Integrand size = 31, antiderivative size = 296 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {2 (e+f x)^2}{a d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{a d}-\frac {(e+f x)^2 \coth (c+d x)}{a d}+\frac {4 f (e+f x) \log \left (1+i e^{c+d x}\right )}{a d^2}+\frac {2 f (e+f x) \log \left (1-e^{2 (c+d x)}\right )}{a d^2}+\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^2}+\frac {4 f^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}-\frac {2 i f (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^2}+\frac {f^2 \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {2 i f^2 \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {2 i f^2 \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}-\frac {(e+f x)^2 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:
-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 = 7.96 (sec) , antiderivative size = 803, normalized size of antiderivative = 2.71 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
Integrate[((e + f*x)^2*Csch[c + d*x]^2)/(a + I*a*Sinh[c + d*x]),x]
Output:
((-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 |
| \(\displaystyle -i \left (\frac {i \left (-\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i f \left (\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (2,e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^2 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \left (\frac {4 i f \left (-2 \left (-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}-\frac {i (e+f x) \log \left (1+i e^{c+d x}\right )}{d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}+\frac {2 (e+f x)^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )}{d}\right )}{2 a}\right )-\frac {\frac {(e+f x)^2 \coth (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{4 d^2}+\frac {(e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}}{a}\) |
Input:
Int[((e + f*x)^2*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. 888 vs. \(2 (275 ) = 550\).
Time = 0.78 (sec) , antiderivative size = 889, normalized size of antiderivative = 3.00
| method | result | size |
| risch | \(-\frac {4 f^{2} c^{2}}{a \,d^{3}}-\frac {4 f^{2} x^{2}}{a d}+\frac {i f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x^{2}}{a d}+\frac {i f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c^{2}}{a \,d^{3}}-\frac {4 i c \,f^{2} \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}+\frac {2 i e f \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right )}{a \,d^{2}}-\frac {2 i e f \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {4 i e f \arctan \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {2 i f^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {i f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x^{2}}{a d}-\frac {2 i f^{2} \operatorname {polylog}\left (2, {\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}-\frac {i c^{2} f^{2} \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\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 ) d a}+\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 {2 f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{a \,d^{2}}+\frac {2 f^{2} \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{a \,d^{3}}-\frac {2 f^{2} c \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{3}}-\frac {2 f^{2} c \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{3}}-\frac {8 e f \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{2}}+\frac {2 e f \ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{a \,d^{2}}+\frac {2 e f \ln \left ({\mathrm e}^{d x +c}+1\right )}{a \,d^{2}}+\frac {2 e f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {8 c \,f^{2} \ln \left ({\mathrm e}^{d x +c}\right )}{a \,d^{3}}-\frac {8 f^{2} c x}{a \,d^{2}}+\frac {2 f^{2} \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a \,d^{2}}-\frac {2 i e f \ln \left (1-{\mathrm e}^{d x +c}\right ) c}{a \,d^{2}}+\frac {2 i e f \ln \left ({\mathrm e}^{d x +c}+1\right ) x}{a d}-\frac {2 i e f \ln \left (1-{\mathrm e}^{d x +c}\right ) x}{a d}+\frac {2 i e c f \ln \left ({\mathrm e}^{d x +c}-1\right )}{a \,d^{2}}+\frac {4 f^{2} \operatorname {polylog}\left (2, -i {\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 {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 )}{a d}+\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 {i e^{2} \ln \left ({\mathrm e}^{d x +c}+1\right )}{a d}\) | \(889\) |
Input:
int((f*x+e)^2*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
2*I*f^2*polylog(3,exp(d*x+c))/a/d^3-4/a/d^3*f^2*c^2-4/a/d*f^2*x^2+I/a/d*f^ 2*ln(exp(d*x+c)+1)*x^2+I/a/d^3*f^2*ln(1-exp(d*x+c))*c^2-2*I/a/d^2*e*f*ln(1 -exp(d*x+c))*c+2*I/a/d*e*f*ln(exp(d*x+c)+1)*x-2*I/a/d*e*f*ln(1-exp(d*x+c)) *x+2*I/a/d^2*e*c*f*ln(exp(d*x+c)-1)-2*I*f^2*polylog(3,-exp(d*x+c))/a/d^3-I /a/d*e^2*ln(exp(d*x+c)-1)+2*f^2*polylog(2,-exp(d*x+c))/a/d^3+2*f^2*polylog (2,exp(d*x+c))/a/d^3+2/a/d^2*f^2*ln(1-exp(d*x+c))*x+4/a/d^2*f^2*ln(1+I*exp (d*x+c))*x+2/a/d^3*f^2*ln(1-exp(d*x+c))*c+4/a/d^3*f^2*ln(1+I*exp(d*x+c))*c -2/a/d^3*f^2*c*ln(1+exp(2*d*x+2*c))-2/a/d^3*f^2*c*ln(exp(d*x+c)-1)-8/a/d^2 *e*f*ln(exp(d*x+c))+2/a/d^2*e*f*ln(1+exp(2*d*x+2*c))+2/a/d^2*e*f*ln(exp(d* x+c)+1)+2/a/d^2*e*f*ln(exp(d*x+c)-1)+8/a/d^3*c*f^2*ln(exp(d*x+c))-8/a/d^2* f^2*c*x+2/a/d^2*f^2*ln(exp(d*x+c)+1)*x+I/a/d*e^2*ln(exp(d*x+c)+1)-4*I/a/d^ 3*c*f^2*arctan(exp(d*x+c))+2*I/a/d^2*e*f*polylog(2,-exp(d*x+c))-2*I/a/d^2* e*f*polylog(2,exp(d*x+c))+4*I/a/d^2*e*f*arctan(exp(d*x+c))+2*I/a/d^2*f^2*p olylog(2,-exp(d*x+c))*x-I/a/d*f^2*ln(1-exp(d*x+c))*x^2-2*I/a/d^2*f^2*polyl og(2,exp(d*x+c))*x-I/a/d^3*c^2*f^2*ln(exp(d*x+c)-1)+4*f^2*polylog(2,-I*exp (d*x+c))/a/d^3-2*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)/d/a
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 = 0.12 (sec) , antiderivative size = 1355, normalized size of antiderivative = 4.58 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^2*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="fricas ")
Output:
(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...
\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{2} x^{2} \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {2 e f x \operatorname {csch}^{2}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \] Input:
integrate((f*x+e)**2*csch(d*x+c)**2/(a+I*a*sinh(d*x+c)),x)
Output:
-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 = 0.29 (sec) , antiderivative size = 601, normalized size of antiderivative = 2.03 \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)^2*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="maxima ")
Output:
-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)
\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{2} \operatorname {csch}\left (d x + c\right )^{2}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^2*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^2*csch(d*x + c)^2/(I*a*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^2}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:
int((e + f*x)^2/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)),x)
Output:
int((e + f*x)^2/(sinh(c + d*x)^2*(a + a*sinh(c + d*x)*1i)), x)
\[ \int \frac {(e+f x)^2 \text {csch}^2(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\int \frac {\mathrm {csch}\left (d x +c \right )^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{2}+\left (\int \frac {\mathrm {csch}\left (d x +c \right )^{2} x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) f^{2}+2 \left (\int \frac {\mathrm {csch}\left (d x +c \right )^{2} x}{\sinh \left (d x +c \right ) i +1}d x \right ) e f}{a} \] Input:
int((f*x+e)^2*csch(d*x+c)^2/(a+I*a*sinh(d*x+c)),x)
Output:
(int(csch(c + d*x)**2/(sinh(c + d*x)*i + 1),x)*e**2 + int((csch(c + d*x)** 2*x**2)/(sinh(c + d*x)*i + 1),x)*f**2 + 2*int((csch(c + d*x)**2*x)/(sinh(c + d*x)*i + 1),x)*e*f)/a