Integrand size = 31, antiderivative size = 546 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {2 i (e+f x)^3}{a d}-\frac {6 f^2 (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{a d}+\frac {i (e+f x)^3 \coth (c+d x)}{a d}-\frac {3 f (e+f x)^2 \text {csch}(c+d x)}{2 a d^2}-\frac {(e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {6 i f (e+f x)^2 \log \left (1+i e^{c+d x}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \log \left (1-e^{2 (c+d x)}\right )}{a d^2}-\frac {3 f^3 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{a d^4}+\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{2 a d^2}-\frac {12 i f^2 (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^3}+\frac {3 f^3 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{a d^4}-\frac {9 f (e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (2,e^{2 (c+d x)}\right )}{a d^3}-\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,-e^{c+d x}\right )}{a d^3}+\frac {12 i f^3 \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^4}+\frac {9 f^2 (e+f x) \operatorname {PolyLog}\left (3,e^{c+d x}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (3,e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 f^3 \operatorname {PolyLog}\left (4,-e^{c+d x}\right )}{a d^4}-\frac {9 f^3 \operatorname {PolyLog}\left (4,e^{c+d x}\right )}{a d^4}+\frac {i (e+f x)^3 \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{a d} \]
-6*I*f*(f*x+e)^2*ln(1+I*exp(d*x+c))/a/d^2-6*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-3*I*f^2*(f*x+e)*polylog(2,exp( 2*d*x+2*c))/a/d^3-3/2*f*(f*x+e)^2*csch(d*x+c)/a/d^2-1/2*(f*x+e)^3*coth(d*x +c)*csch(d*x+c)/a/d-12*I*f^2*(f*x+e)*polylog(2,-I*exp(d*x+c))/a/d^3+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+12*I*f^3*polylog(3,-I*exp(d*x+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+I*(f*x+e)^3*coth(d*x+c)/a/d-9*f^2*(f*x+e)*polylog(3,-exp(d*x+c ))/a/d^3-3*I*f*(f*x+e)^2*ln(1-exp(2*d*x+2*c))/a/d^2+9*f^2*(f*x+e)*polylog( 3,exp(d*x+c))/a/d^3+2*I*(f*x+e)^3/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+I*(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. \(2585\) vs. \(2(546)=1092\).
Time = 90.16 (sec) , antiderivative size = 2585, normalized size of antiderivative = 4.73 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \]
(-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 |
| \(\displaystyle -\frac {i \left (\frac {1}{2} \left (-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d}\right )}{d}+\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d}\right )}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 i (e+f x) \text {arctanh}\left (e^{c+d x}\right )}{d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{c+d x}\right )}{d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{c+d x}\right )}{d^2}\right )}{d^2}-\frac {3 i f (e+f x)^2 \text {csch}(c+d x)}{2 d^2}-\frac {i (e+f x)^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}\right )}{a}-i \left (-i \left (\frac {i \left (\frac {3 i f \int (e+f x)^2 \log \left (1-e^{c+d x}\right )dx}{d}-\frac {3 i f \int (e+f x)^2 \log \left (1+e^{c+d x}\right )dx}{d}+\frac {2 i (e+f x)^3 \text {arctanh}\left (e^{c+d x}\right )}{d}\right )}{a}-\frac {i \int (e+f x)^3 \csc \left (\frac {i c}{2}+\frac {i d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}\right )-\frac {\frac {(e+f x)^3 \coth (c+d x)}{d}+\frac {3 i f \left (2 i \left (\frac {(e+f x)^2 \log \left (1+e^{2 c+2 d x-i \pi }\right )}{2 d}-\frac {f \left (\frac {f \int e^{-2 c-2 d x+i \pi } \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )de^{2 c+2 d x-i \pi }}{4 d^2}-\frac {(e+f x) \operatorname {PolyLog}\left (2,-e^{2 c+2 d x-i \pi }\right )}{2 d}\right )}{d}\right )-\frac {i (e+f x)^3}{3 f}\right )}{d}}{a}\right )\) |
3.3.17.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp [(-(c + d*x)^m)*(Cot[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1) *Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbo l] :> Simp[(-b^2)*(c + d*x)^m*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*m*(c + d*x)^(m - 1)*((b*Csc[e + f*x])^(n - 2)/(f^ 2*(n - 1)*(n - 2))), x] + Simp[b^2*d^2*m*((m - 1)/(f^2*(n - 1)*(n - 2))) Int[(c + d*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Simp[b^2*((n - 2)/ (n - 1)) Int[(c + d*x)^m*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c , d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]
Int[(Csch[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Csch[ c + d*x]^n, x], x] - Simp[b/a Int[(e + f*x)^m*(Csch[c + d*x]^(n - 1)/(a + b*Sinh[c + d*x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2123 vs. \(2 (504 ) = 1008\).
Time = 3.28 (sec) , antiderivative size = 2124, normalized size of antiderivative = 3.89
6/a/d^2*e^2*f*arctan(exp(d*x+c))+6/a/d^4*c^2*f^3*arctan(exp(d*x+c))+12*I*f ^3*polylog(3,-I*exp(d*x+c))/a/d^4-12/a/d^3*c*f^2*e*arctan(exp(d*x+c))-6*I/ a/d^3*e*f^2*polylog(2,-exp(d*x+c))+6*I/a/d^3*c*f^2*e*ln(1+exp(2*d*x+2*c))- 6*I/a/d^3*e*f^2*ln(1-exp(d*x+c))*c+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*f^3*polylog(2,-exp(d*x+c))/a/d^4+3*f^3*pol ylog(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)-(-9*I*d*e^2*f*x*exp(3*d*x+3*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)+3*d*e^3*exp( 4*d*x+4*c)+3*e^2*f*exp(4*d*x+4*c)+4*d*f^3*x^3+I*e^3*d*exp(d*x+c)-9*I*d*e*f ^2*x^2*exp(3*d*x+3*c)+3*I*d*e*f^2*x^2*exp(d*x+c)+3*I*d*e^2*f*x*exp(d*x+c)+ 4*d*e^3+12*d*e*f^2*x^2+12*d*e^2*f*x+3*I*f^3*x^2*exp(d*x+c)+3*I*exp(d*x+c)* e^2*f+3*f^3*x^2*exp(4*d*x+4*c)-6*e*f^2*x*exp(2*d*x+2*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)-6*I*e*f^2*x*exp(3*d*x+3*c)-3*I*d*f^3 *x^3*exp(3*d*x+3*c)-3*f^3*x^2*exp(2*d*x+2*c)-5*f^3*x^3*d*exp(2*d*x+2*c)-3* e^2*f*exp(2*d*x+2*c)-5*e^3*d*exp(2*d*x+2*c)+3*d*f^3*x^3*exp(4*d*x+4*c)+6*e *f^2*x*exp(4*d*x+4*c)-3*I*f^3*x^2*exp(3*d*x+3*c)-15*e*f^2*x^2*d*exp(2*d*x+ 2*c)-15*e^2*f*x*d*exp(2*d*x+2*c))/(exp(2*d*x+2*c)-1)^2/d^2/(exp(d*x+c)-I)/ a-12*I/a/d^3*e*f^2*ln(1+I*exp(d*x+c))*c-24*I/a/d^3*c*e*f^2*ln(exp(d*x+c))- 6*I/a/d^2*e*f^2*ln(1-exp(d*x+c))*x-6*I/a/d^2*e*f^2*ln(exp(d*x+c)+1)*x-12*I /a/d^2*e*f^2*ln(1+I*exp(d*x+c))*x+24*I/a/d^2*e*f^2*c*x+6*I/a/d^3*c*f^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 = 0.32 (sec) , antiderivative size = 4252, normalized size of antiderivative = 7.79 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(8*d^3*e^3 - 24*c*d^2*e^2*f + 24*c^2*d*e*f^2 - 8*c^3*f^3 + 24*(d*f^3* x + d*e*f^2 + (I*d*f^3*x + I*d*e*f^2)*e^(5*d*x + 5*c) + (d*f^3*x + d*e*f^2 )*e^(4*d*x + 4*c) + 2*(-I*d*f^3*x - I*d*e*f^2)*e^(3*d*x + 3*c) - 2*(d*f^3* x + d*e*f^2)*e^(2*d*x + 2*c) + (I*d*f^3*x + I*d*e*f^2)*e^(d*x + c))*dilog( -I*e^(d*x + c)) + 3*(3*I*d^2*f^3*x^2 + 3*I*d^2*e^2*f + 4*d*e*f^2 - 2*I*f^3 + 2*(3*I*d^2*e*f^2 + 2*d*f^3)*x - (3*d^2*f^3*x^2 + 3*d^2*e^2*f - 4*I*d*e* f^2 - 2*f^3 + 2*(3*d^2*e*f^2 - 2*I*d*f^3)*x)*e^(5*d*x + 5*c) + (3*I*d^2*f^ 3*x^2 + 3*I*d^2*e^2*f + 4*d*e*f^2 - 2*I*f^3 + 2*(3*I*d^2*e*f^2 + 2*d*f^3)* x)*e^(4*d*x + 4*c) + 2*(3*d^2*f^3*x^2 + 3*d^2*e^2*f - 4*I*d*e*f^2 - 2*f^3 + 2*(3*d^2*e*f^2 - 2*I*d*f^3)*x)*e^(3*d*x + 3*c) + 2*(-3*I*d^2*f^3*x^2 - 3 *I*d^2*e^2*f - 4*d*e*f^2 + 2*I*f^3 + 2*(-3*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(2* d*x + 2*c) - (3*d^2*f^3*x^2 + 3*d^2*e^2*f - 4*I*d*e*f^2 - 2*f^3 + 2*(3*d^2 *e*f^2 - 2*I*d*f^3)*x)*e^(d*x + c))*dilog(-e^(d*x + c)) + 3*(-3*I*d^2*f^3* x^2 - 3*I*d^2*e^2*f + 4*d*e*f^2 + 2*I*f^3 + 2*(-3*I*d^2*e*f^2 + 2*d*f^3)*x + (3*d^2*f^3*x^2 + 3*d^2*e^2*f + 4*I*d*e*f^2 - 2*f^3 + 2*(3*d^2*e*f^2 + 2 *I*d*f^3)*x)*e^(5*d*x + 5*c) + (-3*I*d^2*f^3*x^2 - 3*I*d^2*e^2*f + 4*d*e*f ^2 + 2*I*f^3 + 2*(-3*I*d^2*e*f^2 + 2*d*f^3)*x)*e^(4*d*x + 4*c) - 2*(3*d^2* f^3*x^2 + 3*d^2*e^2*f + 4*I*d*e*f^2 - 2*f^3 + 2*(3*d^2*e*f^2 + 2*I*d*f^3)* x)*e^(3*d*x + 3*c) + 2*(3*I*d^2*f^3*x^2 + 3*I*d^2*e^2*f - 4*d*e*f^2 - 2*I* f^3 + 2*(3*I*d^2*e*f^2 - 2*d*f^3)*x)*e^(2*d*x + 2*c) + (3*d^2*f^3*x^2 +...
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Timed out} \]
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 = 0.52 (sec) , antiderivative size = 1320, normalized size of antiderivative = 2.42 \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
-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) -...
\[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {csch}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \text {csch}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]