Integrand size = 31, antiderivative size = 546 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x)^3 \coth (c+d x)}{a d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \] Output:
12*I*f^3*polylog(3,-I*exp(d*x+c))/a/d^4-6*f^2*(f*x+e)*arctanh(exp(d*x+c))/ a/d^3+3*(f*x+e)^3*arctanh(exp(d*x+c))/a/d+I*(f*x+e)^3*tanh(1/2*c+1/4*I*Pi+ 1/2*d*x)/a/d-3/2*f*(f*x+e)^2*csch(d*x+c)/a/d^2-1/2*(f*x+e)^3*coth(d*x+c)*c sch(d*x+c)/a/d-6*I*f*(f*x+e)^2*ln(1+I*exp(d*x+c))/a/d^2+2*I*(f*x+e)^3/a/d- 3*f^3*polylog(2,-exp(d*x+c))/a/d^4+9/2*f*(f*x+e)^2*polylog(2,-exp(d*x+c))/ a/d^2+3/2*I*f^3*polylog(3,exp(2*d*x+2*c))/a/d^4+3*f^3*polylog(2,exp(d*x+c) )/a/d^4-9/2*f*(f*x+e)^2*polylog(2,exp(d*x+c))/a/d^2-3*I*f*(f*x+e)^2*ln(1-e xp(2*d*x+2*c))/a/d^2-9*f^2*(f*x+e)*polylog(3,-exp(d*x+c))/a/d^3-12*I*f^2*( f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3+9*f^2*(f*x+e)*polylog(3,exp(d*x+c))/ a/d^3+I*(f*x+e)^3*coth(d*x+c)/a/d+9*f^3*polylog(4,-exp(d*x+c))/a/d^4-9*f^3 *polylog(4,exp(d*x+c))/a/d^4-3*I*f^2*(f*x+e)*polylog(2,exp(2*d*x+2*c))/a/d ^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2585\) vs. \(2(546)=1092\).
Time = 108.77 (sec) , antiderivative size = 2585, normalized size of antiderivative = 4.73 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Csch[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
(-6*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)) + ((12*I)* d^3*e^2*E^(2*c)*f*x + (12*I)*d^3*e*E^(2*c)*f^2*x^2 + (4*I)*d^3*E^(2*c)*f^3 *x^3 - 6*d^3*e^3*ArcTanh[E^(c + d*x)] + 6*d^3*e^3*E^(2*c)*ArcTanh[E^(c + d *x)] + 12*d*e*f^2*ArcTanh[E^(c + d*x)] - 12*d*e*E^(2*c)*f^2*ArcTanh[E^(c + d*x)] + 9*d^3*e^2*f*x*Log[1 - E^(c + d*x)] - 9*d^3*e^2*E^(2*c)*f*x*Log[1 - E^(c + d*x)] - 6*d*f^3*x*Log[1 - E^(c + d*x)] + 6*d*E^(2*c)*f^3*x*Log[1 - E^(c + d*x)] + 9*d^3*e*f^2*x^2*Log[1 - E^(c + d*x)] - 9*d^3*e*E^(2*c)*f^ 2*x^2*Log[1 - E^(c + d*x)] + 3*d^3*f^3*x^3*Log[1 - E^(c + d*x)] - 3*d^3*E^ (2*c)*f^3*x^3*Log[1 - E^(c + d*x)] - 9*d^3*e^2*f*x*Log[1 + E^(c + d*x)] + 9*d^3*e^2*E^(2*c)*f*x*Log[1 + E^(c + d*x)] + 6*d*f^3*x*Log[1 + E^(c + d*x) ] - 6*d*E^(2*c)*f^3*x*Log[1 + E^(c + d*x)] - 9*d^3*e*f^2*x^2*Log[1 + E^(c + d*x)] + 9*d^3*e*E^(2*c)*f^2*x^2*Log[1 + E^(c + d*x)] - 3*d^3*f^3*x^3*Log [1 + E^(c + d*x)] + 3*d^3*E^(2*c)*f^3*x^3*Log[1 + E^(c + d*x)] + (6*I)*d^2 *e^2*f*Log[1 - E^(2*(c + d*x))] - (6*I)*d^2*e^2*E^(2*c)*f*Log[1 - E^(2*(c + d*x))] + (12*I)*d^2*e*f^2*x*Log[1 - E^(2*(c + d*x))] - (12*I)*d^2*e*E^(2 *c)*f^2*x*Log[1 - E^(2*(c + d*x))] + (6*I)*d^2*f^3*x^2*Log[1 - E^(2*(c + d *x))] - (6*I)*d^2*E^(2*c)*f^3*x^2*Log[1 - E^(2*(c + d*x))] + 3*(-1 + E^(2* c))*f*(-2*f^2 + 3*d^2*(e + f*x)^2)*PolyLog[2, -E^(c + d*x)] - 3*(-1 + E...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle \frac {\int (e+f x)^3 \text {csch}^3(c+d x)dx}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -i (e+f x)^3 \csc (i c+i d x)^3dx}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int (e+f x)^3 \csc (i c+i d x)^3dx}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle -\frac {i \left (-\frac {3 f^2 \int -i (e+f x) \text {csch}(c+d x)dx}{d^2}+\frac {1}{2} \int -i (e+f x)^3 \text {csch}(c+d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {3 i f^2 \int (e+f x) \text {csch}(c+d x)dx}{d^2}-\frac {1}{2} i \int (e+f x)^3 \text {csch}(c+d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {3 i f^2 \int i (e+f x) \csc (i c+i d x)dx}{d^2}-\frac {1}{2} i \int i (e+f x)^3 \csc (i c+i d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (-\frac {3 f^2 \int (e+f x) \csc (i c+i d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc (i c+i d x)dx-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {i \left (-\frac {3 f^2 \left (\frac {i f \int \log \left (1-e^{c+d x}\right )dx}{d}-\frac {i f \int \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {i \left (-\frac {3 f^2 \left (\frac {i f \int e^{-c-d x} \log \left (1-e^{c+d x}\right )de^{c+d x}}{d^2}-\frac {i f \int e^{-c-d x} \log \left (1+e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \int \frac {(e+f x)^3 \text {csch}^2(c+d x)}{i \sinh (c+d x) a+a}dx\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (\frac {\int (e+f x)^3 \text {csch}^2(c+d x)dx}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (\frac {\int -(e+f x)^3 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\int (e+f x)^3 \csc (i c+i d x)^2dx}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 i f \int -i (e+f x)^2 \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 f \int (e+f x)^2 \coth (c+d x)dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}-\frac {3 f \int -i (e+f x)^2 \tan \left (i c+i d x+\frac {\pi }{2}\right )dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \int (e+f x)^2 \tan \left (\frac {1}{2} (2 i c+\pi )+i d x\right )dx}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \int \frac {e^{2 c+2 d x-i \pi } (e+f x)^2}{1+e^{2 c+2 d x-i \pi }}dx-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \int (e+f x) \log \left (1+e^{2 c+2 d x-i \pi }\right )dx}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )dx}{2 d}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \int \frac {(e+f x)^3 \text {csch}(c+d x)}{i \sinh (c+d x) a+a}dx\right )\) |
| \(\Big \downarrow \) 6109 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {\int (e+f x)^3 \text {csch}(c+d x)dx}{a}-i \int \frac {(e+f x)^3}{i \sinh (c+d x) a+a}dx\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {\int i (e+f x)^3 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^3}{\sin (i c+i d x) a+a}dx\right )\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-i \int \frac {(e+f x)^3}{\sin (i c+i d x) a+a}dx\right )\right )\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {i \int -(e+f x)^3 \text {csch}^2\left (\frac {c}{2}+\frac {d x}{2}-\frac {i \pi }{4}\right )dx}{2 a}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}+\frac {i \int -(e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^3 \text {sech}^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right )dx}{2 a}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}-i \left (\frac {i \int (e+f x)^3 \csc (i c+i d x)dx}{a}-\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )\right )\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {i \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )\) |
Input:
Int[((e + f*x)^3*Csch[c + d*x]^3)/(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. 2123 vs. \(2 (504 ) = 1008\).
Time = 1.40 (sec) , antiderivative size = 2124, normalized size of antiderivative = 3.89
Input:
int((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
3*I/a/d^4*c^2*f^3*ln(1-exp(d*x+c))+6*I/a/d^4*c^2*f^3*ln(1+I*exp(d*x+c))+12 *I/a/d^2*e^2*f*ln(exp(d*x+c))-3*I/a/d^2*e^2*f*ln(1+exp(2*d*x+2*c))-3*I/a/d ^2*e^2*f*ln(exp(d*x+c)+1)-3*I/a/d^2*e^2*f*ln(exp(d*x+c)-1)+12*I/a/d^3*e*f^ 2*c^2-12*I/a/d^3*c^2*f^3*x-6*I/a/d^3*e*f^2*polylog(2,-exp(d*x+c))-12*I/a/d ^3*e*f^2*polylog(2,-I*exp(d*x+c))-6*I/a/d^3*e*f^2*polylog(2,exp(d*x+c))+12 *I/a/d*e*f^2*x^2+12*I/a/d^4*c^2*f^3*ln(exp(d*x+c))-3*I/a/d^2*f^3*ln(exp(d* x+c)+1)*x^2-6*I/a/d^3*f^3*polylog(2,-exp(d*x+c))*x-3*I/a/d^2*f^3*ln(1-exp( d*x+c))*x^2-6*I/a/d^3*f^3*polylog(2,exp(d*x+c))*x-6*I/a/d^2*f^3*ln(1+I*exp (d*x+c))*x^2-12*I/a/d^3*f^3*polylog(2,-I*exp(d*x+c))*x-3*I/a/d^4*c^2*f^3*l n(1+exp(2*d*x+2*c))-3*I/a/d^4*c^2*f^3*ln(exp(d*x+c)-1)-12/a/d^3*c*f^2*e*ar ctan(exp(d*x+c))-3*f^3*polylog(2,-exp(d*x+c))/a/d^4+3*f^3*polylog(2,exp(d* x+c))/a/d^4+3/2/a/d*e^3*ln(exp(d*x+c)+1)-3/2/a/d*e^3*ln(exp(d*x+c)-1)-(I*d *e^3*exp(d*x+c)+3*I*f^3*x^2*exp(d*x+c)+3*I*e^2*f*exp(d*x+c)+3*d*e^3*exp(4* d*x+4*c)+3*e^2*f*exp(4*d*x+4*c)-6*I*e*f^2*x*exp(3*d*x+3*c)-3*I*d*f^3*x^3*e xp(3*d*x+3*c)+9*d*e*f^2*x^2*exp(4*d*x+4*c)+9*d*e^2*f*x*exp(4*d*x+4*c)-9*I* d*e*f^2*x^2*exp(3*d*x+3*c)+12*d*e*f^2*x^2+12*d*e^2*f*x+3*I*d*e*f^2*x^2*exp (d*x+c)+3*I*d*e^2*f*x*exp(d*x+c)-9*I*d*e^2*f*x*exp(3*d*x+3*c)+6*e*f^2*x*ex p(4*d*x+4*c)-3*I*f^3*x^2*exp(3*d*x+3*c)+3*d*f^3*x^3*exp(4*d*x+4*c)-3*I*d*e ^3*exp(3*d*x+3*c)-3*I*e^2*f*exp(3*d*x+3*c)+6*I*e*f^2*x*exp(d*x+c)+I*d*f^3* x^3*exp(d*x+c)+4*d*e^3-3*f^3*x^2*exp(2*d*x+2*c)-5*f^3*x^3*d*exp(2*d*x+2...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4252 vs. \(2 (485) = 970\).
Time = 0.19 (sec) , antiderivative size = 4252, normalized size of antiderivative = 7.79 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas ")
Output:
Too large to include
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3*csch(d*x+c)**3/(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. 1320 vs. \(2 (485) = 970\).
Time = 0.39 (sec) , antiderivative size = 1320, normalized size of antiderivative = 2.42 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima ")
Output:
-1/2*e^3*(2*(-I*e^(-d*x - c) - 5*e^(-2*d*x - 2*c) + 3*I*e^(-3*d*x - 3*c) + 3*e^(-4*d*x - 4*c) + 4)/((a*e^(-d*x - c) - 2*I*a*e^(-2*d*x - 2*c) - 2*a*e ^(-3*d*x - 3*c) + I*a*e^(-4*d*x - 4*c) + a*e^(-5*d*x - 5*c) + I*a)*d) - 3* log(e^(-d*x - c) + 1)/(a*d) + 3*log(e^(-d*x - c) - 1)/(a*d)) + 6*I*e^2*f*x /(a*d) - 6*I*e^2*f*log(I*e^(d*x + c) + 1)/(a*d^2) - (4*d*f^3*x^3 + 12*d*e* f^2*x^2 + 12*d*e^2*f*x + 3*(d*f^3*x^3*e^(4*c) + e^2*f*e^(4*c) + (3*d*e*f^2 + f^3)*x^2*e^(4*c) + (3*d*e^2*f + 2*e*f^2)*x*e^(4*c))*e^(4*d*x) - 3*(I*d* f^3*x^3*e^(3*c) + I*e^2*f*e^(3*c) + (3*I*d*e*f^2 + I*f^3)*x^2*e^(3*c) + (3 *I*d*e^2*f + 2*I*e*f^2)*x*e^(3*c))*e^(3*d*x) - (5*d*f^3*x^3*e^(2*c) + 3*e^ 2*f*e^(2*c) + 3*(5*d*e*f^2 + f^3)*x^2*e^(2*c) + 3*(5*d*e^2*f + 2*e*f^2)*x* e^(2*c))*e^(2*d*x) + (I*d*f^3*x^3*e^c + 3*I*e^2*f*e^c - 3*(-I*d*e*f^2 - I* f^3)*x^2*e^c - 3*(-I*d*e^2*f - 2*I*e*f^2)*x*e^c)*e^(d*x))/(a*d^2*e^(5*d*x + 5*c) - I*a*d^2*e^(4*d*x + 4*c) - 2*a*d^2*e^(3*d*x + 3*c) + 2*I*a*d^2*e^( 2*d*x + 2*c) + a*d^2*e^(d*x + c) - I*a*d^2) - 12*I*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))*e*f^2/(a*d^3) + 3/2*(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) - 3/2*(d^3*x^3*log(-e^(d*x + c) + 1 ) + 3*d^2*x^2*dilog(e^(d*x + c)) - 6*d*x*polylog(3, e^(d*x + c)) + 6*polyl og(4, e^(d*x + c)))*f^3/(a*d^4) - 6*I*(d^2*x^2*log(I*e^(d*x + c) + 1) + 2* d*x*dilog(-I*e^(d*x + c)) - 2*polylog(3, -I*e^(d*x + c)))*f^3/(a*d^4) -...
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:
int((e + f*x)^3/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)
Output:
int((e + f*x)^3/(sinh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)), x)
\[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\int \frac {\mathrm {csch}\left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{3}+\left (\int \frac {\mathrm {csch}\left (d x +c \right )^{3} x^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) f^{3}+3 \left (\int \frac {\mathrm {csch}\left (d x +c \right )^{3} x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) e \,f^{2}+3 \left (\int \frac {\mathrm {csch}\left (d x +c \right )^{3} x}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{2} f}{a} \] Input:
int((f*x+e)^3*csch(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)
Output:
(int(csch(c + d*x)**3/(sinh(c + d*x)*i + 1),x)*e**3 + int((csch(c + d*x)** 3*x**3)/(sinh(c + d*x)*i + 1),x)*f**3 + 3*int((csch(c + d*x)**3*x**2)/(sin h(c + d*x)*i + 1),x)*e*f**2 + 3*int((csch(c + d*x)**3*x)/(sinh(c + d*x)*i + 1),x)*e**2*f)/a