\(\int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx\) [206]

Optimal result

Integrand size = 29, antiderivative size = 224 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i (e+f x)^2}{a d}-\frac {2 (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {4 i f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {2 f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {2 f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {2 f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:

-I*(f*x+e)^2/a/d-2*(f*x+e)^2*arctanh(exp(d*x+c))/a/d+4*I*f*(f*x+e)*ln(1+I* 
exp(d*x+c))/a/d^2-2*f*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^2+4*I*f^2*polylog 
(2,-I*exp(d*x+c))/a/d^3+2*f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2+2*f^2*poly 
log(3,-exp(d*x+c))/a/d^3-2*f^2*polylog(3,exp(d*x+c))/a/d^3-I*(f*x+e)^2*tan 
h(1/2*c+1/4*I*Pi+1/2*d*x)/a/d
 

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.23 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {(e+f x)^2 \log \left (1-e^{c+d x}\right )-(e+f x)^2 \log \left (1+e^{c+d x}\right )+\frac {2 d (e+f x) \left (-i d (e+f x)+2 \left (-i+e^c\right ) f \log \left (1-i e^{-c-d x}\right )\right )-4 \left (-i+e^c\right ) f^2 \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )}{d^2 \left (-1-i e^c\right )}-\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,-e^{c+d x}\right )\right )}{d^2}+\frac {2 f \left (d (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )-f \operatorname {PolyLog}\left (3,e^{c+d x}\right )\right )}{d^2}-\frac {2 i (e+f x)^2 \sinh \left (\frac {d x}{2}\right )}{\left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}}{a d} \] Input:

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

Output:

((e + f*x)^2*Log[1 - E^(c + d*x)] - (e + f*x)^2*Log[1 + E^(c + d*x)] + (2* 
d*(e + f*x)*((-I)*d*(e + f*x) + 2*(-I + E^c)*f*Log[1 - I*E^(-c - d*x)]) - 
4*(-I + E^c)*f^2*PolyLog[2, I*E^(-c - d*x)])/(d^2*(-1 - I*E^c)) - (2*f*(d* 
(e + f*x)*PolyLog[2, -E^(c + d*x)] - f*PolyLog[3, -E^(c + d*x)]))/d^2 + (2 
*f*(d*(e + f*x)*PolyLog[2, E^(c + d*x)] - f*PolyLog[3, E^(c + d*x)]))/d^2 
- ((2*I)*(e + f*x)^2*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 = 1.44 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.05, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.690, Rules used = {6109, 3042, 26, 3799, 25, 25, 3042, 4670, 3011, 2720, 4672, 26, 3042, 26, 4199, 26, 2620, 2715, 2838, 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.

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

\(\Big \downarrow \) 6109

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3799

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4670

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 4672

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4199

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

(I*(((2*I)*(e + f*x)^2*ArcTanh[E^(c + d*x)])/d - ((2*I)*f*(-(((e + f*x)*Po 
lyLog[2, -E^(c + d*x)])/d) + (f*PolyLog[3, -E^(c + d*x)])/d^2))/d + ((2*I) 
*f*(-(((e + f*x)*PolyLog[2, E^(c + d*x)])/d) + (f*PolyLog[3, E^(c + d*x)]) 
/d^2))/d))/a - ((I/2)*(((4*I)*f*(((-1/2*I)*(e + f*x)^2)/f - 2*(((-I)*(e + 
f*x)*Log[1 + I*E^(c + d*x)])/d - (I*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2))) 
/d + (2*(e + f*x)^2*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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 572 vs. \(2 (204 ) = 408\).

Time = 0.55 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.56

method	result	size
risch	\(-\frac {f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{3}}-\frac {2 e f \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {2 e f \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {c^{2} f^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\frac {f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x^{2}}{a d}-\frac {2 f^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x^{2}}{a d}+\frac {2 f^{2} \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {4 i f^{2} c x}{a \,d^{2}}-\frac {4 i e f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {2 i f^{2} c^{2}}{a \,d^{3}}+\frac {4 i f^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 f^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {2 f^{2} \operatorname {polylog}\left (3, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i c \,f^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {4 i f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {4 i e f \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{2}}-\frac {4 i c \,f^{2} \ln \left ({\mathrm e}^{d x +c}-i\right )}{a \,d^{3}}-\frac {2 i f^{2} x^{2}}{a d}-\frac {e^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{a d}+\frac {e^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a d}+\frac {2 x^{2} f^{2}+4 e f x +2 e^{2}}{d a \left ({\mathrm e}^{d x +c}-i\right )}-\frac {2 e f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a d}+\frac {2 e f \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{a d}-\frac {2 e c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {2 e f \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{a \,d^{2}}\)	\(573\)

Input:

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

Output:

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

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. 559 vs. \(2 (194) = 388\).

Time = 0.11 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.50 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 \, d^{2} e^{2} - 4 \, c d e f + 2 \, c^{2} f^{2} - 4 \, {\left (-i \, f^{2} e^{\left (d x + c\right )} - f^{2}\right )} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\left (-i \, d f^{2} x - i \, d e f + {\left (d f^{2} x + d e f\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\left (i \, d f^{2} x + i \, d e f - {\left (d f^{2} x + d e f\right )} e^{\left (d x + c\right )}\right )} {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\left (i \, d^{2} f^{2} x^{2} + 2 i \, d^{2} e f x + 2 i \, c d e f - i \, c^{2} f^{2}\right )} e^{\left (d x + c\right )} + {\left (i \, d^{2} f^{2} x^{2} + 2 i \, d^{2} e f x + i \, d^{2} e^{2} - {\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + d^{2} e^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} + 1\right ) + 4 \, {\left (d e f - c f^{2} - {\left (-i \, d e f + i \, c f^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - i\right ) + {\left (-i \, d^{2} e^{2} + 2 i \, c d e f - i \, c^{2} f^{2} + {\left (d^{2} e^{2} - 2 \, c d e f + c^{2} f^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (e^{\left (d x + c\right )} - 1\right ) + 4 \, {\left (d f^{2} x + c f^{2} - {\left (-i \, d f^{2} x - i \, c f^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\left (-i \, d^{2} f^{2} x^{2} - 2 i \, d^{2} e f x - 2 i \, c d e f + i \, c^{2} f^{2} + {\left (d^{2} f^{2} x^{2} + 2 \, d^{2} e f x + 2 \, c d e f - c^{2} f^{2}\right )} e^{\left (d x + c\right )}\right )} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, {\left (f^{2} e^{\left (d x + c\right )} - i \, f^{2}\right )} {\rm polylog}\left (3, -e^{\left (d x + c\right )}\right ) - 2 \, {\left (f^{2} e^{\left (d x + c\right )} - i \, f^{2}\right )} {\rm polylog}\left (3, e^{\left (d x + c\right )}\right )}{a d^{3} e^{\left (d x + c\right )} - i \, a d^{3}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.55 \[ \int \frac {(e+f x)^2 \text {csch}(c+d x)}{a+i a \sinh (c+d x)} \, dx=-e^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{a d} - \frac {2}{{\left (a e^{\left (-d x - c\right )} + i \, a\right )} d}\right )} - \frac {2 i \, f^{2} x^{2}}{a d} - \frac {4 i \, e f x}{a d} + \frac {2 \, {\left (f^{2} x^{2} + 2 \, e f x\right )}}{a d e^{\left (d x + c\right )} - i \, a d} - \frac {2 \, {\left (d x \log \left (e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac {2 \, {\left (d x \log \left (-e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (d x + c\right )}\right )\right )} e f}{a d^{2}} + \frac {4 i \, e f \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d^{2}} - \frac {{\left (d^{2} x^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} + \frac {{\left (d^{2} x^{2} \log \left (-e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (d x + c\right )})\right )} f^{2}}{a d^{3}} + \frac {4 i \, {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )} f^{2}}{a d^{3}} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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