Integrand size = 20, antiderivative size = 420 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))^2} \, dx=-\frac {1}{4 a^2 d (c+d x)}-\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Chi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}+\frac {f \text {Chi}\left (\frac {4 c f}{d}+4 f x\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}-\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a^2 d^2}+\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2}-\frac {f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (\frac {4 c f}{d}+4 f x\right )}{a^2 d^2} \]
-1/4/a^2/d/(d*x+c)-f*Chi(4*c*f/d+4*f*x)*cosh(-4*e+4*c*f/d)/a^2/d^2-f*Chi(2 *c*f/d+2*f*x)*cosh(-2*e+2*c*f/d)/a^2/d^2-1/2*cosh(2*f*x+2*e)/a^2/d/(d*x+c) -1/4*cosh(2*f*x+2*e)^2/a^2/d/(d*x+c)+f*cosh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*f* x)/a^2/d^2+f*cosh(-4*e+4*c*f/d)*Shi(4*c*f/d+4*f*x)/a^2/d^2-f*Chi(4*c*f/d+4 *f*x)*sinh(-4*e+4*c*f/d)/a^2/d^2+f*Shi(4*c*f/d+4*f*x)*sinh(-4*e+4*c*f/d)/a ^2/d^2-f*Chi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/a^2/d^2+f*Shi(2*c*f/d+2*f*x )*sinh(-2*e+2*c*f/d)/a^2/d^2+1/2*sinh(2*f*x+2*e)/a^2/d/(d*x+c)-1/4*sinh(2* f*x+2*e)^2/a^2/d/(d*x+c)+1/4*sinh(4*f*x+4*e)/a^2/d/(d*x+c)
Time = 1.48 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.05 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))^2} \, dx=\frac {\left (-\cosh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )+\sinh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )\right ) \left (2 d \cosh \left (\frac {2 c f}{d}\right )+d \cosh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )+d \cosh \left (2 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )-2 d \sinh \left (\frac {2 c f}{d}\right )+4 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) (\cosh (2 f x)+\sinh (2 f x))+d \sinh \left (2 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )-d \sinh \left (2 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+4 f (c+d x) \text {Chi}\left (\frac {4 f (c+d x)}{d}\right ) \left (\cosh \left (2 e-\frac {2 f (c+d x)}{d}\right )-\sinh \left (2 e-\frac {2 f (c+d x)}{d}\right )\right )-4 c f \cosh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-4 d f x \cosh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-4 c f \sinh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-4 d f x \sinh (2 f x) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-4 c f \cosh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )-4 d f x \cosh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )+4 c f \sinh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )+4 d f x \sinh \left (2 e-\frac {2 f (c+d x)}{d}\right ) \text {Shi}\left (\frac {4 f (c+d x)}{d}\right )\right )}{4 a^2 d^2 (c+d x)} \]
((-Cosh[2*(e + f*(-(c/d) + x))] + Sinh[2*(e + f*(-(c/d) + x))])*(2*d*Cosh[ (2*c*f)/d] + d*Cosh[2*(e + f*(-(c/d) + x))] + d*Cosh[2*(e + f*(c/d + x))] - 2*d*Sinh[(2*c*f)/d] + 4*f*(c + d*x)*CoshIntegral[(2*f*(c + d*x))/d]*(Cos h[2*f*x] + Sinh[2*f*x]) + d*Sinh[2*(e + f*(-(c/d) + x))] - d*Sinh[2*(e + f *(c/d + x))] + 4*f*(c + d*x)*CoshIntegral[(4*f*(c + d*x))/d]*(Cosh[2*e - ( 2*f*(c + d*x))/d] - Sinh[2*e - (2*f*(c + d*x))/d]) - 4*c*f*Cosh[2*f*x]*Sin hIntegral[(2*f*(c + d*x))/d] - 4*d*f*x*Cosh[2*f*x]*SinhIntegral[(2*f*(c + d*x))/d] - 4*c*f*Sinh[2*f*x]*SinhIntegral[(2*f*(c + d*x))/d] - 4*d*f*x*Sin h[2*f*x]*SinhIntegral[(2*f*(c + d*x))/d] - 4*c*f*Cosh[2*e - (2*f*(c + d*x) )/d]*SinhIntegral[(4*f*(c + d*x))/d] - 4*d*f*x*Cosh[2*e - (2*f*(c + d*x))/ d]*SinhIntegral[(4*f*(c + d*x))/d] + 4*c*f*Sinh[2*e - (2*f*(c + d*x))/d]*S inhIntegral[(4*f*(c + d*x))/d] + 4*d*f*x*Sinh[2*e - (2*f*(c + d*x))/d]*Sin hIntegral[(4*f*(c + d*x))/d]))/(4*a^2*d^2*(c + d*x))
Time = 1.14 (sec) , antiderivative size = 420, 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.
| \(\displaystyle \int \frac {1}{(c+d x)^2 (a \tanh (e+f x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c+d x)^2 (a-i a \tan (i e+i f x))^2}dx\) |
| \(\Big \downarrow \) 4211 |
| \(\displaystyle \int \left (\frac {\sinh ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}-\frac {\sinh (2 e+2 f x)}{2 a^2 (c+d x)^2}-\frac {\sinh (4 e+4 f x)}{4 a^2 (c+d x)^2}+\frac {\cosh ^2(2 e+2 f x)}{4 a^2 (c+d x)^2}+\frac {\cosh (2 e+2 f x)}{2 a^2 (c+d x)^2}+\frac {1}{4 a^2 (c+d x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \sinh \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {f \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {f \text {Chi}\left (4 x f+\frac {4 c f}{d}\right ) \cosh \left (4 e-\frac {4 c f}{d}\right )}{a^2 d^2}-\frac {f \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a^2 d^2}-\frac {f \sinh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{a^2 d^2}+\frac {f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{a^2 d^2}+\frac {f \cosh \left (4 e-\frac {4 c f}{d}\right ) \text {Shi}\left (4 x f+\frac {4 c f}{d}\right )}{a^2 d^2}-\frac {\sinh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}+\frac {\sinh (2 e+2 f x)}{2 a^2 d (c+d x)}+\frac {\sinh (4 e+4 f x)}{4 a^2 d (c+d x)}-\frac {\cosh ^2(2 e+2 f x)}{4 a^2 d (c+d x)}-\frac {\cosh (2 e+2 f x)}{2 a^2 d (c+d x)}-\frac {1}{4 a^2 d (c+d x)}\) |
-1/4*1/(a^2*d*(c + d*x)) - Cosh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Cosh[2* e + 2*f*x]^2/(4*a^2*d*(c + d*x)) - (f*Cosh[2*e - (2*c*f)/d]*CoshIntegral[( 2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Cosh[4*e - (4*c*f)/d]*CoshIntegral[(4*c* f)/d + 4*f*x])/(a^2*d^2) + (f*CoshIntegral[(4*c*f)/d + 4*f*x]*Sinh[4*e - ( 4*c*f)/d])/(a^2*d^2) + (f*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c* f)/d])/(a^2*d^2) + Sinh[2*e + 2*f*x]/(2*a^2*d*(c + d*x)) - Sinh[2*e + 2*f* x]^2/(4*a^2*d*(c + d*x)) + Sinh[4*e + 4*f*x]/(4*a^2*d*(c + d*x)) + (f*Cosh [2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) - (f*Sinh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/(a^2*d^2) + (f*Cosh[4*e - ( 4*c*f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2) - (f*Sinh[4*e - (4*c* f)/d]*SinhIntegral[(4*c*f)/d + 4*f*x])/(a^2*d^2)
3.1.42.3.1 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 = 0.43 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(-\frac {1}{4 a^{2} d \left (d x +c \right )}-\frac {f \,{\mathrm e}^{-4 f x -4 e}}{4 a^{2} d \left (d x f +c f \right )}+\frac {f \,{\mathrm e}^{\frac {4 c f -4 d e}{d}} \operatorname {Ei}_{1}\left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right )}{a^{2} d^{2}}-\frac {f \,{\mathrm e}^{-2 f x -2 e}}{2 a^{2} d \left (d x f +c f \right )}+\frac {f \,{\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {Ei}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a^{2} d^{2}}\) | \(163\) |
-1/4/a^2/d/(d*x+c)-1/4*f/a^2*exp(-4*f*x-4*e)/d/(d*f*x+c*f)+f/a^2/d^2*exp(4 *(c*f-d*e)/d)*Ei(1,4*f*x+4*e+4*(c*f-d*e)/d)-1/2*f/a^2*exp(-2*f*x-2*e)/d/(d *f*x+c*f)+f/a^2/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 = 609, normalized size of antiderivative = 1.45 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))^2} \, dx=-\frac {2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right )^{2} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right )^{2} \sinh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + {\left (2 \, {\left (d f x + c f\right )} {\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 ) + 2 \, {\left (d f x + c f\right )} {\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 ) + d\right )} \cosh \left (f x + e\right )^{2} + {\left (2 \, {\left (d f x + c f\right )} {\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 ) + 2 \, {\left (d f x + c f\right )} {\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 ) + 2 \, {\left (d f x + c f\right )} {\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 ) + 2 \, {\left (d f x + c f\right )} {\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 ) + d\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left ({\left (d f x + c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (d f x + c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + {\left ({\left (d f x + c f\right )} {\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 ) + {\left (d f x + c f\right )} {\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 )\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + d}{2 \, {\left ({\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + {\left (a^{2} d^{3} x + a^{2} c d^{2}\right )} \sinh \left (f x + e\right )^{2}\right )}} \]
-1/2*(2*(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) + 2*(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) + (2*(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 )*cosh(f*x + e)^2 + (2*(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) + 2*(d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*sinh(-2*(d*e - c*f)/d) + 2*(d*f*x + c*f)*Ei(-4*(d*f*x + c*f)/d)*sinh(-4*(d*e - c*f)/d) + d)*sinh(f*x + e)^2 + 4*((d*f*x + c*f)*Ei(-2*(d*f*x + c*f)/d)*cosh(f*x + e)*sinh(-2*(d*e - c*f )/d) + (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(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) + (d *f*x + c*f)*Ei(-4*(d*f*x + c*f)/d)*cosh(-4*(d*e - c*f)/d))*cosh(f*x + e))* sinh(f*x + e) + d)/((a^2*d^3*x + a^2*c*d^2)*cosh(f*x + e)^2 + 2*(a^2*d^3*x + a^2*c*d^2)*cosh(f*x + e)*sinh(f*x + e) + (a^2*d^3*x + a^2*c*d^2)*sinh(f *x + e)^2)
\[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))^2} \, dx=\frac {\int \frac {1}{c^{2} \tanh ^{2}{\left (e + f x \right )} + 2 c^{2} \tanh {\left (e + f x \right )} + c^{2} + 2 c d x \tanh ^{2}{\left (e + f x \right )} + 4 c d x \tanh {\left (e + f x \right )} + 2 c d x + d^{2} x^{2} \tanh ^{2}{\left (e + f x \right )} + 2 d^{2} x^{2} \tanh {\left (e + f x \right )} + d^{2} x^{2}}\, dx}{a^{2}} \]
Integral(1/(c**2*tanh(e + f*x)**2 + 2*c**2*tanh(e + f*x) + c**2 + 2*c*d*x* tanh(e + f*x)**2 + 4*c*d*x*tanh(e + f*x) + 2*c*d*x + d**2*x**2*tanh(e + f* x)**2 + 2*d**2*x**2*tanh(e + f*x) + d**2*x**2), x)/a**2
Time = 1.35 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.24 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))^2} \, dx=-\frac {1}{4 \, {\left (a^{2} d^{2} x + a^{2} c d\right )}} - \frac {e^{\left (-4 \, e + \frac {4 \, c f}{d}\right )} E_{2}\left (\frac {4 \, {\left (d x + c\right )} f}{d}\right )}{4 \, {\left (d x + c\right )} a^{2} d} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )} a^{2} d} \]
-1/4/(a^2*d^2*x + a^2*c*d) - 1/4*e^(-4*e + 4*c*f/d)*exp_integral_e(2, 4*(d *x + c)*f/d)/((d*x + c)*a^2*d) - 1/2*e^(-2*e + 2*c*f/d)*exp_integral_e(2, 2*(d*x + c)*f/d)/((d*x + c)*a^2*d)
Time = 0.31 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.39 \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))^2} \, dx=-\frac {{\left (4 \, {\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 )} - 4 \, 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 )} + 4 \, 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 )} + 4 \, {\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 )} - 4 \, 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 )} + 4 \, 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 )} + 2 \, 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 )} + 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}\right )} d^{2}}{4 \, {\left ({\left (d x + c\right )} a^{2} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - a^{2} d^{5} e + a^{2} c d^{4} f\right )} f} \]
-1/4*(4*(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) - 4*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) + 4*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) + 4*(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) - 4*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) + 4*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) + 2*d*f^2*e^(-2*(d*x + c)*(d*e/(d*x + c) - c *f/(d*x + c) + f)/d) + d*f^2*e^(-4*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c ) + f)/d) + d*f^2)*d^2/(((d*x + c)*a^2*d^4*(d*e/(d*x + c) - c*f/(d*x + c) + f) - a^2*d^5*e + a^2*c*d^4*f)*f)
Timed out. \[ \int \frac {1}{(c+d x)^2 (a+a \tanh (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \]