (\frac {3 f \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{16 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{16 a^3 d^2}+\frac {\left (9 f \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}{16 a^3 d}-\frac {\left (9 f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}-\frac {\left (9 f \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}{16 a^3 d}+\frac {\left (9 f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Chi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+2 \left (\frac {3 f \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}\\ \end {align*}

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Mathematica [A]  time = 3.12, size = 796, normalized size = 1.15 \[ \frac {\text {csch}^3(e+f x) \left (\cosh \left (\frac {3 c f}{d}\right )+\sinh \left (\frac {3 c f}{d}\right )\right ) \left (3 d \cosh \left (e+f \left (x-\frac {3 c}{d}\right )\right )-d \cosh \left (3 \left (e+f \left (x-\frac {c}{d}\right )\right )\right )+d \cosh \left (3 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )-3 d \cosh \left (e+f \left (\frac {3 c}{d}+x\right )\right )+6 c f \cosh \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Chi}\left (\frac {6 f (c+d x)}{d}\right )+6 d f x \cosh \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Chi}\left (\frac {6 f (c+d x)}{d}\right )+6 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac {c f}{d}+3 f x\right )+\sinh \left (e-\frac {c f}{d}+3 f x\right )\right )+3 d \sinh \left (e+f \left (x-\frac {3 c}{d}\right )\right )-d \sinh \left (3 \left (e+f \left (x-\frac {c}{d}\right )\right )\right )-d \sinh \left (3 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+3 d \sinh \left (e+f \left (\frac {3 c}{d}+x\right )\right )-6 c f \text {Chi}\left (\frac {6 f (c+d x)}{d}\right ) \sinh \left (3 e-\frac {3 f (c+d x)}{d}\right )-6 d f x \text {Chi}\left (\frac {6 f (c+d x)}{d}\right ) \sinh \left (3 e-\frac {3 f (c+d x)}{d}\right )+12 f (c+d x) \text {Chi}\left (\frac {4 f (c+d x)}{d}\right ) \left (\sinh \left (e-\frac {f (c+3 d x)}{d}\right )-\cosh \left (e-\frac {f (c+3 d x)}{d}\right )\right )-6 c f \cosh \left (e-\frac {c f}{d}+3 f x\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-6 d f x \cosh \left (e-\frac {c f}{d}+3 f x\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-6 c f \sinh \left (e-\frac {c f}{d}+3 f x\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-6 d f x \sinh \left (e-\frac {c f}{d}+3 f x\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+12 c f \cosh \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )+12 d f x \cosh \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )-12 c f \sinh \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )-12 d f x \sinh \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )-6 c f \cosh \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {6 f (c+d x)}{d}\right )-6 d f x \cosh \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {6 f (c+d x)}{d}\right )+6 c f \sinh \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {6 f (c+d x)}{d}\right )+6 d f x \sinh \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {6 f (c+d x)}{d}\right )\right )}{8 a^3 d^2 (c+d x) (\coth (e+f x)+1)^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((c + d*x)^2*(a + a*Coth[e + f*x])^3),x]

[Out]

(Csch[e + f*x]^3*(Cosh[(3*c*f)/d] + Sinh[(3*c*f)/d])*(3*d*Cosh[e + f*((-3*c)/d + x)] - d*Cosh[3*(e + f*(-(c/d)
+ x))] + d*Cosh[3*(e + f*(c/d + x))] - 3*d*Cosh[e + f*((3*c)/d + x)] + 6*c*f*Cosh[3*e - (3*f*(c + d*x))/d]*Co
shIntegral[(6*f*(c + d*x))/d] + 6*d*f*x*Cosh[3*e - (3*f*(c + d*x))/d]*CoshIntegral[(6*f*(c + d*x))/d] + 6*f*(c
+ d*x)*CoshIntegral[(2*f*(c + d*x))/d]*(Cosh[e - (c*f)/d + 3*f*x] + Sinh[e - (c*f)/d + 3*f*x]) + 3*d*Sinh[e +
f*((-3*c)/d + x)] - d*Sinh[3*(e + f*(-(c/d) + x))] - d*Sinh[3*(e + f*(c/d + x))] + 3*d*Sinh[e + f*((3*c)/d +
x)] - 6*c*f*CoshIntegral[(6*f*(c + d*x))/d]*Sinh[3*e - (3*f*(c + d*x))/d] - 6*d*f*x*CoshIntegral[(6*f*(c + d*x
))/d]*Sinh[3*e - (3*f*(c + d*x))/d] + 12*f*(c + d*x)*CoshIntegral[(4*f*(c + d*x))/d]*(-Cosh[e - (f*(c + 3*d*x)
)/d] + Sinh[e - (f*(c + 3*d*x))/d]) - 6*c*f*Cosh[e - (c*f)/d + 3*f*x]*SinhIntegral[(2*f*(c + d*x))/d] - 6*d*f*
x*Cosh[e - (c*f)/d + 3*f*x]*SinhIntegral[(2*f*(c + d*x))/d] - 6*c*f*Sinh[e - (c*f)/d + 3*f*x]*SinhIntegral[(2*
f*(c + d*x))/d] - 6*d*f*x*Sinh[e - (c*f)/d + 3*f*x]*SinhIntegral[(2*f*(c + d*x))/d] + 12*c*f*Cosh[e - (f*(c +
3*d*x))/d]*SinhIntegral[(4*f*(c + d*x))/d] + 12*d*f*x*Cosh[e - (f*(c + 3*d*x))/d]*SinhIntegral[(4*f*(c + d*x))
/d] - 12*c*f*Sinh[e - (f*(c + 3*d*x))/d]*SinhIntegral[(4*f*(c + d*x))/d] - 12*d*f*x*Sinh[e - (f*(c + 3*d*x))/d
]*SinhIntegral[(4*f*(c + d*x))/d] - 6*c*f*Cosh[3*e - (3*f*(c + d*x))/d]*SinhIntegral[(6*f*(c + d*x))/d] - 6*d*
f*x*Cosh[3*e - (3*f*(c + d*x))/d]*SinhIntegral[(6*f*(c + d*x))/d] + 6*c*f*Sinh[3*e - (3*f*(c + d*x))/d]*SinhIn
tegral[(6*f*(c + d*x))/d] + 6*d*f*x*Sinh[3*e - (3*f*(c + d*x))/d]*SinhIntegral[(6*f*(c + d*x))/d]))/(8*a^3*d^2
*(c + d*x)*(1 + Coth[e + f*x])^3)

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fricas [A]  time = 0.41, size = 1162, normalized size = 1.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="fricas")

[Out]

1/4*(3*(d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(f*x + e)^3*sinh(-2*(d*e - c*f)/d) - 6*(d*f*x + c*f)*Ei(-4*(d*
f*x + c*f)/d)*cosh(f*x + e)^3*sinh(-4*(d*e - c*f)/d) + 3*(d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*cosh(f*x + e)^3*
sinh(-6*(d*e - c*f)/d) + 3*((d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) - 2*(d*f*x + c*f)*Ei(-
4*(d*f*x + c*f)/d)*cosh(-4*(d*e - c*f)/d) + (d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*cosh(-6*(d*e - c*f)/d))*cosh(
f*x + e)^3 + (3*(d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) - 6*(d*f*x + c*f)*Ei(-4*(d*f*x + c
*f)/d)*cosh(-4*(d*e - c*f)/d) + 3*(d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*cosh(-6*(d*e - c*f)/d) + 3*(d*f*x + c*f
)*Ei(-2*(d*f*x + c*f)/d)*sinh(-2*(d*e - c*f)/d) - 6*(d*f*x + c*f)*Ei(-4*(d*f*x + c*f)/d)*sinh(-4*(d*e - c*f)/d
) + 3*(d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*sinh(-6*(d*e - c*f)/d) - d)*sinh(f*x + e)^3 + 9*((d*f*x + c*f)*Ei(-
2*(d*f*x + c*f)/d)*cosh(f*x + e)*sinh(-2*(d*e - c*f)/d) - 2*(d*f*x + c*f)*Ei(-4*(d*f*x + c*f)/d)*cosh(f*x + e)
*sinh(-4*(d*e - c*f)/d) + (d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*cosh(f*x + e)*sinh(-6*(d*e - c*f)/d) + ((d*f*x
+ c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) - 2*(d*f*x + c*f)*Ei(-4*(d*f*x + c*f)/d)*cosh(-4*(d*e - c
*f)/d) + (d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*cosh(-6*(d*e - c*f)/d))*cosh(f*x + e))*sinh(f*x + e)^2 + 3*(3*(d
*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(f*x + e)^2*sinh(-2*(d*e - c*f)/d) - 6*(d*f*x + c*f)*Ei(-4*(d*f*x + c*f
)/d)*cosh(f*x + e)^2*sinh(-4*(d*e - c*f)/d) + 3*(d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*cosh(f*x + e)^2*sinh(-6*(
d*e - c*f)/d) + (3*(d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) - 6*(d*f*x + c*f)*Ei(-4*(d*f*x
+ c*f)/d)*cosh(-4*(d*e - c*f)/d) + 3*(d*f*x + c*f)*Ei(-6*(d*f*x + c*f)/d)*cosh(-6*(d*e - c*f)/d) - d)*cosh(f*x
+ e)^2 + d)*sinh(f*x + e))/((a^3*d^3*x + a^3*c*d^2)*cosh(f*x + e)^3 + 3*(a^3*d^3*x + a^3*c*d^2)*cosh(f*x + e)
^2*sinh(f*x + e) + 3*(a^3*d^3*x + a^3*c*d^2)*cosh(f*x + e)*sinh(f*x + e)^2 + (a^3*d^3*x + a^3*c*d^2)*sinh(f*x
+ e)^3)

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giac [A]  time = 0.24, size = 911, normalized size = 1.32 \[ \frac {{\left (6 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (\frac {6 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {6 \, {\left (c f - d e\right )}}{d}\right )} - 6 \, c f^{3} {\rm Ei}\left (\frac {6 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {6 \, {\left (c f - d e\right )}}{d}\right )} - 12 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {4 \, {\left (c f - d e\right )}}{d}\right )} + 12 \, c f^{3} {\rm Ei}\left (\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {4 \, {\left (c f - d e\right )}}{d}\right )} + 6 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{d}\right )} - 6 \, c f^{3} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{d}\right )} + 6 \, d f^{2} {\rm Ei}\left (\frac {6 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {6 \, {\left (c f - d e\right )}}{d} + 1\right )} - 12 \, d f^{2} {\rm Ei}\left (\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {4 \, {\left (c f - d e\right )}}{d} + 1\right )} + 6 \, d f^{2} {\rm Ei}\left (\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - c f + d e\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (c f - d e\right )}}{d} + 1\right )} - d f^{2} e^{\left (\frac {6 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )} + 3 \, d f^{2} e^{\left (\frac {4 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )} - 3 \, d f^{2} e^{\left (\frac {2 \, {\left (d x + c\right )} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )}}{d}\right )} + d f^{2}\right )} d^{2}}{8 \, {\left ({\left (d x + c\right )} a^{3} d^{4} {\left (\frac {c f}{d x + c} - f - \frac {d e}{d x + c}\right )} - a^{3} c d^{4} f + a^{3} d^{5} e\right )} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="giac")

[Out]

1/8*(6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*Ei(6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) -
c*f + d*e)/d)*e^(6*(c*f - d*e)/d) - 6*c*f^3*Ei(6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/
d)*e^(6*(c*f - d*e)/d) - 12*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*Ei(4*((d*x + c)*(c*f/(d*x + c) -
f - d*e/(d*x + c)) - c*f + d*e)/d)*e^(4*(c*f - d*e)/d) + 12*c*f^3*Ei(4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d
*x + c)) - c*f + d*e)/d)*e^(4*(c*f - d*e)/d) + 6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*Ei(2*((d*x
+ c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*e^(2*(c*f - d*e)/d) - 6*c*f^3*Ei(2*((d*x + c)*(c*f/(d
*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*e^(2*(c*f - d*e)/d) + 6*d*f^2*Ei(6*((d*x + c)*(c*f/(d*x + c) - f
- d*e/(d*x + c)) - c*f + d*e)/d)*e^(6*(c*f - d*e)/d + 1) - 12*d*f^2*Ei(4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(
d*x + c)) - c*f + d*e)/d)*e^(4*(c*f - d*e)/d + 1) + 6*d*f^2*Ei(2*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)
) - c*f + d*e)/d)*e^(2*(c*f - d*e)/d + 1) - d*f^2*e^(6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))/d) + 3*d*
f^2*e^(4*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))/d) - 3*d*f^2*e^(2*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d
*x + c))/d) + d*f^2)*d^2/(((d*x + c)*a^3*d^4*(c*f/(d*x + c) - f - d*e/(d*x + c)) - a^3*c*d^4*f + a^3*d^5*e)*f)

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maple [A]  time = 4.39, size = 239, normalized size = 0.35 \[ -\frac {1}{8 a^{3} d \left (d x +c \right )}+\frac {f \,{\mathrm e}^{-6 f x -6 e}}{8 a^{3} d \left (d f x +c f \right )}-\frac {3 f \,{\mathrm e}^{\frac {6 c f -6 d e}{d}} \Ei \left (1, 6 f x +6 e +\frac {6 c f -6 d e}{d}\right )}{4 a^{3} d^{2}}-\frac {3 f \,{\mathrm e}^{-4 f x -4 e}}{8 a^{3} d \left (d f x +c f \right )}+\frac {3 f \,{\mathrm e}^{\frac {4 c f -4 d e}{d}} \Ei \left (1, 4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{2 a^{3} d^{2}}+\frac {3 f \,{\mathrm e}^{-2 f x -2 e}}{8 a^{3} d \left (d f x +c f \right )}-\frac {3 f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \Ei \left (1, 2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{4 a^{3} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8/a^3/d/(d*x+c)+1/8*f/a^3*exp(-6*f*x-6*e)/d/(d*f*x+c*f)-3/4*f/a^3/d^2*exp(6*(c*f-d*e)/d)*Ei(1,6*f*x+6*e+6*(
c*f-d*e)/d)-3/8*f/a^3*exp(-4*f*x-4*e)/d/(d*f*x+c*f)+3/2*f/a^3/d^2*exp(4*(c*f-d*e)/d)*Ei(1,4*f*x+4*e+4*(c*f-d*e
)/d)+3/8*f/a^3*exp(-2*f*x-2*e)/d/(d*f*x+c*f)-3/4*f/a^3/d^2*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 = 9.93, size = 140, normalized size = 0.20 \[ -\frac {1}{8 \, {\left (a^{3} d^{2} x + a^{3} c d\right )}} + \frac {e^{\left (-6 \, e + \frac {6 \, c f}{d}\right )} E_{2}\left (\frac {6 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} - \frac {3 \, e^{\left (-4 \, e + \frac {4 \, c f}{d}\right )} E_{2}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} + \frac {3 \, e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{c^{2} \coth ^{3}{\left (e + f x \right )} + 3 c^{2} \coth ^{2}{\left (e + f x \right )} + 3 c^{2} \coth {\left (e + f x \right )} + c^{2} + 2 c d x \coth ^{3}{\left (e + f x \right )} + 6 c d x \coth ^{2}{\left (e + f x \right )} + 6 c d x \coth {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth ^{3}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth ^{2}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(c**2*coth(e + f*x)**3 + 3*c**2*coth(e + f*x)**2 + 3*c**2*coth(e + f*x) + c**2 + 2*c*d*x*coth(e + f
*x)**3 + 6*c*d*x*coth(e + f*x)**2 + 6*c*d*x*coth(e + f*x) + 2*c*d*x + d**2*x**2*coth(e + f*x)**3 + 3*d**2*x**2
*coth(e + f*x)**2 + 3*d**2*x**2*coth(e + f*x) + d**2*x**2), x)/a**3

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