Integrand size = 20, antiderivative size = 692 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Chi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2} \]
-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*S hi(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)
Time = 3.77 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\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 (-\frac {3 c}{d}+x\right )\right )-d \cosh \left (3 \left (e+f \left (-\frac {c}{d}+x\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 (-\frac {3 c}{d}+x\right )\right )-d \sinh \left (3 \left (e+f \left (-\frac {c}{d}+x\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 (-\cosh \left (e-\frac {f (c+3 d x)}{d}\right )+\sinh \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) (1+\coth (e+f x))^3} \]
(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]*Cosh Integral[(6*f*(c + d*x))/d] + 6*d*f*x*Cosh[3*e - (3*f*(c + d*x))/d]*CoshIn tegral[(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]*Si nhIntegral[(4*f*(c + d*x))/d] - 12*c*f*Sinh[e - (f*(c + 3*d*x))/d]*SinhInt egral[(4*f*(c + d*x))/d] - 12*d*f*x*Sinh[e - (f*(c + 3*d*x))/d]*SinhIntegr al[(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[...
Time = 2.22 (sec) , antiderivative size = 692, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4211, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{(c+d x)^2 (a \coth (e+f x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{(c+d x)^2 \left (a-i a \tan \left (i e+i f x+\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4211 |
\(\displaystyle \int \left (\frac {\sinh ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \sinh (2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 \sinh (2 e+2 f x) \sinh (4 e+4 f x)}{16 a^3 (c+d x)^2}-\frac {3 \sinh (4 e+4 f x)}{8 a^3 (c+d x)^2}-\frac {\cosh ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 \cosh (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \sinh (2 e+2 f x) \cosh ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {1}{8 a^3 (c+d x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \cosh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {1}{8 a^3 d (c+d x)}\) |
-1/8*1/(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*CoshIntegral [(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*Sinh[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)
3.1.31.3.1 Defintions of rubi rules used
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]
Time = 0.47 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.35
method | result | size |
risch | \(-\frac {1}{8 a^{3} d \left (d x +c \right )}+\frac {f \,{\mathrm e}^{-6 f x -6 e}}{8 a^{3} d \left (d x f +c f \right )}-\frac {3 f \,{\mathrm e}^{\frac {6 c f -6 d e}{d}} \operatorname {Ei}_{1}\left (6 f x +6 e +\frac {6 c f -6 d e}{d}\right )}{4 a^{3} d^{2}}-\frac {3 f \,{\mathrm e}^{-4 f x -4 e}}{8 a^{3} d \left (d x f +c f \right )}+\frac {3 f \,{\mathrm e}^{\frac {4 c f -4 d e}{d}} \operatorname {Ei}_{1}\left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{2 a^{3} d^{2}}+\frac {3 f \,{\mathrm e}^{-2 f x -2 e}}{8 a^{3} d \left (d x f +c f \right )}-\frac {3 f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{4 a^{3} d^{2}}\) | \(239\) |
-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*e xp(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)
Time = 0.26 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\text {Too large to display} \]
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*si nh(-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)*s inh(-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)*co sh(-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...
\[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\frac {\int \frac {1}{c^{2} \coth ^{3}{\left (e + f x \right )} + 3 c^{2} \coth ^{2}{\left (e + f x \right )} + 3 c^{2} \coth {\left (e + f x \right )} + c^{2} + 2 c d x \coth ^{3}{\left (e + f x \right )} + 6 c d x \coth ^{2}{\left (e + f x \right )} + 6 c d x \coth {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth ^{3}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth ^{2}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{3}} \]
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*c oth(e + f*x)**2 + 3*d**2*x**2*coth(e + f*x) + d**2*x**2), x)/a**3
Time = 5.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.20 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=-\frac {1}{8 \, {\left (a^{3} d^{2} x + a^{3} c d\right )}} + \frac {e^{\left (-6 \, e + \frac {6 \, c f}{d}\right )} E_{2}\left (\frac {6 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} - \frac {3 \, e^{\left (-4 \, e + \frac {4 \, c f}{d}\right )} E_{2}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} + \frac {3 \, e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} \]
-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)
Time = 0.32 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\frac {{\left (6 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 6 \, d e f^{2} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 6 \, c f^{3} {\rm Ei}\left (-\frac {2 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )} - 12 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right )} + 12 \, d e f^{2} {\rm Ei}\left (-\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right )} - 12 \, c f^{3} {\rm Ei}\left (-\frac {4 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right )} + 6 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {6 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right )} - 6 \, d e f^{2} {\rm Ei}\left (-\frac {6 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right )} + 6 \, c f^{3} {\rm Ei}\left (-\frac {6 \, {\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f\right )}}{d}\right ) e^{\left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right )} + 3 \, d f^{2} e^{\left (-\frac {2 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - 3 \, d f^{2} e^{\left (-\frac {4 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} + d f^{2} e^{\left (-\frac {6 \, {\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )} - d f^{2}\right )} d^{2}}{8 \, {\left ({\left (d x + c\right )} a^{3} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - a^{3} d^{5} e + a^{3} c d^{4} f\right )} f} \]
1/8*(6*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(-2*((d*x + c)* (d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f)/d) - 6*d*e*f^2*Ei(-2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c* f)/d)*e^(-2*(d*e - c*f)/d) + 6*c*f^3*Ei(-2*((d*x + c)*(d*e/(d*x + c) - c*f /(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f)/d) - 12*(d*x + c)*(d*e/( d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(-4*((d*x + c)*(d*e/(d*x + c) - c*f/(d *x + c) + f) - d*e + c*f)/d)*e^(-4*(d*e - c*f)/d) + 12*d*e*f^2*Ei(-4*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-4*(d*e - c*f )/d) - 12*c*f^3*Ei(-4*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-4*(d*e - c*f)/d) + 6*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(-6*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c *f)/d)*e^(-6*(d*e - c*f)/d) - 6*d*e*f^2*Ei(-6*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-6*(d*e - c*f)/d) + 6*c*f^3*Ei(-6*(( d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-6*(d*e - c*f)/d) + 3*d*f^2*e^(-2*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d) - 3*d*f^2*e^(-4*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d) + d*f^2*e^ (-6*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d) - d*f^2)*d^2/(((d*x + c)*a^3*d^4*(d*e/(d*x + c) - c*f/(d*x + c) + f) - a^3*d^5*e + a^3*c*d^4*f) *f)
Timed out. \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]