\(\int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [205]

Optimal result

Integrand size = 29, antiderivative size = 313 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i (e+f x)^3}{a d}-\frac {2 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:

-I*(f*x+e)^3/a/d-2*(f*x+e)^3*arctanh(exp(d*x+c))/a/d+6*I*f*(f*x+e)^2*ln(1+ 
I*exp(d*x+c))/a/d^2-3*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a/d^2+12*I*f^2*(f 
*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3+3*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a 
/d^2+6*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3-12*I*f^3*polylog(3,-I*exp( 
d*x+c))/a/d^4-6*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a/d^3-6*f^3*polylog(4,-e 
xp(d*x+c))/a/d^4+6*f^3*polylog(4,exp(d*x+c))/a/d^4-I*(f*x+e)^3*tanh(1/2*c+ 
1/4*I*Pi+1/2*d*x)/a/d
 

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.09 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\frac {2 (e+f x)^3}{-i+e^c}+\frac {6 i f (e+f x)^2 \log \left (1-i e^{-c-d x}\right )}{d}+(e+f x)^3 \log \left (1-e^{c+d x}\right )-(e+f x)^3 \log \left (1+e^{c+d x}\right )-\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}+\frac {3 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}-\frac {12 i f^2 \left (d (e+f x) \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )+f \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )\right )}{d^3}+\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{d^2}-\frac {6 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{d^2}-\frac {6 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{d^3}+\frac {6 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{d^3}-\frac {2 i (e+f x)^3 \sinh \left (\frac {d x}{2}\right )}{\left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}}{a d} \] Input:

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

Output:

((2*(e + f*x)^3)/(-I + E^c) + ((6*I)*f*(e + f*x)^2*Log[1 - I*E^(-c - d*x)] 
)/d + (e + f*x)^3*Log[1 - E^(c + d*x)] - (e + f*x)^3*Log[1 + E^(c + d*x)] 
- (3*f*(e + f*x)^2*PolyLog[2, -E^(c + d*x)])/d + (3*f*(e + f*x)^2*PolyLog[ 
2, E^(c + d*x)])/d - ((12*I)*f^2*(d*(e + f*x)*PolyLog[2, I*E^(-c - d*x)] + 
 f*PolyLog[3, I*E^(-c - d*x)]))/d^3 + (6*f^2*(e + f*x)*PolyLog[3, -E^(c + 
d*x)])/d^2 - (6*f^2*(e + f*x)*PolyLog[3, E^(c + d*x)])/d^2 - (6*f^3*PolyLo 
g[4, -E^(c + d*x)])/d^3 + (6*f^3*PolyLog[4, E^(c + d*x)])/d^3 - ((2*I)*(e 
+ f*x)^3*Sinh[(d*x)/2])/((Cosh[c/2] + I*Sinh[c/2])*(Cosh[(c + d*x)/2] + I* 
Sinh[(c + d*x)/2])))/(a*d)
 

Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.04, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {6109, 3042, 26, 3799, 25, 25, 3042, 4670, 3011, 4672, 26, 3042, 26, 4199, 26, 2620, 3011, 2720, 7143, 7163, 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.

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

\(\Big \downarrow \) 6109

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3799

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

\(\Big \downarrow \) 7163

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(I*(((2*I)*(e + f*x)^3*ArcTanh[E^(c + d*x)])/d - ((3*I)*f*(-(((e + f*x)^2* 
PolyLog[2, -E^(c + d*x)])/d) + (2*f*(((e + f*x)*PolyLog[3, -E^(c + d*x)])/ 
d - (f*PolyLog[4, -E^(c + d*x)])/d^2))/d))/d + ((3*I)*f*(-(((e + f*x)^2*Po 
lyLog[2, E^(c + d*x)])/d) + (2*f*(((e + f*x)*PolyLog[3, E^(c + d*x)])/d - 
(f*PolyLog[4, E^(c + d*x)])/d^2))/d))/d))/a - ((I/2)*(((6*I)*f*(((-1/3*I)* 
(e + f*x)^3)/f - 2*(((-I)*(e + f*x)^2*Log[1 + I*E^(c + d*x)])/d + ((2*I)*f 
*(-(((e + f*x)*PolyLog[2, (-I)*E^(c + d*x)])/d) + (f*PolyLog[3, (-I)*E^(c 
+ d*x)])/d^2))/d)))/d + (2*(e + f*x)^3*Tanh[c/2 + (I/4)*Pi + (d*x)/2])/d)) 
/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 4670

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x 
_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] 
 + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x 
)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e 
+ f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && 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 6109

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

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]
 

rule 7163

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1033 vs. \(2 (288 ) = 576\).

Time = 0.70 (sec) , antiderivative size = 1034, normalized size of antiderivative = 3.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1034\)

Input:

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

Output:

-1/a/d*e^3*ln(exp(d*x+c)+1)+1/a/d*e^3*ln(exp(d*x+c)-1)+6*I/a/d^3*f^3*c^2*x 
-6*I/a/d*e*f^2*x^2-6*I/a/d^3*e*f^2*c^2-6*I/a/d^4*c^2*f^3*ln(exp(d*x+c))-6* 
I/a/d^2*e^2*f*ln(exp(d*x+c))-6*I/a/d^4*f^3*ln(1+I*exp(d*x+c))*c^2+6*I/a/d^ 
2*f^3*ln(1+I*exp(d*x+c))*x^2+12*I/a/d^3*e*f^2*polylog(2,-I*exp(d*x+c))+12* 
I/a/d^3*f^3*polylog(2,-I*exp(d*x+c))*x-12*I/a/d^2*e*f^2*c*x+12*I/a/d^3*c*f 
^2*e*ln(exp(d*x+c))+12*I/a/d^3*e*f^2*ln(1+I*exp(d*x+c))*c-6/a/d^3*f^3*poly 
log(3,exp(d*x+c))*x-1/a/d*f^3*ln(exp(d*x+c)+1)*x^3-3/a/d^2*f^3*polylog(2,- 
exp(d*x+c))*x^2+6/a/d^3*f^3*polylog(3,-exp(d*x+c))*x+6/a/d^3*e*f^2*polylog 
(3,-exp(d*x+c))-6/a/d^3*e*f^2*polylog(3,exp(d*x+c))-1/a/d^4*c^3*f^3*ln(exp 
(d*x+c)-1)-3/a/d^2*e^2*f*polylog(2,-exp(d*x+c))+3/a/d^2*e^2*f*polylog(2,ex 
p(d*x+c))+1/a/d^4*c^3*f^3*ln(1-exp(d*x+c))+1/a/d*f^3*ln(1-exp(d*x+c))*x^3+ 
3/a/d^2*f^3*polylog(2,exp(d*x+c))*x^2+2*(f^3*x^3+3*e*f^2*x^2+3*e^2*f*x+e^3 
)/d/a/(exp(d*x+c)-I)-6*f^3*polylog(4,-exp(d*x+c))/a/d^4+6*f^3*polylog(4,ex 
p(d*x+c))/a/d^4-12*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4-12*I/a/d^3*c*e*f^2 
*ln(exp(d*x+c)-I)+4*I/a/d^4*f^3*c^3-2*I/a/d*f^3*x^3+12*I/a/d^2*e*f^2*ln(1+ 
I*exp(d*x+c))*x+3/a/d*e^2*f*ln(1-exp(d*x+c))*x-3/a/d*e^2*f*ln(exp(d*x+c)+1 
)*x-3/a/d^2*e^2*c*f*ln(exp(d*x+c)-1)+3/a/d^3*c^2*e*f^2*ln(exp(d*x+c)-1)+3/ 
a/d^2*e^2*f*ln(1-exp(d*x+c))*c+6*I/a/d^4*c^2*f^3*ln(exp(d*x+c)-I)+6*I/a/d^ 
2*e^2*f*ln(exp(d*x+c)-I)+3/a/d*e*f^2*ln(1-exp(d*x+c))*x^2+6/a/d^2*e*f^2*po 
lylog(2,exp(d*x+c))*x-3/a/d^3*e*f^2*ln(1-exp(d*x+c))*c^2-3/a/d*e*f^2*ln...
 

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. 1000 vs. \(2 (276) = 552\).

Time = 0.14 (sec) , antiderivative size = 1000, normalized size of antiderivative = 3.19 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

(2*d^3*e^3 - 6*c*d^2*e^2*f + 6*c^2*d*e*f^2 - 2*c^3*f^3 + 12*(d*f^3*x + d*e 
*f^2 - (-I*d*f^3*x - I*d*e*f^2)*e^(d*x + c))*dilog(-I*e^(d*x + c)) - 3*(-I 
*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^2*f + (d^2*f^3*x^2 + 2*d^2*e*f^2* 
x + d^2*e^2*f)*e^(d*x + c))*dilog(-e^(d*x + c)) - 3*(I*d^2*f^3*x^2 + 2*I*d 
^2*e*f^2*x + I*d^2*e^2*f - (d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*e^(d* 
x + c))*dilog(e^(d*x + c)) - 2*(I*d^3*f^3*x^3 + 3*I*d^3*e*f^2*x^2 + 3*I*d^ 
3*e^2*f*x + 3*I*c*d^2*e^2*f - 3*I*c^2*d*e*f^2 + I*c^3*f^3)*e^(d*x + c) + ( 
I*d^3*f^3*x^3 + 3*I*d^3*e*f^2*x^2 + 3*I*d^3*e^2*f*x + I*d^3*e^3 - (d^3*f^3 
*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + d^3*e^3)*e^(d*x + c))*log(e^(d*x 
+ c) + 1) + 6*(d^2*e^2*f - 2*c*d*e*f^2 + c^2*f^3 - (-I*d^2*e^2*f + 2*I*c*d 
*e*f^2 - I*c^2*f^3)*e^(d*x + c))*log(e^(d*x + c) - I) + (-I*d^3*e^3 + 3*I* 
c*d^2*e^2*f - 3*I*c^2*d*e*f^2 + I*c^3*f^3 + (d^3*e^3 - 3*c*d^2*e^2*f + 3*c 
^2*d*e*f^2 - c^3*f^3)*e^(d*x + c))*log(e^(d*x + c) - 1) + 6*(d^2*f^3*x^2 + 
 2*d^2*e*f^2*x + 2*c*d*e*f^2 - c^2*f^3 - (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x 
 - 2*I*c*d*e*f^2 + I*c^2*f^3)*e^(d*x + c))*log(I*e^(d*x + c) + 1) + (-I*d^ 
3*f^3*x^3 - 3*I*d^3*e*f^2*x^2 - 3*I*d^3*e^2*f*x - 3*I*c*d^2*e^2*f + 3*I*c^ 
2*d*e*f^2 - I*c^3*f^3 + (d^3*f^3*x^3 + 3*d^3*e*f^2*x^2 + 3*d^3*e^2*f*x + 3 
*c*d^2*e^2*f - 3*c^2*d*e*f^2 + c^3*f^3)*e^(d*x + c))*log(-e^(d*x + c) + 1) 
 - 6*(f^3*e^(d*x + c) - I*f^3)*polylog(4, -e^(d*x + c)) + 6*(f^3*e^(d*x + 
c) - I*f^3)*polylog(4, e^(d*x + c)) - 12*(I*f^3*e^(d*x + c) + f^3)*poly...
 

Sympy [F]

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

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

Output:

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

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. 580 vs. \(2 (276) = 552\).

Time = 0.25 (sec) , antiderivative size = 580, normalized size of antiderivative = 1.85 \[ \int \frac {(e+f x)^3 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-e^3*(log(e^(-d*x - c) + 1)/(a*d) - log(e^(-d*x - c) - 1)/(a*d) - 2/((a*e^ 
(-d*x - c) + I*a)*d)) - 6*I*e^2*f*x/(a*d) - 3*(d*x*log(e^(d*x + c) + 1) + 
dilog(-e^(d*x + c)))*e^2*f/(a*d^2) + 3*(d*x*log(-e^(d*x + c) + 1) + dilog( 
e^(d*x + c)))*e^2*f/(a*d^2) + 6*I*e^2*f*log(I*e^(d*x + c) + 1)/(a*d^2) + 2 
*(f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x)/(a*d*e^(d*x + c) - I*a*d) - 3*(d^2*x^ 
2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*polylog(3, -e^(d*x 
+ c)))*e*f^2/(a*d^3) + 3*(d^2*x^2*log(-e^(d*x + c) + 1) + 2*d*x*dilog(e^(d 
*x + c)) - 2*polylog(3, e^(d*x + c)))*e*f^2/(a*d^3) + 12*I*(d*x*log(I*e^(d 
*x + c) + 1) + dilog(-I*e^(d*x + c)))*e*f^2/(a*d^3) - (d^3*x^3*log(e^(d*x 
+ c) + 1) + 3*d^2*x^2*dilog(-e^(d*x + c)) - 6*d*x*polylog(3, -e^(d*x + c)) 
 + 6*polylog(4, -e^(d*x + c)))*f^3/(a*d^4) + (d^3*x^3*log(-e^(d*x + c) + 1 
) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polyl 
og(4, e^(d*x + c)))*f^3/(a*d^4) + 6*I*(d^2*x^2*log(I*e^(d*x + c) + 1) + 2* 
d*x*dilog(-I*e^(d*x + c)) - 2*polylog(3, -I*e^(d*x + c)))*f^3/(a*d^4) + 2* 
(-I*d^3*f^3*x^3 - 3*I*d^3*e*f^2*x^2)/(a*d^4)
                                                                                    
                                                                                    
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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