Integrand size = 20, antiderivative size = 211 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=-\frac {f}{2 a d^2 (c+d x)}+\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Chi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{a d^3}-\frac {f^2 \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}+\frac {f^2 \sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+a \tanh (e+f x))}+\frac {f}{d^2 (c+d x) (a+a \tanh (e+f x))} \] Output:
-1/2*f/a/d^2/(d*x+c)+f^2*cosh(-2*e+2*c*f/d)*Chi(2*c*f/d+2*f*x)/a/d^3+f^2*C hi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/a/d^3-f^2*cosh(-2*e+2*c*f/d)*Shi(2*c* f/d+2*f*x)/a/d^3-f^2*sinh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*f*x)/a/d^3-1/2/d/(d* x+c)^2/(a+a*tanh(f*x+e))+f/d^2/(d*x+c)/(a+a*tanh(f*x+e))
Time = 0.78 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=-\frac {\text {sech}(e+f x) \left (\cosh \left (\frac {c f}{d}\right )+\sinh \left (\frac {c f}{d}\right )\right ) \left (d \left (d \cosh \left (e+f \left (-\frac {c}{d}+x\right )\right )+(d-2 c f-2 d f x) \cosh \left (e+f \left (\frac {c}{d}+x\right )\right )+d \sinh \left (e+f \left (-\frac {c}{d}+x\right )\right )-d \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )+2 c f \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )+2 d f x \sinh \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+4 f^2 (c+d x)^2 \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \left (-\cosh \left (e-\frac {f (c+d x)}{d}\right )+\sinh \left (e-\frac {f (c+d x)}{d}\right )\right )+4 f^2 (c+d x)^2 \left (\cosh \left (e-\frac {f (c+d x)}{d}\right )-\sinh \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{4 a d^3 (c+d x)^2 (1+\tanh (e+f x))} \] Input:
Integrate[1/((c + d*x)^3*(a + a*Tanh[e + f*x])),x]
Output:
-1/4*(Sech[e + f*x]*(Cosh[(c*f)/d] + Sinh[(c*f)/d])*(d*(d*Cosh[e + f*(-(c/ d) + x)] + (d - 2*c*f - 2*d*f*x)*Cosh[e + f*(c/d + x)] + d*Sinh[e + f*(-(c /d) + x)] - d*Sinh[e + f*(c/d + x)] + 2*c*f*Sinh[e + f*(c/d + x)] + 2*d*f* x*Sinh[e + f*(c/d + x)]) + 4*f^2*(c + d*x)^2*CoshIntegral[(2*f*(c + d*x))/ d]*(-Cosh[e - (f*(c + d*x))/d] + Sinh[e - (f*(c + d*x))/d]) + 4*f^2*(c + d *x)^2*(Cosh[e - (f*(c + d*x))/d] - Sinh[e - (f*(c + d*x))/d])*SinhIntegral [(2*f*(c + d*x))/d]))/(a*d^3*(c + d*x)^2*(1 + Tanh[e + f*x]))
Result contains complex when optimal does not.
Time = 1.07 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {3042, 4208, 3042, 4207, 26, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {1}{(c+d x)^3 (a \tanh (e+f x)+a)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c+d x)^3 (a-i a \tan (i e+i f x))}dx\) |
| \(\Big \downarrow \) 4208 |
| \(\displaystyle -\frac {f \int \frac {1}{(c+d x)^2 (\tanh (e+f x) a+a)}dx}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {f \int \frac {1}{(c+d x)^2 (a-i a \tan (i e+i f x))}dx}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 4207 |
| \(\displaystyle -\frac {f \left (-\frac {i f \int \frac {i \sinh (2 e+2 f x)}{c+d x}dx}{a d}-\frac {f \int \frac {\cosh (2 e+2 f x)}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {f \left (\frac {f \int \frac {\sinh (2 e+2 f x)}{c+d x}dx}{a d}-\frac {f \int \frac {\cosh (2 e+2 f x)}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {f \left (\frac {f \int -\frac {i \sin (2 i e+2 i f x)}{c+d x}dx}{a d}-\frac {f \int \frac {\sin \left (2 i e+2 i f x+\frac {\pi }{2}\right )}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {f \left (-\frac {i f \int \frac {\sin (2 i e+2 i f x)}{c+d x}dx}{a d}-\frac {f \int \frac {\sin \left (2 i e+2 i f x+\frac {\pi }{2}\right )}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle -\frac {f \left (-\frac {i f \left (i \sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx+\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {i \sinh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {f \left (\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx-i \sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {i \sinh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {f \left (-\frac {i f \left (i \sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx+i \cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sinh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {f \left (\sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sinh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx+\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cosh \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {f \left (-\frac {f \left (\sinh \left (2 e-\frac {2 c f}{d}\right ) \int -\frac {i \sin \left (2 i x f+\frac {2 i c f}{d}\right )}{c+d x}dx+\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx\right )}{a d}-\frac {i f \left (i \sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx+i \cosh \left (2 e-\frac {2 c f}{d}\right ) \int -\frac {i \sin \left (2 i x f+\frac {2 i c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {f \left (-\frac {f \left (\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx-i \sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {i f \left (i \sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx+\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle -\frac {f \left (-\frac {f \left (\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d}+\cosh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx\right )}{a d}-\frac {i f \left (i \sinh \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 i x f+\frac {2 i c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx+\frac {i \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle -\frac {f \left (-\frac {i f \left (\frac {i \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d}+\frac {i \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {f \left (\frac {\text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \cosh \left (2 e-\frac {2 c f}{d}\right )}{d}+\frac {\sinh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {1}{d (c+d x) (a \tanh (e+f x)+a)}\right )}{d}-\frac {f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a \tanh (e+f x)+a)}\) |
Input:
Int[1/((c + d*x)^3*(a + a*Tanh[e + f*x])),x]
Output:
-1/2*f/(a*d^2*(c + d*x)) - 1/(2*d*(c + d*x)^2*(a + a*Tanh[e + f*x])) - (f* (((-I)*f*((I*CoshIntegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d + (I *Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d))/(a*d) - (f*((C osh[2*e - (2*c*f)/d]*CoshIntegral[(2*c*f)/d + 2*f*x])/d + (Sinh[2*e - (2*c *f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d))/(a*d) - 1/(d*(c + d*x)*(a + a* Tanh[e + f*x]))))/d
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> 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]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> 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]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
Int[1/(((c_.) + (d_.)*(x_))^2*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])), x_Sy mbol] :> -Simp[(d*(c + d*x)*(a + b*Tan[e + f*x]))^(-1), x] + (-Simp[f/(a*d) Int[Sin[2*e + 2*f*x]/(c + d*x), x], x] + Simp[f/(b*d) Int[Cos[2*e + 2* f*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0 ]
Int[((c_.) + (d_.)*(x_))^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sym bol] :> Simp[f*((c + d*x)^(m + 2)/(b*d^2*(m + 1)*(m + 2))), x] + (Simp[2*b* (f/(a*d*(m + 1))) Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x], x] + Si mp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[m, -1] && NeQ[m, -2]
Time = 0.73 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(-\frac {1}{4 a d \left (d x +c \right )^{2}}+\frac {f^{3} {\mathrm e}^{-2 f x -2 e} x}{2 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}+\frac {f^{3} {\mathrm e}^{-2 f x -2 e} c}{2 a \,d^{2} \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} {\mathrm e}^{-2 f x -2 e}}{4 a d \left (d^{2} x^{2} f^{2}+2 c d \,f^{2} x +c^{2} f^{2}\right )}-\frac {f^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {expIntegral}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{a \,d^{3}}\) | \(211\) |
Input:
int(1/(d*x+c)^3/(a+tanh(f*x+e)*a),x,method=_RETURNVERBOSE)
Output:
-1/4/a/d/(d*x+c)^2+1/2*f^3/a*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^ 2*f^2)*x+1/2*f^3/a*exp(-2*f*x-2*e)/d^2/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)*c -1/4*f^2/a*exp(-2*f*x-2*e)/d/(d^2*f^2*x^2+2*c*d*f^2*x+c^2*f^2)-f^2/a/d^3*e xp(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 = 342, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\frac {2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (f x + e\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + {\left (d^{2} f x + c d f + 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - d^{2}\right )} \cosh \left (f x + e\right ) - {\left (d^{2} f x + c d f - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - 2 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )\right )} \sinh \left (f x + e\right )}{2 \, {\left ({\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \cosh \left (f x + e\right ) + {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )} \sinh \left (f x + e\right )\right )}} \] Input:
integrate(1/(d*x+c)^3/(a+a*tanh(f*x+e)),x, algorithm="fricas")
Output:
1/2*(2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(f *x + e)*sinh(-2*(d*e - c*f)/d) + (d^2*f*x + c*d*f + 2*(d^2*f^2*x^2 + 2*c*d *f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) - d^2)*cos h(f*x + e) - (d^2*f*x + c*d*f - 2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei (-2*(d*f*x + c*f)/d)*cosh(-2*(d*e - c*f)/d) - 2*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(d*f*x + c*f)/d)*sinh(-2*(d*e - c*f)/d))*sinh(f*x + e))/ ((a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)*cosh(f*x + e) + (a*d^5*x^2 + 2*a*c* d^4*x + a*c^2*d^3)*sinh(f*x + e))
\[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\frac {\int \frac {1}{c^{3} \tanh {\left (e + f x \right )} + c^{3} + 3 c^{2} d x \tanh {\left (e + f x \right )} + 3 c^{2} d x + 3 c d^{2} x^{2} \tanh {\left (e + f x \right )} + 3 c d^{2} x^{2} + d^{3} x^{3} \tanh {\left (e + f x \right )} + d^{3} x^{3}}\, dx}{a} \] Input:
integrate(1/(d*x+c)**3/(a+a*tanh(f*x+e)),x)
Output:
Integral(1/(c**3*tanh(e + f*x) + c**3 + 3*c**2*d*x*tanh(e + f*x) + 3*c**2* d*x + 3*c*d**2*x**2*tanh(e + f*x) + 3*c*d**2*x**2 + d**3*x**3*tanh(e + f*x ) + d**3*x**3), x)/a
Time = 0.57 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.32 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=-\frac {1}{4 \, {\left (a d^{3} x^{2} + 2 \, a c d^{2} x + a c^{2} d\right )}} - \frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{2 \, {\left (d x + c\right )}^{2} a d} \] Input:
integrate(1/(d*x+c)^3/(a+a*tanh(f*x+e)),x, algorithm="maxima")
Output:
-1/4/(a*d^3*x^2 + 2*a*c*d^2*x + a*c^2*d) - 1/2*e^(-2*e + 2*c*f/d)*exp_inte gral_e(3, 2*(d*x + c)*f/d)/((d*x + c)^2*a*d)
Time = 0.13 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\frac {4 \, d^{2} f^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 8 \, c d f^{2} x {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 4 \, c^{2} f^{2} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, c f}{d}\right )} + 2 \, d^{2} f x e^{\left (-2 \, f x\right )} + 2 \, c d f e^{\left (-2 \, f x\right )} - d^{2} e^{\left (-2 \, f x\right )} - d^{2} e^{\left (2 \, e\right )}}{4 \, {\left (a d^{5} x^{2} e^{\left (2 \, e\right )} + 2 \, a c d^{4} x e^{\left (2 \, e\right )} + a c^{2} d^{3} e^{\left (2 \, e\right )}\right )}} \] Input:
integrate(1/(d*x+c)^3/(a+a*tanh(f*x+e)),x, algorithm="giac")
Output:
1/4*(4*d^2*f^2*x^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d) + 8*c*d*f^2*x*Ei(-2* (d*f*x + c*f)/d)*e^(2*c*f/d) + 4*c^2*f^2*Ei(-2*(d*f*x + c*f)/d)*e^(2*c*f/d ) + 2*d^2*f*x*e^(-2*f*x) + 2*c*d*f*e^(-2*f*x) - d^2*e^(-2*f*x) - d^2*e^(2* e))/(a*d^5*x^2*e^(2*e) + 2*a*c*d^4*x*e^(2*e) + a*c^2*d^3*e^(2*e))
Timed out. \[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {tanh}\left (e+f\,x\right )\right )\,{\left (c+d\,x\right )}^3} \,d x \] Input:
int(1/((a + a*tanh(e + f*x))*(c + d*x)^3),x)
Output:
int(1/((a + a*tanh(e + f*x))*(c + d*x)^3), x)
\[ \int \frac {1}{(c+d x)^3 (a+a \tanh (e+f x))} \, dx=\frac {2 \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) c^{2} d +4 \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) c \,d^{2} x +2 \left (\int \frac {1}{e^{2 f x +2 e} c^{3}+3 e^{2 f x +2 e} c^{2} d x +3 e^{2 f x +2 e} c \,d^{2} x^{2}+e^{2 f x +2 e} d^{3} x^{3}}d x \right ) d^{3} x^{2}-1}{4 a d \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int(1/(d*x+c)^3/(a+a*tanh(f*x+e)),x)
Output:
(2*int(1/(e**(2*e + 2*f*x)*c**3 + 3*e**(2*e + 2*f*x)*c**2*d*x + 3*e**(2*e + 2*f*x)*c*d**2*x**2 + e**(2*e + 2*f*x)*d**3*x**3),x)*c**2*d + 4*int(1/(e* *(2*e + 2*f*x)*c**3 + 3*e**(2*e + 2*f*x)*c**2*d*x + 3*e**(2*e + 2*f*x)*c*d **2*x**2 + e**(2*e + 2*f*x)*d**3*x**3),x)*c*d**2*x + 2*int(1/(e**(2*e + 2* f*x)*c**3 + 3*e**(2*e + 2*f*x)*c**2*d*x + 3*e**(2*e + 2*f*x)*c*d**2*x**2 + e**(2*e + 2*f*x)*d**3*x**3),x)*d**3*x**2 - 1)/(4*a*d*(c**2 + 2*c*d*x + d* *2*x**2))