Integrand size = 31, antiderivative size = 419 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {2 (e+f x)^3}{a d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^3 \coth (c+d x)}{a d}+\frac {6 f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {3 f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {12 f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {3 f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}-\frac {12 f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {6 i f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {3 f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {6 i f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {6 i f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}-\frac {(e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:
-2*(f*x+e)^3/a/d+6*I*f^2*(f*x+e)*polylog(3,exp(d*x+c))/a/d^3-(f*x+e)^3*cot h(d*x+c)/a/d+6*f*(f*x+e)^2*ln(1+I*exp(d*x+c))/a/d^2+3*f*(f*x+e)^2*ln(1-exp (2*d*x+2*c))/a/d^2-6*I*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3+12*f^2*(f* x+e)*polylog(2,-I*exp(d*x+c))/a/d^3-6*I*f^3*polylog(4,exp(d*x+c))/a/d^4+3* f^2*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d^3+6*I*f^3*polylog(4,-exp(d*x+c)) /a/d^4-12*f^3*polylog(3,-I*exp(d*x+c))/a/d^4+2*I*(f*x+e)^3*arctanh(exp(d*x +c))/a/d-3/2*f^3*polylog(3,exp(2*d*x+2*c))/a/d^4-3*I*f*(f*x+e)^2*polylog(2 ,exp(d*x+c))/a/d^2+3*I*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/a/d^2-(f*x+e)^3* 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. \(1205\) vs. \(2(419)=838\).
Time = 8.47 (sec) , antiderivative size = 1205, normalized size of antiderivative = 2.88 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^3*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]
Output:
((-6*I)*E^c*f*((e + f*x)^3/(3*E^c*f) + ((I + E^(-c))*(e + f*x)^2*Log[1 - I *E^(-c - d*x)])/d - ((2*I)*(-I + E^c)*f*(d*(e + f*x)*PolyLog[2, I*E^(-c - d*x)] + f*PolyLog[3, I*E^(-c - d*x)]))/(d^3*E^c)))/(a*d*(-I + E^c)) + (I*d ^3*e^2*(-1 + E^(2*c))*(d*e + (3*I)*f)*x + d^3*e^2*(1 - E^(2*c))*(I*d*e + 3 *f)*x - 2*d^3*(e + f*x)^3 + 3*d^2*e*(-1 + E^(2*c))*f*((-I)*d*e + 2*f)*x*Lo g[1 - E^(-c - d*x)] + 3*d^2*(-1 + E^(2*c))*f^2*((-I)*d*e + f)*x^2*Log[1 - E^(-c - d*x)] - I*d^3*(-1 + E^(2*c))*f^3*x^3*Log[1 - E^(-c - d*x)] + 3*d^2 *e*(-1 + E^(2*c))*f*(I*d*e + 2*f)*x*Log[1 + E^(-c - d*x)] + 3*d^2*(-1 + E^ (2*c))*f^2*(I*d*e + f)*x^2*Log[1 + E^(-c - d*x)] + I*d^3*(-1 + E^(2*c))*f^ 3*x^3*Log[1 + E^(-c - d*x)] + d^2*e^2*(-1 + E^(2*c))*((-I)*d*e + 3*f)*Log[ 1 - E^(c + d*x)] + d^2*e^2*(-1 + E^(2*c))*(I*d*e + 3*f)*Log[1 + E^(c + d*x )] + 3*d*e*(1 - E^(2*c))*f*(I*d*e + 2*f)*PolyLog[2, -E^(-c - d*x)] + 6*d*( 1 - E^(2*c))*f^2*(I*d*e + f)*x*PolyLog[2, -E^(-c - d*x)] - (3*I)*d^2*(-1 + E^(2*c))*f^3*x^2*PolyLog[2, -E^(-c - d*x)] + (3*I)*d*e*(-1 + E^(2*c))*(d* e + (2*I)*f)*f*PolyLog[2, E^(-c - d*x)] + (6*I)*d*(-1 + E^(2*c))*(d*e + I* f)*f^2*x*PolyLog[2, E^(-c - d*x)] + (3*I)*d^2*(-1 + E^(2*c))*f^3*x^2*PolyL og[2, E^(-c - d*x)] - 6*(-1 + E^(2*c))*f^2*(I*d*e + f)*PolyLog[3, -E^(-c - d*x)] - (6*I)*d*(-1 + E^(2*c))*f^3*x*PolyLog[3, -E^(-c - d*x)] + (6*I)*(- 1 + E^(2*c))*(d*e + I*f)*f^2*PolyLog[3, E^(-c - d*x)] + (6*I)*d*(-1 + E^(2 *c))*f^3*x*PolyLog[3, E^(-c - d*x)] - (6*I)*(-1 + E^(2*c))*f^3*PolyLog[...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x)^3 \text {csch}^2(c+d x)dx}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -(e+f x)^3 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int (e+f x)^3 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^2}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {\int (e+f x)^3 \text {csch}(c+d x)dx}{a}-i \int \frac {(e+f x)^3}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {\int i (e+f x)^3 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^3}{\sin (i c+i d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^3}{\sin (i c+i d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {i \int -(e+f x)^3 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}+\frac {i \int -(e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -i \left (\frac {i \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 i f \int -i (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 f \int (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}-\frac {6 f \int -i (e+f x)^2 \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {6 i f \int (e+f x)^2 \tan \left (\frac {i c}{2}+\frac {i d x}{2}-\frac {\pi }{4}\right )dx}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {6 i f \left (2 i \int \frac {i e^{c+d x} (e+f x)^2}{1+i e^{c+d x}}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {6 i f \left (-2 \int \frac {e^{c+d x} (e+f x)^2}{1+i e^{c+d x}}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \int (e+f x) \log \left (1+i e^{c+d x}\right )dx}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \left (\frac {f \int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {6 i f \left (-2 \left (\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {i (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}+\frac {2 (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \operatorname {PolyLog}\left (3,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\) |
Input:
Int[((e + f*x)^3*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]
Output:
$Aborted
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1603 vs. \(2 (390 ) = 780\).
Time = 1.10 (sec) , antiderivative size = 1604, normalized size of antiderivative = 3.83
Input:
int((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-4/a/d*f^3*x^3+8/a/d^4*c^3*f^3+6*I*f^3*polylog(4,-exp(d*x+c))/a/d^4+I/a/d* f^3*ln(exp(d*x+c)+1)*x^3+12/a/d^2*e*f^2*ln(1+I*exp(d*x+c))*x-24/a/d^2*e*f^ 2*c*x+6/a/d^2*e*f^2*ln(exp(d*x+c)+1)*x+6/a/d^2*e*f^2*ln(1-exp(d*x+c))*x+6/ a/d^3*e*f^2*ln(1-exp(d*x+c))*c+12/a/d^3*e*f^2*ln(1+I*exp(d*x+c))*c+24/a/d^ 3*c*e*f^2*ln(exp(d*x+c))-6/a/d^3*c*e*f^2*ln(1+exp(2*d*x+2*c))-6/a/d^3*c*e* f^2*ln(exp(d*x+c)-1)+I/a/d^4*c^3*f^3*ln(exp(d*x+c)-1)-2*I*(f^3*x^3*exp(2*d *x+2*c)+3*e*f^2*x^2*exp(2*d*x+2*c)+3*e^2*f*x*exp(2*d*x+2*c)-2*x^3*f^3-I*ex p(d*x+c)*f^3*x^3+e^3*exp(2*d*x+2*c)-6*e*f^2*x^2-3*I*exp(d*x+c)*e*f^2*x^2-6 *e^2*f*x-3*I*exp(d*x+c)*e^2*f*x-2*e^3-I*exp(d*x+c)*e^3)/(exp(2*d*x+2*c)-1) /(exp(d*x+c)-I)/d/a-12/a/d*e*f^2*x^2+I/a/d*e^3*ln(exp(d*x+c)+1)+3/a/d^2*e^ 2*f*ln(1+exp(2*d*x+2*c))+3/a/d^2*e^2*f*ln(exp(d*x+c)+1)-6*I*f^3*polylog(4, exp(d*x+c))/a/d^4-I/a/d*f^3*ln(1-exp(d*x+c))*x^3-3*I/a/d^2*f^3*polylog(2,e xp(d*x+c))*x^2+6*I/a/d^3*f^3*polylog(3,exp(d*x+c))*x+3*I/a/d^2*f^3*polylog (2,-exp(d*x+c))*x^2-6*I/a/d^3*f^3*polylog(3,-exp(d*x+c))*x+6*I/a/d^4*c^2*f ^3*arctan(exp(d*x+c))-I/a/d^4*c^3*f^3*ln(1-exp(d*x+c))+3*I/a/d^2*e^2*f*pol ylog(2,-exp(d*x+c))-3*I/a/d^2*e^2*f*polylog(2,exp(d*x+c))+6*I/a/d^2*e^2*f* arctan(exp(d*x+c))-6*I/a/d^3*e*f^2*polylog(3,-exp(d*x+c))+6*I/a/d^3*e*f^2* polylog(3,exp(d*x+c))-6*f^3*polylog(3,-exp(d*x+c))/a/d^4-6*f^3*polylog(3,e xp(d*x+c))/a/d^4-12*f^3*polylog(3,-I*exp(d*x+c))/a/d^4+3/a/d^2*e^2*f*ln(ex p(d*x+c)-1)+12/a/d^3*c^2*f^3*x-3/a/d^4*c^2*f^3*ln(1-exp(d*x+c))-6/a/d^4...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2562 vs. \(2 (375) = 750\).
Time = 0.14 (sec) , antiderivative size = 2562, normalized size of antiderivative = 6.11 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas ")
Output:
(4*I*d^3*e^3 - 12*I*c*d^2*e^2*f + 12*I*c^2*d*e*f^2 - 4*I*c^3*f^3 - 12*(-I* d*f^3*x - I*d*e*f^2 - (d*f^3*x + d*e*f^2)*e^(3*d*x + 3*c) + (I*d*f^3*x + I *d*e*f^2)*e^(2*d*x + 2*c) + (d*f^3*x + d*e*f^2)*e^(d*x + c))*dilog(-I*e^(d *x + c)) - 3*(d^2*f^3*x^2 + d^2*e^2*f - 2*I*d*e*f^2 + 2*(d^2*e*f^2 - I*d*f ^3)*x + (-I*d^2*f^3*x^2 - I*d^2*e^2*f - 2*d*e*f^2 + 2*(-I*d^2*e*f^2 - d*f^ 3)*x)*e^(3*d*x + 3*c) - (d^2*f^3*x^2 + d^2*e^2*f - 2*I*d*e*f^2 + 2*(d^2*e* f^2 - I*d*f^3)*x)*e^(2*d*x + 2*c) + (I*d^2*f^3*x^2 + I*d^2*e^2*f + 2*d*e*f ^2 + 2*(I*d^2*e*f^2 + d*f^3)*x)*e^(d*x + c))*dilog(-e^(d*x + c)) + 3*(d^2* f^3*x^2 + d^2*e^2*f + 2*I*d*e*f^2 + 2*(d^2*e*f^2 + I*d*f^3)*x - (I*d^2*f^3 *x^2 + I*d^2*e^2*f - 2*d*e*f^2 + 2*(I*d^2*e*f^2 - d*f^3)*x)*e^(3*d*x + 3*c ) - (d^2*f^3*x^2 + d^2*e^2*f + 2*I*d*e*f^2 + 2*(d^2*e*f^2 + I*d*f^3)*x)*e^ (2*d*x + 2*c) - (-I*d^2*f^3*x^2 - I*d^2*e^2*f + 2*d*e*f^2 + 2*(-I*d^2*e*f^ 2 + d*f^3)*x)*e^(d*x + c))*dilog(e^(d*x + c)) - 4*(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^(3*d*x + 3*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 + I*d^3* e^3 - 6*I*c*d^2*e^2*f + 6*I*c^2*d*e*f^2 - 2*I*c^3*f^3)*e^(2*d*x + 2*c) + 2 *(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 + 6*c*d^2*e^2*f - 6*c^2*d*e*f^2 + 2*c^3*f^3)*e^(d*x + c) - (d^3*f^3*x^3 + d^3*e^3 - 3*I*d^ 2*e^2*f + 3*(d^3*e*f^2 - I*d^2*f^3)*x^2 + 3*(d^3*e^2*f - 2*I*d^2*e*f^2)*x - (I*d^3*f^3*x^3 + I*d^3*e^3 + 3*d^2*e^2*f - 3*(-I*d^3*e*f^2 - d^2*f^3)...
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3*csch(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)
Output:
Timed out
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 939 vs. \(2 (375) = 750\).
Time = 0.29 (sec) , antiderivative size = 939, normalized size of antiderivative = 2.24 \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima ")
Output:
-e^3*(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)) - 12*e^2*f*x/(a*d) + 3*e^2*f*log(e^( d*x + c) + 1)/(a*d^2) + 6*e^2*f*log(e^(d*x + c) - I)/(a*d^2) + 3*e^2*f*log (e^(d*x + c) - 1)/(a*d^2) - 2*(-2*I*f^3*x^3 - 6*I*e*f^2*x^2 - 6*I*e^2*f*x - (-I*f^3*x^3*e^(2*c) - 3*I*e*f^2*x^2*e^(2*c) - 3*I*e^2*f*x*e^(2*c))*e^(2* d*x) + (f^3*x^3*e^c + 3*e*f^2*x^2*e^c + 3*e^2*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) + 12*(d*x*lo g(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))*e*f^2/(a*d^3) + I*(d^3*x^3*l og(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) - I*(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) + 6*(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) - 3*(-I*d*e^2*f - 2*e*f^2)*(d*x*log(e^(d*x + c) + 1) + dilog(-e^(d*x + c)))/(a*d^3) + 3*(-I*d*e^2*f + 2*e*f^2)*(d*x*log(-e^(d*x + c) + 1) + di log(e^(d*x + c)))/(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)))*(-I*d*e*f^2 + f^3)/(a*d^4) - 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)))*(-I*d*e*f^2 - f^3)/(a*d^4) + 1/4*(I*d^4*f^3*x^4 - 4*(-I*d...
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:
int((e + f*x)^3/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)),x)
Output:
int((e + f*x)^3/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)), x)
\[ \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\int \frac {\mathrm {csch}\left (d x +c \right )^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{3}+\left (\int \frac {\mathrm {csch}\left (d x +c \right )^{2} x^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) f^{3}+3 \left (\int \frac {\mathrm {csch}\left (d x +c \right )^{2} x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) e \,f^{2}+3 \left (\int \frac {\mathrm {csch}\left (d x +c \right )^{2} x}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{2} f}{a} \] Input:
int((f*x+e)^3*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x)
Output:
(int(csch(c + d*x)**2/(sinh(c + d*x)*i + 1),x)*e**3 + int((csch(c + d*x)** 2*x**3)/(sinh(c + d*x)*i + 1),x)*f**3 + 3*int((csch(c + d*x)**2*x**2)/(sin h(c + d*x)*i + 1),x)*e*f**2 + 3*int((csch(c + d*x)**2*x)/(sinh(c + d*x)*i + 1),x)*e**2*f)/a