\(\int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx\) [113]

Optimal result

Integrand size = 23, antiderivative size = 305 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\frac {(c+d x)^3}{3 a^2 f}-\frac {2 d (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{a^2 f^2}+\frac {4 d^3 \log \left (\cosh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )\right )}{a^2 f^4}-\frac {4 d^2 (c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{a^2 f^3}+\frac {4 d^3 \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{a^2 f^4}+\frac {d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{2 a^2 f^2}-\frac {2 d^2 (c+d x) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{a^2 f^3}+\frac {(c+d x)^3 \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{3 a^2 f}+\frac {(c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \tanh \left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{6 a^2 f} \] Output:

1/3*(d*x+c)^3/a^2/f-2*d*(d*x+c)^2*ln(1+I*exp(f*x+e))/a^2/f^2+4*d^3*ln(cosh 
(1/2*e+1/4*I*Pi+1/2*f*x))/a^2/f^4-4*d^2*(d*x+c)*polylog(2,-I*exp(f*x+e))/a 
^2/f^3+4*d^3*polylog(3,-I*exp(f*x+e))/a^2/f^4+1/2*d*(d*x+c)^2*sech(1/2*e+1 
/4*I*Pi+1/2*f*x)^2/a^2/f^2-2*d^2*(d*x+c)*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/a^2/ 
f^3+1/3*(d*x+c)^3*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/a^2/f+1/6*(d*x+c)^3*sech(1/ 
2*e+1/4*I*Pi+1/2*f*x)^2*tanh(1/2*e+1/4*I*Pi+1/2*f*x)/a^2/f
 

Mathematica [A] (verified)

Time = 2.66 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.55 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\frac {\frac {2 d \left (-6 d^2 x+3 c^2 f^2 x+3 \left (1+i e^e\right ) \left (2 d^2-c^2 f^2\right ) x+3 c d f^2 x^2+d^2 f^2 x^3+6 c d \left (1+i e^e\right ) f x \log \left (1-i e^{-e-f x}\right )+3 d^2 \left (1+i e^e\right ) f x^2 \log \left (1-i e^{-e-f x}\right )+\frac {3 \left (1+i e^e\right ) \left (-2 d^2+c^2 f^2\right ) \log \left (i-e^{e+f x}\right )}{f}-6 c d \left (1+i e^e\right ) \operatorname {PolyLog}\left (2,i e^{-e-f x}\right )-6 d^2 \left (1+i e^e\right ) x \operatorname {PolyLog}\left (2,i e^{-e-f x}\right )-\frac {6 d^2 \left (1+i e^e\right ) \operatorname {PolyLog}\left (3,i e^{-e-f x}\right )}{f}\right )}{-1-i e^e}+\frac {(c+d x) \left (3 d f (c+d x) \cosh \left (\frac {f x}{2}\right )+6 i d^2 \cosh \left (e+\frac {f x}{2}\right )+i \left (c^2 f^2+2 c d f^2 x+d^2 \left (-6+f^2 x^2\right )\right ) \cosh \left (e+\frac {3 f x}{2}\right )+3 \left (c^2 f^2+2 c d f^2 x+d^2 \left (-4+f^2 x^2\right )\right ) \sinh \left (\frac {f x}{2}\right )+3 i d f (c+d x) \sinh \left (e+\frac {f x}{2}\right )\right )}{\left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^3}}{3 a^2 f^3} \] Input:

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

Output:

((2*d*(-6*d^2*x + 3*c^2*f^2*x + 3*(1 + I*E^e)*(2*d^2 - c^2*f^2)*x + 3*c*d* 
f^2*x^2 + d^2*f^2*x^3 + 6*c*d*(1 + I*E^e)*f*x*Log[1 - I*E^(-e - f*x)] + 3* 
d^2*(1 + I*E^e)*f*x^2*Log[1 - I*E^(-e - f*x)] + (3*(1 + I*E^e)*(-2*d^2 + c 
^2*f^2)*Log[I - E^(e + f*x)])/f - 6*c*d*(1 + I*E^e)*PolyLog[2, I*E^(-e - f 
*x)] - 6*d^2*(1 + I*E^e)*x*PolyLog[2, I*E^(-e - f*x)] - (6*d^2*(1 + I*E^e) 
*PolyLog[3, I*E^(-e - f*x)])/f))/(-1 - I*E^e) + ((c + d*x)*(3*d*f*(c + d*x 
)*Cosh[(f*x)/2] + (6*I)*d^2*Cosh[e + (f*x)/2] + I*(c^2*f^2 + 2*c*d*f^2*x + 
 d^2*(-6 + f^2*x^2))*Cosh[e + (3*f*x)/2] + 3*(c^2*f^2 + 2*c*d*f^2*x + d^2* 
(-4 + f^2*x^2))*Sinh[(f*x)/2] + (3*I)*d*f*(c + d*x)*Sinh[e + (f*x)/2]))/(( 
Cosh[e/2] + I*Sinh[e/2])*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^3))/(3* 
a^2*f^3)
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.01, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.696, Rules used = {3042, 3799, 3042, 4674, 3042, 4672, 26, 3042, 26, 3956, 4199, 26, 2620, 3011, 2720, 7143}

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.

\(\displaystyle \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(c+d x)^3}{(a+a \sin (i e+i f x))^2}dx\)

\(\Big \downarrow \) 3799

\(\displaystyle \frac {\int (c+d x)^3 \text {csch}^4\left (\frac {e}{2}+\frac {f x}{2}-\frac {i \pi }{4}\right )dx}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (c+d x)^3 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^4dx}{4 a^2}\)

\(\Big \downarrow \) 4674

\(\displaystyle \frac {-\frac {4 d^2 \int (c+d x) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f^2}+\frac {2}{3} \int (c+d x)^3 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {4 d^2 \int (c+d x) \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^2dx}{f^2}+\frac {2}{3} \int (c+d x)^3 \csc \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^2dx+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 4672

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 i d \int -i \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 i d \int -i (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 d \int \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 d \int (c+d x)^2 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )dx}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {2 d \int -i \tan \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {6 d \int -i (c+d x)^2 \tan \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 i d \int \tan \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )}{f^2}+\frac {2}{3} \left (\frac {6 i d \int (c+d x)^2 \tan \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3956

\(\displaystyle \frac {\frac {2}{3} \left (\frac {6 i d \int (c+d x)^2 \tan \left (\frac {i e}{2}+\frac {i f x}{2}-\frac {\pi }{4}\right )dx}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 4199

\(\displaystyle \frac {\frac {2}{3} \left (\frac {6 i d \left (2 i \int \frac {i e^{e+f x} (c+d x)^2}{1+i e^{e+f x}}dx-\frac {i (c+d x)^3}{3 d}\right )}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\frac {2}{3} \left (\frac {6 i d \left (-2 \int \frac {e^{e+f x} (c+d x)^2}{1+i e^{e+f x}}dx-\frac {i (c+d x)^3}{3 d}\right )}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {\frac {2}{3} \left (\frac {6 i d \left (-2 \left (\frac {2 i d \int (c+d x) \log \left (1+i e^{e+f x}\right )dx}{f}-\frac {i (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {\frac {2}{3} \left (\frac {6 i d \left (-2 \left (\frac {2 i d \left (\frac {d \int \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )dx}{f}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{f}\right )}{f}-\frac {i (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {\frac {2}{3} \left (\frac {6 i d \left (-2 \left (\frac {2 i d \left (\frac {d \int e^{-e-f x} \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )de^{e+f x}}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{f}\right )}{f}-\frac {i (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{f^2}\right )}{f^2}+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {4 d^2 \left (\frac {2 (c+d x) \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}-\frac {4 d \log \left (\cosh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )\right )}{f^2}\right )}{f^2}+\frac {2}{3} \left (\frac {6 i d \left (-2 \left (\frac {2 i d \left (\frac {d \operatorname {PolyLog}\left (3,-i e^{e+f x}\right )}{f^2}-\frac {(c+d x) \operatorname {PolyLog}\left (2,-i e^{e+f x}\right )}{f}\right )}{f}-\frac {i (c+d x)^2 \log \left (1+i e^{e+f x}\right )}{f}\right )-\frac {i (c+d x)^3}{3 d}\right )}{f}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f}\right )+\frac {2 d (c+d x)^2 \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{f^2}+\frac {2 (c+d x)^3 \tanh \left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \text {sech}^2\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{3 f}}{4 a^2}\)

Input:

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

Output:

((2*d*(c + d*x)^2*Sech[e/2 + (I/4)*Pi + (f*x)/2]^2)/f^2 + (2*(c + d*x)^3*S 
ech[e/2 + (I/4)*Pi + (f*x)/2]^2*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/(3*f) - (4 
*d^2*((-4*d*Log[Cosh[e/2 + (I/4)*Pi + (f*x)/2]])/f^2 + (2*(c + d*x)*Tanh[e 
/2 + (I/4)*Pi + (f*x)/2])/f))/f^2 + (2*(((6*I)*d*(((-1/3*I)*(c + d*x)^3)/d 
 - 2*(((-I)*(c + d*x)^2*Log[1 + I*E^(e + f*x)])/f + ((2*I)*d*(-(((c + d*x) 
*PolyLog[2, (-I)*E^(e + f*x)])/f) + (d*PolyLog[3, (-I)*E^(e + f*x)])/f^2)) 
/f)))/f + (2*(c + d*x)^3*Tanh[e/2 + (I/4)*Pi + (f*x)/2])/f))/3)/(4*a^2)
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Simp[(2*a)^n   Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) 
+ f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 
2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
 

rule 3956

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d 
*x], x]]/d, x] /; FreeQ[{c, d}, x]
 

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 4672

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1) 
*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 4674

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo 
l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n 
- 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 
2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) 
Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ 
(n - 1))   Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c 
, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 722 vs. \(2 (249 ) = 498\).

Time = 1.01 (sec) , antiderivative size = 723, normalized size of antiderivative = 2.37

method	result	size
risch	\(\frac {-2 i f^{2} c^{2} d x +4 i c \,d^{2}-4 i d^{3} x \,{\mathrm e}^{2 f x +2 e}-\frac {2 i f^{2} d^{3} x^{3}}{3}+6 f^{2} c \,d^{2} x^{2} {\mathrm e}^{f x +e}+6 f^{2} c^{2} d x \,{\mathrm e}^{f x +e}-4 f c \,d^{2} x \,{\mathrm e}^{f x +e}+4 i d^{3} x -2 i f \,d^{3} x^{2} {\mathrm e}^{2 f x +2 e}-4 i c \,d^{2} {\mathrm e}^{2 f x +2 e}+2 f^{2} d^{3} x^{3} {\mathrm e}^{f x +e}-2 f \,d^{3} x^{2} {\mathrm e}^{f x +e}-2 f \,c^{2} d \,{\mathrm e}^{f x +e}-2 i f^{2} c \,d^{2} x^{2}+2 f^{2} c^{3} {\mathrm e}^{f x +e}-\frac {2 i c^{3} f^{2}}{3}-4 i f c \,d^{2} x \,{\mathrm e}^{2 f x +2 e}-2 i f \,c^{2} d \,{\mathrm e}^{2 f x +2 e}-8 d^{3} x \,{\mathrm e}^{f x +e}-8 c \,d^{2} {\mathrm e}^{f x +e}}{\left ({\mathrm e}^{f x +e}-i\right )^{3} f^{3} a^{2}}+\frac {2 d^{3} x^{3}}{3 a^{2} f}+\frac {2 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) e^{2}}{a^{2} f^{4}}-\frac {2 d \ln \left ({\mathrm e}^{f x +e}-i\right ) c^{2}}{a^{2} f^{2}}+\frac {2 d \ln \left ({\mathrm e}^{f x +e}\right ) c^{2}}{a^{2} f^{2}}-\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}-i\right ) e^{2}}{a^{2} f^{4}}+\frac {2 d^{3} \ln \left ({\mathrm e}^{f x +e}\right ) e^{2}}{a^{2} f^{4}}-\frac {4 d^{3} e^{3}}{3 a^{2} f^{4}}+\frac {4 d^{3} \operatorname {polylog}\left (3, -i {\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}+\frac {4 d^{2} c e x}{a^{2} f^{2}}-\frac {4 d^{2} c \ln \left (1+i {\mathrm e}^{f x +e}\right ) x}{a^{2} f^{2}}-\frac {4 d^{2} c \ln \left (1+i {\mathrm e}^{f x +e}\right ) e}{a^{2} f^{3}}+\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}-i\right ) c e}{a^{2} f^{3}}-\frac {4 d^{2} \ln \left ({\mathrm e}^{f x +e}\right ) c e}{a^{2} f^{3}}-\frac {2 d^{3} e^{2} x}{a^{2} f^{3}}-\frac {2 d^{3} \ln \left (1+i {\mathrm e}^{f x +e}\right ) x^{2}}{a^{2} f^{2}}-\frac {4 d^{3} \operatorname {polylog}\left (2, -i {\mathrm e}^{f x +e}\right ) x}{a^{2} f^{3}}+\frac {2 d^{2} c \,x^{2}}{a^{2} f}+\frac {2 d^{2} c \,e^{2}}{a^{2} f^{3}}-\frac {4 d^{2} c \operatorname {polylog}\left (2, -i {\mathrm e}^{f x +e}\right )}{a^{2} f^{3}}+\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}-i\right )}{a^{2} f^{4}}-\frac {4 d^{3} \ln \left ({\mathrm e}^{f x +e}\right )}{a^{2} f^{4}}\)	\(723\)

Input:

int((d*x+c)^3/(a+I*a*sinh(f*x+e))^2,x,method=_RETURNVERBOSE)
 

Output:

2/3*(-3*I*f^2*c^2*d*x+6*I*c*d^2-6*I*d^3*x*exp(2*f*x+2*e)-I*f^2*d^3*x^3+9*f 
^2*c*d^2*x^2*exp(f*x+e)+9*f^2*c^2*d*x*exp(f*x+e)-6*f*c*d^2*x*exp(f*x+e)+6* 
I*d^3*x-3*I*f*d^3*x^2*exp(2*f*x+2*e)-6*I*c*d^2*exp(2*f*x+2*e)+3*f^2*d^3*x^ 
3*exp(f*x+e)-3*f*d^3*x^2*exp(f*x+e)-3*f*c^2*d*exp(f*x+e)-3*I*f^2*c*d^2*x^2 
+3*f^2*c^3*exp(f*x+e)-I*c^3*f^2-6*I*f*c*d^2*x*exp(2*f*x+2*e)-3*I*f*c^2*d*e 
xp(2*f*x+2*e)-12*d^3*x*exp(f*x+e)-12*c*d^2*exp(f*x+e))/(exp(f*x+e)-I)^3/f^ 
3/a^2+2/3/a^2/f*d^3*x^3+2/a^2/f^4*d^3*ln(1+I*exp(f*x+e))*e^2-2/a^2/f^2*d*l 
n(exp(f*x+e)-I)*c^2+2/a^2/f^2*d*ln(exp(f*x+e))*c^2-2/a^2/f^4*d^3*ln(exp(f* 
x+e)-I)*e^2+2/a^2/f^4*d^3*ln(exp(f*x+e))*e^2-4/3/a^2/f^4*d^3*e^3+4*d^3*pol 
ylog(3,-I*exp(f*x+e))/a^2/f^4+4/a^2/f^2*d^2*c*e*x-4/a^2/f^2*d^2*c*ln(1+I*e 
xp(f*x+e))*x-4/a^2/f^3*d^2*c*ln(1+I*exp(f*x+e))*e+4/a^2/f^3*d^2*ln(exp(f*x 
+e)-I)*c*e-4/a^2/f^3*d^2*ln(exp(f*x+e))*c*e-2/a^2/f^3*d^3*e^2*x-2/a^2/f^2* 
d^3*ln(1+I*exp(f*x+e))*x^2-4/a^2/f^3*d^3*polylog(2,-I*exp(f*x+e))*x+2/a^2/ 
f*d^2*c*x^2+2/a^2/f^3*d^2*c*e^2-4/a^2/f^3*d^2*c*polylog(2,-I*exp(f*x+e))+4 
/a^2/f^4*d^3*ln(exp(f*x+e)-I)-4/a^2/f^4*d^3*ln(exp(f*x+e))
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 917 vs. \(2 (239) = 478\).

Time = 0.10 (sec) , antiderivative size = 917, normalized size of antiderivative = 3.01 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-2/3*(-I*d^3*e^3 - 3*I*c^2*d*e*f^2 + I*c^3*f^3 + 6*I*d^3*e + 3*(I*c*d^2*e^ 
2 - 2*I*c*d^2)*f + 6*(I*d^3*f*x + I*c*d^2*f + (d^3*f*x + c*d^2*f)*e^(3*f*x 
 + 3*e) + 3*(-I*d^3*f*x - I*c*d^2*f)*e^(2*f*x + 2*e) - 3*(d^3*f*x + c*d^2* 
f)*e^(f*x + e))*dilog(-I*e^(f*x + e)) - (d^3*f^3*x^3 + 3*c*d^2*f^3*x^2 + d 
^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - 6*d^3*e + 3*(c^2*d*f^3 - 2*d^3*f) 
*x)*e^(3*f*x + 3*e) + 3*(I*d^3*f^3*x^3 + I*d^3*e^3 - 6*I*d^3*e + (3*I*c^2* 
d*e + I*c^2*d)*f^2 + (3*I*c*d^2*f^3 + I*d^3*f^2)*x^2 + (-3*I*c*d^2*e^2 + 2 
*I*c*d^2)*f + (3*I*c^2*d*f^3 + 2*I*c*d^2*f^2 - 4*I*d^3*f)*x)*e^(2*f*x + 2* 
e) + 3*(d^3*f^2*x^2 + d^3*e^3 - c^3*f^3 - 6*d^3*e + (3*c^2*d*e + c^2*d)*f^ 
2 - (3*c*d^2*e^2 - 4*c*d^2)*f + 2*(c*d^2*f^2 - d^3*f)*x)*e^(f*x + e) + 3*( 
I*d^3*e^2 - 2*I*c*d^2*e*f + I*c^2*d*f^2 - 2*I*d^3 + (d^3*e^2 - 2*c*d^2*e*f 
 + c^2*d*f^2 - 2*d^3)*e^(3*f*x + 3*e) + 3*(-I*d^3*e^2 + 2*I*c*d^2*e*f - I* 
c^2*d*f^2 + 2*I*d^3)*e^(2*f*x + 2*e) - 3*(d^3*e^2 - 2*c*d^2*e*f + c^2*d*f^ 
2 - 2*d^3)*e^(f*x + e))*log(e^(f*x + e) - I) + 3*(I*d^3*f^2*x^2 + 2*I*c*d^ 
2*f^2*x - I*d^3*e^2 + 2*I*c*d^2*e*f + (d^3*f^2*x^2 + 2*c*d^2*f^2*x - d^3*e 
^2 + 2*c*d^2*e*f)*e^(3*f*x + 3*e) + 3*(-I*d^3*f^2*x^2 - 2*I*c*d^2*f^2*x + 
I*d^3*e^2 - 2*I*c*d^2*e*f)*e^(2*f*x + 2*e) - 3*(d^3*f^2*x^2 + 2*c*d^2*f^2* 
x - d^3*e^2 + 2*c*d^2*e*f)*e^(f*x + e))*log(I*e^(f*x + e) + 1) - 6*(d^3*e^ 
(3*f*x + 3*e) - 3*I*d^3*e^(2*f*x + 2*e) - 3*d^3*e^(f*x + e) + I*d^3)*polyl 
og(3, -I*e^(f*x + e)))/(a^2*f^4*e^(3*f*x + 3*e) - 3*I*a^2*f^4*e^(2*f*x ...
 

Sympy [F]

\[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\frac {- 2 i c^{3} f^{2} - 6 i c^{2} d f^{2} x - 6 i c d^{2} f^{2} x^{2} + 12 i c d^{2} - 2 i d^{3} f^{2} x^{3} + 12 i d^{3} x + \left (- 6 i c^{2} d f e^{2 e} - 12 i c d^{2} f x e^{2 e} - 12 i c d^{2} e^{2 e} - 6 i d^{3} f x^{2} e^{2 e} - 12 i d^{3} x e^{2 e}\right ) e^{2 f x} + \left (6 c^{3} f^{2} e^{e} + 18 c^{2} d f^{2} x e^{e} - 6 c^{2} d f e^{e} + 18 c d^{2} f^{2} x^{2} e^{e} - 12 c d^{2} f x e^{e} - 24 c d^{2} e^{e} + 6 d^{3} f^{2} x^{3} e^{e} - 6 d^{3} f x^{2} e^{e} - 24 d^{3} x e^{e}\right ) e^{f x}}{3 a^{2} f^{3} e^{3 e} e^{3 f x} - 9 i a^{2} f^{3} e^{2 e} e^{2 f x} - 9 a^{2} f^{3} e^{e} e^{f x} + 3 i a^{2} f^{3}} - \frac {2 i d \left (\int \left (- \frac {2 d^{2}}{e^{e} e^{f x} - i}\right )\, dx + \int \frac {c^{2} f^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {d^{2} f^{2} x^{2}}{e^{e} e^{f x} - i}\, dx + \int \frac {2 c d f^{2} x}{e^{e} e^{f x} - i}\, dx\right )}{a^{2} f^{3}} \] Input:

integrate((d*x+c)**3/(a+I*a*sinh(f*x+e))**2,x)
 

Output:

(-2*I*c**3*f**2 - 6*I*c**2*d*f**2*x - 6*I*c*d**2*f**2*x**2 + 12*I*c*d**2 - 
 2*I*d**3*f**2*x**3 + 12*I*d**3*x + (-6*I*c**2*d*f*exp(2*e) - 12*I*c*d**2* 
f*x*exp(2*e) - 12*I*c*d**2*exp(2*e) - 6*I*d**3*f*x**2*exp(2*e) - 12*I*d**3 
*x*exp(2*e))*exp(2*f*x) + (6*c**3*f**2*exp(e) + 18*c**2*d*f**2*x*exp(e) - 
6*c**2*d*f*exp(e) + 18*c*d**2*f**2*x**2*exp(e) - 12*c*d**2*f*x*exp(e) - 24 
*c*d**2*exp(e) + 6*d**3*f**2*x**3*exp(e) - 6*d**3*f*x**2*exp(e) - 24*d**3* 
x*exp(e))*exp(f*x))/(3*a**2*f**3*exp(3*e)*exp(3*f*x) - 9*I*a**2*f**3*exp(2 
*e)*exp(2*f*x) - 9*a**2*f**3*exp(e)*exp(f*x) + 3*I*a**2*f**3) - 2*I*d*(Int 
egral(-2*d**2/(exp(e)*exp(f*x) - I), x) + Integral(c**2*f**2/(exp(e)*exp(f 
*x) - I), x) + Integral(d**2*f**2*x**2/(exp(e)*exp(f*x) - I), x) + Integra 
l(2*c*d*f**2*x/(exp(e)*exp(f*x) - I), x))/(a**2*f**3)
 

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (239) = 478\).

Time = 0.26 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.08 \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx =\text {Too large to display} \] Input:

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

Output:

2*c^2*d*((f*x*e^(3*f*x + 3*e) - (3*I*f*x*e^(2*e) + I*e^(2*e))*e^(2*f*x) - 
e^(f*x + e))/(a^2*f^2*e^(3*f*x + 3*e) - 3*I*a^2*f^2*e^(2*f*x + 2*e) - 3*a^ 
2*f^2*e^(f*x + e) + I*a^2*f^2) - log(-I*(I*e^(f*x + e) + 1)*e^(-e))/(a^2*f 
^2)) + 2/3*c^3*(3*e^(-f*x - e)/((3*a^2*e^(-f*x - e) - 3*I*a^2*e^(-2*f*x - 
2*e) - a^2*e^(-3*f*x - 3*e) + I*a^2)*f) + I/((3*a^2*e^(-f*x - e) - 3*I*a^2 
*e^(-2*f*x - 2*e) - a^2*e^(-3*f*x - 3*e) + I*a^2)*f)) - 2/3*(I*d^3*f^2*x^3 
 + 3*I*c*d^2*f^2*x^2 - 6*I*d^3*x - 6*I*c*d^2 - 3*(-I*d^3*f*x^2*e^(2*e) - 2 
*I*c*d^2*e^(2*e) + 2*(-I*c*d^2*f*e^(2*e) - I*d^3*e^(2*e))*x)*e^(2*f*x) - 3 
*(d^3*f^2*x^3*e^e - 4*c*d^2*e^e + (3*c*d^2*f^2*e^e - d^3*f*e^e)*x^2 - 2*(c 
*d^2*f*e^e + 2*d^3*e^e)*x)*e^(f*x))/(a^2*f^3*e^(3*f*x + 3*e) - 3*I*a^2*f^3 
*e^(2*f*x + 2*e) - 3*a^2*f^3*e^(f*x + e) + I*a^2*f^3) - 4*(f*x*log(I*e^(f* 
x + e) + 1) + dilog(-I*e^(f*x + e)))*c*d^2/(a^2*f^3) - 4*d^3*x/(a^2*f^3) - 
 2*(f^2*x^2*log(I*e^(f*x + e) + 1) + 2*f*x*dilog(-I*e^(f*x + e)) - 2*polyl 
og(3, -I*e^(f*x + e)))*d^3/(a^2*f^4) + 4*d^3*log(e^(f*x + e) - I)/(a^2*f^4 
) + 2/3*(d^3*f^3*x^3 + 3*c*d^2*f^3*x^2)/(a^2*f^4)
 

Giac [F]

\[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\int { \frac {{\left (d x + c\right )}^{3}}{{\left (i \, a \sinh \left (f x + e\right ) + a\right )}^{2}} \,d x } \] Input:

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

Output:

integrate((d*x + c)^3/(I*a*sinh(f*x + e) + a)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\int \frac {{\left (c+d\,x\right )}^3}{{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(c+d x)^3}{(a+i a \sinh (e+f x))^2} \, dx=\frac {-\left (\int \frac {x^{3}}{\sinh \left (f x +e \right )^{2}-2 \sinh \left (f x +e \right ) i -1}d x \right ) d^{3}-3 \left (\int \frac {x^{2}}{\sinh \left (f x +e \right )^{2}-2 \sinh \left (f x +e \right ) i -1}d x \right ) c \,d^{2}-3 \left (\int \frac {x}{\sinh \left (f x +e \right )^{2}-2 \sinh \left (f x +e \right ) i -1}d x \right ) c^{2} d -\left (\int \frac {1}{\sinh \left (f x +e \right )^{2}-2 \sinh \left (f x +e \right ) i -1}d x \right ) c^{3}}{a^{2}} \] Input:

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

Output:

( - int(x**3/(sinh(e + f*x)**2 - 2*sinh(e + f*x)*i - 1),x)*d**3 - 3*int(x* 
*2/(sinh(e + f*x)**2 - 2*sinh(e + f*x)*i - 1),x)*c*d**2 - 3*int(x/(sinh(e 
+ f*x)**2 - 2*sinh(e + f*x)*i - 1),x)*c**2*d - int(1/(sinh(e + f*x)**2 - 2 
*sinh(e + f*x)*i - 1),x)*c**3)/a**2