Integrand size = 20, antiderivative size = 692 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Chi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {6 c f}{d}+6 f x\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2} \] Output:
-1/8/a^3/d/(d*x+c)+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)*Chi(2*c*f/d+2*f*x)/a^3/d^2-3/2*f*cos h(-4*e+4*c*f/d)*Chi(4*c*f/d+4*f*x)/a^3/d^2+3/4*f*cosh(-6*e+6*c*f/d)*Chi(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 /2*f*Chi(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^2-15/32*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)-3/4*f*cosh(-2*e+ 2*c*f/d)*Shi(2*c*f/d+2*f*x)/a^3/d^2-3/4*f*sinh(-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+3/2*f*si nh(-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*sinh(-6*e+6*c*f/d)*Shi(6*c*f/d+6*f*x)/a^3/d^2
Time = 2.46 (sec) , antiderivative size = 796, normalized size of antiderivative = 1.15 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx =\text {Too large to display} \] Input:
Integrate[1/((c + d*x)^2*(a + a*Coth[e + f*x])^3),x]
Output:
(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 = 2.27 (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 |
| \(\displaystyle -\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \sinh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \text {Chi}\left (6 x f+\frac {6 c f}{d}\right ) \cosh \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \sinh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \cosh \left (6 e-\frac {6 c f}{d}\right ) \text {Shi}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {\sinh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {15 \sinh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sinh (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {3 \sinh (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {\cosh ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cosh ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {9 \cosh (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \cosh (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {1}{8 a^3 d (c+d x)}\) |
Input:
Int[1/((c + d*x)^2*(a + a*Coth[e + f*x])^3),x]
Output:
-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)
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 = 0.28 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(-\frac {1}{8 a^{3} d \left (d x +c \right )}+\frac {f \,{\mathrm e}^{-6 f x -6 e}}{8 a^{3} d \left (d f x +c f \right )}-\frac {3 f \,{\mathrm e}^{\frac {6 c f -6 d e}{d}} \operatorname {expIntegral}_{1}\left (6 f x +6 e +\frac {6 c f -6 d e}{d}\right )}{4 a^{3} d^{2}}-\frac {3 f \,{\mathrm e}^{-4 f x -4 e}}{8 a^{3} d \left (d f x +c f \right )}+\frac {3 f \,{\mathrm e}^{\frac {4 c f -4 d e}{d}} \operatorname {expIntegral}_{1}\left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{2 a^{3} d^{2}}+\frac {3 f \,{\mathrm e}^{-2 f x -2 e}}{8 a^{3} d \left (d f x +c f \right )}-\frac {3 f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {expIntegral}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{4 a^{3} d^{2}}\) | \(239\) |
Input:
int(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
-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 = 0.11 (sec) , antiderivative size = 1162, normalized size of antiderivative = 1.68 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\text {Too large to display} \] Input:
integrate(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="fricas")
Output:
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...
\[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\frac {\int \frac {1}{c^{2} \coth ^{3}{\left (e + f x \right )} + 3 c^{2} \coth ^{2}{\left (e + f x \right )} + 3 c^{2} \coth {\left (e + f x \right )} + c^{2} + 2 c d x \coth ^{3}{\left (e + f x \right )} + 6 c d x \coth ^{2}{\left (e + f x \right )} + 6 c d x \coth {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \coth ^{3}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth ^{2}{\left (e + f x \right )} + 3 d^{2} x^{2} \coth {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{3}} \] Input:
integrate(1/(d*x+c)**2/(a+a*coth(f*x+e))**3,x)
Output:
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 = 13.50 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.20 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=-\frac {1}{8 \, {\left (a^{3} d^{2} x + a^{3} c d\right )}} + \frac {e^{\left (-6 \, e + \frac {6 \, c f}{d}\right )} E_{2}\left (\frac {6 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} - \frac {3 \, e^{\left (-4 \, e + \frac {4 \, c f}{d}\right )} E_{2}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} + \frac {3 \, e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{8 \, {\left (d x + c\right )} a^{3} d} \] Input:
integrate(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="maxima")
Output:
-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 = 0.16 (sec) , antiderivative size = 841, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x, algorithm="giac")
Output:
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. \[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {coth}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \] Input:
int(1/((a + a*coth(e + f*x))^3*(c + d*x)^2),x)
Output:
int(1/((a + a*coth(e + f*x))^3*(c + d*x)^2), x)
\[ \int \frac {1}{(c+d x)^2 (a+a \coth (e+f x))^3} \, dx=\frac {-3 e^{4 e} \left (\int \frac {1}{e^{2 f x +6 e} c^{2}+2 e^{2 f x +6 e} c d x +e^{2 f x +6 e} d^{2} x^{2}}d x \right ) c^{2}-3 e^{4 e} \left (\int \frac {1}{e^{2 f x +6 e} c^{2}+2 e^{2 f x +6 e} c d x +e^{2 f x +6 e} d^{2} x^{2}}d x \right ) c d x +3 e^{2 e} \left (\int \frac {1}{e^{4 f x +6 e} c^{2}+2 e^{4 f x +6 e} c d x +e^{4 f x +6 e} d^{2} x^{2}}d x \right ) c^{2}+3 e^{2 e} \left (\int \frac {1}{e^{4 f x +6 e} c^{2}+2 e^{4 f x +6 e} c d x +e^{4 f x +6 e} d^{2} x^{2}}d x \right ) c d x -\left (\int \frac {1}{e^{6 f x +6 e} c^{2}+2 e^{6 f x +6 e} c d x +e^{6 f x +6 e} d^{2} x^{2}}d x \right ) c^{2}-\left (\int \frac {1}{e^{6 f x +6 e} c^{2}+2 e^{6 f x +6 e} c d x +e^{6 f x +6 e} d^{2} x^{2}}d x \right ) c d x +x}{8 a^{3} c \left (d x +c \right )} \] Input:
int(1/(d*x+c)^2/(a+a*coth(f*x+e))^3,x)
Output:
( - 3*e**(4*e)*int(1/(e**(6*e + 2*f*x)*c**2 + 2*e**(6*e + 2*f*x)*c*d*x + e **(6*e + 2*f*x)*d**2*x**2),x)*c**2 - 3*e**(4*e)*int(1/(e**(6*e + 2*f*x)*c* *2 + 2*e**(6*e + 2*f*x)*c*d*x + e**(6*e + 2*f*x)*d**2*x**2),x)*c*d*x + 3*e **(2*e)*int(1/(e**(6*e + 4*f*x)*c**2 + 2*e**(6*e + 4*f*x)*c*d*x + e**(6*e + 4*f*x)*d**2*x**2),x)*c**2 + 3*e**(2*e)*int(1/(e**(6*e + 4*f*x)*c**2 + 2* e**(6*e + 4*f*x)*c*d*x + e**(6*e + 4*f*x)*d**2*x**2),x)*c*d*x - int(1/(e** (6*e + 6*f*x)*c**2 + 2*e**(6*e + 6*f*x)*c*d*x + e**(6*e + 6*f*x)*d**2*x**2 ),x)*c**2 - int(1/(e**(6*e + 6*f*x)*c**2 + 2*e**(6*e + 6*f*x)*c*d*x + e**( 6*e + 6*f*x)*d**2*x**2),x)*c*d*x + x)/(8*a**3*c*(c + d*x))