Integrand size = 31, antiderivative size = 296 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {2 (e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {4 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
-2*(f*x+e)^2/a/d+2*I*(f*x+e)^2*arctanh(exp(d*x+c))/a/d-(f*x+e)^2*coth(d*x+ c)/a/d+4*f*(f*x+e)*ln(1+I*exp(d*x+c))/a/d^2+2*f*(f*x+e)*ln(1-exp(2*d*x+2*c ))/a/d^2+2*I*f*(f*x+e)*polylog(2,-exp(d*x+c))/a/d^2+4*f^2*polylog(2,-I*exp (d*x+c))/a/d^3-2*I*f*(f*x+e)*polylog(2,exp(d*x+c))/a/d^2+f^2*polylog(2,exp (2*d*x+2*c))/a/d^3-2*I*f^2*polylog(3,-exp(d*x+c))/a/d^3+2*I*f^2*polylog(3, exp(d*x+c))/a/d^3-(f*x+e)^2*tanh(1/2*c+1/4*I*Pi+1/2*d*x)/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(803\) vs. \(2(296)=592\).
Time = 8.08 (sec) , antiderivative size = 803, normalized size of antiderivative = 2.71 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {4 i e^c f \left (\frac {e^{-c} (e+f x)^2}{2 f}+\frac {\left (i+e^{-c}\right ) (e+f x) \log \left (1-i e^{-c-d x}\right )}{d}-\frac {e^{-c} \left (1+i e^c\right ) f \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )}{d^2}\right )}{a d \left (-i+e^c\right )}+\frac {i d^2 e \left (-1+e^{2 c}\right ) (d e+2 i f) x+d^2 e \left (1-e^{2 c}\right ) (i d e+2 f) x-2 d^2 (e+f x)^2+2 d \left (-1+e^{2 c}\right ) f (-i d e+f) x \log \left (1-e^{-c-d x}\right )-i d^2 \left (-1+e^{2 c}\right ) f^2 x^2 \log \left (1-e^{-c-d x}\right )+2 d \left (-1+e^{2 c}\right ) f (i d e+f) x \log \left (1+e^{-c-d x}\right )+i d^2 \left (-1+e^{2 c}\right ) f^2 x^2 \log \left (1+e^{-c-d x}\right )+d e \left (-1+e^{2 c}\right ) (-i d e+2 f) \log \left (1-e^{c+d x}\right )+d e \left (-1+e^{2 c}\right ) (i d e+2 f) \log \left (1+e^{c+d x}\right )-2 \left (-1+e^{2 c}\right ) f (i d e+f) \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )-2 i d \left (-1+e^{2 c}\right ) f^2 x \operatorname {PolyLog}\left (2,-e^{-c-d x}\right )+2 i \left (-1+e^{2 c}\right ) (d e+i f) f \operatorname {PolyLog}\left (2,e^{-c-d x}\right )+2 i d \left (-1+e^{2 c}\right ) f^2 x \operatorname {PolyLog}\left (2,e^{-c-d x}\right )-2 i \left (-1+e^{2 c}\right ) f^2 \operatorname {PolyLog}\left (3,-e^{-c-d x}\right )+2 i \left (-1+e^{2 c}\right ) f^2 \operatorname {PolyLog}\left (3,e^{-c-d x}\right )}{a d^3 \left (-1+e^{2 c}\right )}+\frac {\text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-e^2 \sinh \left (\frac {d x}{2}\right )-2 e f x \sinh \left (\frac {d x}{2}\right )-f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}+\frac {\text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{2 a d}-\frac {2 \left (e^2 \sinh \left (\frac {d x}{2}\right )+2 e f x \sinh \left (\frac {d x}{2}\right )+f^2 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{a d \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]
((-4*I)*E^c*f*((e + f*x)^2/(2*E^c*f) + ((I + E^(-c))*(e + f*x)*Log[1 - I*E ^(-c - d*x)])/d - ((1 + I*E^c)*f*PolyLog[2, I*E^(-c - d*x)])/(d^2*E^c)))/( a*d*(-I + E^c)) + (I*d^2*e*(-1 + E^(2*c))*(d*e + (2*I)*f)*x + d^2*e*(1 - E ^(2*c))*(I*d*e + 2*f)*x - 2*d^2*(e + f*x)^2 + 2*d*(-1 + E^(2*c))*f*((-I)*d *e + f)*x*Log[1 - E^(-c - d*x)] - I*d^2*(-1 + E^(2*c))*f^2*x^2*Log[1 - E^( -c - d*x)] + 2*d*(-1 + E^(2*c))*f*(I*d*e + f)*x*Log[1 + E^(-c - d*x)] + I* d^2*(-1 + E^(2*c))*f^2*x^2*Log[1 + E^(-c - d*x)] + d*e*(-1 + E^(2*c))*((-I )*d*e + 2*f)*Log[1 - E^(c + d*x)] + d*e*(-1 + E^(2*c))*(I*d*e + 2*f)*Log[1 + E^(c + d*x)] - 2*(-1 + E^(2*c))*f*(I*d*e + f)*PolyLog[2, -E^(-c - d*x)] - (2*I)*d*(-1 + E^(2*c))*f^2*x*PolyLog[2, -E^(-c - d*x)] + (2*I)*(-1 + E^ (2*c))*(d*e + I*f)*f*PolyLog[2, E^(-c - d*x)] + (2*I)*d*(-1 + E^(2*c))*f^2 *x*PolyLog[2, E^(-c - d*x)] - (2*I)*(-1 + E^(2*c))*f^2*PolyLog[3, -E^(-c - d*x)] + (2*I)*(-1 + E^(2*c))*f^2*PolyLog[3, E^(-c - d*x)])/(a*d^3*(-1 + E ^(2*c))) + (Sech[c/2]*Sech[c/2 + (d*x)/2]*(-(e^2*Sinh[(d*x)/2]) - 2*e*f*x* Sinh[(d*x)/2] - f^2*x^2*Sinh[(d*x)/2]))/(2*a*d) + (Csch[c/2]*Csch[c/2 + (d *x)/2]*(e^2*Sinh[(d*x)/2] + 2*e*f*x*Sinh[(d*x)/2] + f^2*x^2*Sinh[(d*x)/2]) )/(2*a*d) - (2*(e^2*Sinh[(d*x)/2] + 2*e*f*x*Sinh[(d*x)/2] + f^2*x^2*Sinh[( d*x)/2]))/(a*d*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2]))
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.
| \(\displaystyle \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x)^2 \text {csch}^2(c+d x)dx}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -(e+f x)^2 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (e+f x)^2 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int (e+f x) \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}-\frac {2 f \int -i (e+f x) \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \int (e+f x) \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int e^{-2 c-2 d x+i \pi } \log \left (1+e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}-i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -i \int \frac {(e+f x)^2 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -i \left (\frac {\int (e+f x)^2 \text {csch}(c+d x)dx}{a}-i \int \frac {(e+f x)^2}{i \sinh (c+d x) a+a}dx\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {\int i (e+f x)^2 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^2}{\sin (i c+i d x) a+a}dx\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^2}{\sin (i c+i d x) a+a}dx\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle -i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {i \int -(e+f x)^2 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}+\frac {i \int -(e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^2 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \int (e+f x)^2 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -i \left (\frac {i \left (\frac {2 i f \int (e+f x) \log \left (1-e^{c+d x}\right )dx}{d}-\frac {2 i f \int (e+f x) \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^2 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 i f \int -i (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 f \int (e+f x) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {4 f \int -i (e+f x) \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {4 i f \int (e+f x) \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {4 i f \left (2 i \int \frac {i e^{c+d x} (e+f x)}{1+i e^{c+d x}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {4 i f \left (-2 \int \frac {e^{c+d x} (e+f x)}{1+i e^{c+d x}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {4 i f \left (-2 \left (\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {i (e+f x) \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {4 i f \left (-2 \left (\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i (e+f x) \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {4 i f \left (-2 \left (-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i (e+f x) \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
3.3.12.3.1 Defintions of rubi rules used
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[(((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]
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]
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]]]
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]
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]
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])
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]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
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]
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 888 vs. \(2 (275 ) = 550\).
Time = 2.52 (sec) , antiderivative size = 889, normalized size of antiderivative = 3.00
| method | result | size |
| risch | \(-\frac {4 f^{2} x^{2}}{a d}+\frac {8 f^{2} c \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 f^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 f^{2} \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {8 f \ln \left ({\mathrm e}^{d x +c}\right ) e}{a \,d^{2}}-\frac {8 f^{2} c x}{a \,d^{2}}+\frac {2 f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{d^{2} a}+\frac {2 f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d^{2} a}+\frac {2 f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{d^{3} a}-\frac {2 f^{2} c \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d^{3} a}-\frac {2 f^{2} c \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{3} a}+\frac {2 e f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}+\frac {2 e f \ln \left ({\mathrm e}^{d x +c}+1\right )}{d^{2} a}+\frac {2 e f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d^{2} a}-\frac {2 i e f \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{d a}+\frac {2 i e f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{d a}-\frac {2 i e f \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{d^{2} a}+\frac {2 i e c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{2} a}-\frac {4 f^{2} c^{2}}{a \,d^{3}}-\frac {i c^{2} f^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d^{3} a}+\frac {4 i e f \arctan \left ({\mathrm e}^{d x +c}\right )}{d^{2} a}+\frac {2 i e f \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {2 i e f \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right )}{d^{2} a}-\frac {2 i \left (f^{2} x^{2} {\mathrm e}^{2 d x +2 c}+2 e f x \,{\mathrm e}^{2 d x +2 c}+e^{2} {\mathrm e}^{2 d x +2 c}-2 x^{2} f^{2}-i {\mathrm e}^{d x +c} f^{2} x^{2}-4 e f x -2 i {\mathrm e}^{d x +c} e f x -2 e^{2}-i {\mathrm e}^{d x +c} e^{2}\right )}{\left ({\mathrm e}^{2 d x +2 c}-1\right ) \left ({\mathrm e}^{d x +c}-i\right ) a d}+\frac {i f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{2}}{d^{3} a}-\frac {i f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x^{2}}{d a}-\frac {2 i f^{2} \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right ) x}{d^{2} a}+\frac {i f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x^{2}}{d a}+\frac {2 i f^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right ) x}{d^{2} a}-\frac {4 i c \,f^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{d^{3} a}-\frac {2 i f^{2} \operatorname {polylog}\left (3, -{\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 i f^{2} \operatorname {polylog}\left (3, {\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {i e^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}-\frac {i e^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}+\frac {4 f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {4 f^{2} \ln \left (1+i {\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}+\frac {4 f^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{d x +c}\right )}{a \,d^{3}}\) | \(889\) |
-4*f^2*x^2/a/d+8/a/d^3*f^2*c*ln(exp(d*x+c))-2*I*f^2*polylog(3,-exp(d*x+c)) /a/d^3+2*f^2*polylog(2,-exp(d*x+c))/a/d^3+2*f^2*polylog(2,exp(d*x+c))/a/d^ 3+4*f^2*polylog(2,-I*exp(d*x+c))/a/d^3-8/a/d^2*f*ln(exp(d*x+c))*e-8/a/d^2* f^2*c*x+4/a/d^2*f^2*ln(1+I*exp(d*x+c))*x+4/a/d^3*f^2*ln(1+I*exp(d*x+c))*c+ 2*I*f^2*polylog(3,exp(d*x+c))/a/d^3+2/d^2/a*f^2*ln(1-exp(d*x+c))*x+2/d^2/a *f^2*ln(exp(d*x+c)+1)*x+2/d^3/a*f^2*ln(1-exp(d*x+c))*c-2/d^3/a*f^2*c*ln(1+ exp(2*d*x+2*c))-2/d^3/a*f^2*c*ln(exp(d*x+c)-1)+I/d/a*e^2*ln(exp(d*x+c)+1)+ 2/d^2/a*e*f*ln(exp(d*x+c)-1)+2/d^2/a*e*f*ln(exp(d*x+c)+1)+2/d^2/a*e*f*ln(1 +exp(2*d*x+2*c))-I/d/a*e^2*ln(exp(d*x+c)-1)+I/d/a*f^2*ln(exp(d*x+c)+1)*x^2 +2*I/d^2/a*f^2*polylog(2,-exp(d*x+c))*x-4*I/d^3/a*c*f^2*arctan(exp(d*x+c)) -I/d^3/a*c^2*f^2*ln(exp(d*x+c)-1)+4*I/d^2/a*e*f*arctan(exp(d*x+c))+2*I/d^2 /a*e*f*polylog(2,-exp(d*x+c))-2*I/d^2/a*e*f*polylog(2,exp(d*x+c))-4/a/d^3* f^2*c^2-2*I/d/a*e*f*ln(1-exp(d*x+c))*x+2*I/d/a*e*f*ln(exp(d*x+c)+1)*x-2*I/ d^2/a*e*f*ln(1-exp(d*x+c))*c+2*I/d^2/a*e*c*f*ln(exp(d*x+c)-1)-2*I*(f^2*x^2 *exp(2*d*x+2*c)+2*e*f*x*exp(2*d*x+2*c)+e^2*exp(2*d*x+2*c)-2*x^2*f^2-I*exp( d*x+c)*f^2*x^2-4*e*f*x-2*I*exp(d*x+c)*e*f*x-2*e^2-I*exp(d*x+c)*e^2)/(exp(2 *d*x+2*c)-1)/(exp(d*x+c)-I)/a/d+I/d^3/a*f^2*ln(1-exp(d*x+c))*c^2-I/d/a*f^2 *ln(1-exp(d*x+c))*x^2-2*I/d^2/a*f^2*polylog(2,exp(d*x+c))*x
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1355 vs. \(2 (263) = 526\).
Time = 0.28 (sec) , antiderivative size = 1355, normalized size of antiderivative = 4.58 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
(4*I*d^2*e^2 - 8*I*c*d*e*f + 4*I*c^2*f^2 + 4*(f^2*e^(3*d*x + 3*c) - I*f^2* e^(2*d*x + 2*c) - f^2*e^(d*x + c) + I*f^2)*dilog(-I*e^(d*x + c)) - 2*(d*f^ 2*x + d*e*f - I*f^2 + (-I*d*f^2*x - I*d*e*f - f^2)*e^(3*d*x + 3*c) - (d*f^ 2*x + d*e*f - I*f^2)*e^(2*d*x + 2*c) + (I*d*f^2*x + I*d*e*f + f^2)*e^(d*x + c))*dilog(-e^(d*x + c)) + 2*(d*f^2*x + d*e*f + I*f^2 - (I*d*f^2*x + I*d* e*f - f^2)*e^(3*d*x + 3*c) - (d*f^2*x + d*e*f + I*f^2)*e^(2*d*x + 2*c) - ( -I*d*f^2*x - I*d*e*f + f^2)*e^(d*x + c))*dilog(e^(d*x + c)) - 4*(d^2*f^2*x ^2 + 2*d^2*e*f*x + 2*c*d*e*f - c^2*f^2)*e^(3*d*x + 3*c) - 2*(-I*d^2*f^2*x^ 2 - 2*I*d^2*e*f*x + I*d^2*e^2 - 4*I*c*d*e*f + 2*I*c^2*f^2)*e^(2*d*x + 2*c) + 2*(d^2*f^2*x^2 + 2*d^2*e*f*x - d^2*e^2 + 4*c*d*e*f - 2*c^2*f^2)*e^(d*x + c) - (d^2*f^2*x^2 + d^2*e^2 - 2*I*d*e*f + 2*(d^2*e*f - I*d*f^2)*x - (I*d ^2*f^2*x^2 + I*d^2*e^2 + 2*d*e*f - 2*(-I*d^2*e*f - d*f^2)*x)*e^(3*d*x + 3* c) - (d^2*f^2*x^2 + d^2*e^2 - 2*I*d*e*f + 2*(d^2*e*f - I*d*f^2)*x)*e^(2*d* x + 2*c) - (-I*d^2*f^2*x^2 - I*d^2*e^2 - 2*d*e*f - 2*(I*d^2*e*f + d*f^2)*x )*e^(d*x + c))*log(e^(d*x + c) + 1) - 4*(-I*d*e*f + I*c*f^2 - (d*e*f - c*f ^2)*e^(3*d*x + 3*c) + (I*d*e*f - I*c*f^2)*e^(2*d*x + 2*c) + (d*e*f - c*f^2 )*e^(d*x + c))*log(e^(d*x + c) - I) + (d^2*e^2 - 2*(c - I)*d*e*f + (c^2 - 2*I*c)*f^2 + (-I*d^2*e^2 - 2*(-I*c - 1)*d*e*f + (-I*c^2 - 2*c)*f^2)*e^(3*d *x + 3*c) - (d^2*e^2 - 2*(c - I)*d*e*f + (c^2 - 2*I*c)*f^2)*e^(2*d*x + 2*c ) + (I*d^2*e^2 - 2*(I*c + 1)*d*e*f + (I*c^2 + 2*c)*f^2)*e^(d*x + c))*lo...
\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{2} x^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {2 e f x \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
-I*(Integral(e**2*csch(c + d*x)**2/(sinh(c + d*x) - I), x) + Integral(f**2 *x**2*csch(c + d*x)**2/(sinh(c + d*x) - I), x) + Integral(2*e*f*x*csch(c + d*x)**2/(sinh(c + d*x) - I), x))/a
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 601 vs. \(2 (263) = 526\).
Time = 0.42 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.03 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-e^{2} {\left (\frac {2 \, {\left (e^{\left (-d x - c\right )} - i \, e^{\left (-2 \, d x - 2 \, c\right )} + 2 i\right )}}{{\left (a e^{\left (-d x - c\right )} - i \, a e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-3 \, d x - 3 \, c\right )} + i \, a\right )} d} - \frac {i \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{a d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{a d}\right )} - \frac {2 \, f^{2} x^{2}}{a d} - \frac {8 \, e f x}{a d} - \frac {2 \, {\left (-2 i \, f^{2} x^{2} - 4 i \, e f x - {\left (-i \, f^{2} x^{2} e^{\left (2 \, c\right )} - 2 i \, e f x e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} + {\left (f^{2} x^{2} e^{c} + 2 \, e f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a d e^{\left (3 \, d x + 3 \, c\right )} - i \, a d e^{\left (2 \, d x + 2 \, c\right )} - a d e^{\left (d x + c\right )} + i \, a d} + \frac {2 \, e f \log \left (e^{\left (d x + c\right )} + 1\right )}{a d^{2}} + \frac {4 \, e f \log \left (e^{\left (d x + c\right )} - i\right )}{a d^{2}} + \frac {2 \, e f \log \left (e^{\left (d x + c\right )} - 1\right )}{a d^{2}} + \frac {i \, {\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 {i \, {\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 \, {\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}} - \frac {2 \, {\left (-i \, d e f - f^{2}\right )} {\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 )}}{a d^{3}} + \frac {2 \, {\left (-i \, d e f + f^{2}\right )} {\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 )}}{a d^{3}} + \frac {i \, d^{3} f^{2} x^{3} - 3 \, {\left (-i \, d e f + f^{2}\right )} d^{2} x^{2}}{3 \, a d^{3}} - \frac {i \, d^{3} f^{2} x^{3} - 3 \, {\left (-i \, d e f - f^{2}\right )} d^{2} x^{2}}{3 \, a d^{3}} \]
-e^2*(2*(e^(-d*x - c) - I*e^(-2*d*x - 2*c) + 2*I)/((a*e^(-d*x - c) - I*a*e ^(-2*d*x - 2*c) - a*e^(-3*d*x - 3*c) + I*a)*d) - I*log(e^(-d*x - c) + 1)/( a*d) + I*log(e^(-d*x - c) - 1)/(a*d)) - 2*f^2*x^2/(a*d) - 8*e*f*x/(a*d) - 2*(-2*I*f^2*x^2 - 4*I*e*f*x - (-I*f^2*x^2*e^(2*c) - 2*I*e*f*x*e^(2*c))*e^( 2*d*x) + (f^2*x^2*e^c + 2*e*f*x*e^c)*e^(d*x))/(a*d*e^(3*d*x + 3*c) - I*a*d *e^(2*d*x + 2*c) - a*d*e^(d*x + c) + I*a*d) + 2*e*f*log(e^(d*x + c) + 1)/( a*d^2) + 4*e*f*log(e^(d*x + c) - I)/(a*d^2) + 2*e*f*log(e^(d*x + c) - 1)/( a*d^2) + I*(d^2*x^2*log(e^(d*x + c) + 1) + 2*d*x*dilog(-e^(d*x + c)) - 2*p olylog(3, -e^(d*x + c)))*f^2/(a*d^3) - I*(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*(d*x *log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))*f^2/(a*d^3) - 2*(-I*d*e*f - f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a*d^3) + 2*(-I*d *e*f + f^2)*(d*x*log(-e^(d*x + c) + 1) + dilog(e^(d*x + c)))/(a*d^3) + 1/3 *(I*d^3*f^2*x^3 - 3*(-I*d*e*f + f^2)*d^2*x^2)/(a*d^3) - 1/3*(I*d^3*f^2*x^3 - 3*(-I*d*e*f - f^2)*d^2*x^2)/(a*d^3)
\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]