Integrand size = 31, antiderivative size = 667 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 f^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{4 a d}+\frac {i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{4 a d^4}-\frac {f^3 \text {sech}(c+d x)}{4 a d^4}+\frac {9 f (e+f x)^2 \text {sech}(c+d x)}{8 a d^2}-\frac {i f^2 (e+f x) \text {sech}^2(c+d x)}{4 a d^3}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 a d^2}+\frac {i (e+f x)^3 \text {sech}^4(c+d x)}{4 a d}+\frac {i f^3 \tanh (c+d x)}{4 a d^4}-\frac {i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac {f^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d} \] Output:
9/4*I*f^3*polylog(4,I*exp(d*x+c))/a/d^4-5*f^2*(f*x+e)*arctan(exp(d*x+c))/a /d^3+3/4*(f*x+e)^3*arctan(exp(d*x+c))/a/d-5/2*I*f^3*polylog(2,I*exp(d*x+c) )/a/d^4-1/2*I*f*(f*x+e)^2*tanh(d*x+c)/a/d^2+9/4*I*f^2*(f*x+e)*polylog(3,-I *exp(d*x+c))/a/d^3+1/2*I*f^3*polylog(2,-exp(2*d*x+2*c))/a/d^4-9/4*I*f^2*(f *x+e)*polylog(3,I*exp(d*x+c))/a/d^3+9/8*I*f*(f*x+e)^2*polylog(2,I*exp(d*x+ c))/a/d^2+1/4*I*f^3*tanh(d*x+c)/a/d^4+1/4*I*(f*x+e)^3*sech(d*x+c)^4/a/d-1/ 4*I*f*(f*x+e)^2*sech(d*x+c)^2*tanh(d*x+c)/a/d^2-9/4*I*f^3*polylog(4,-I*exp (d*x+c))/a/d^4-1/4*f^3*sech(d*x+c)/a/d^4+9/8*f*(f*x+e)^2*sech(d*x+c)/a/d^2 -1/4*I*f^2*(f*x+e)*sech(d*x+c)^2/a/d^3+1/4*f*(f*x+e)^2*sech(d*x+c)^3/a/d^2 +5/2*I*f^3*polylog(2,-I*exp(d*x+c))/a/d^4-9/8*I*f*(f*x+e)^2*polylog(2,-I*e xp(d*x+c))/a/d^2-1/2*I*f*(f*x+e)^2/a/d^2-1/4*f^2*(f*x+e)*sech(d*x+c)*tanh( d*x+c)/a/d^3+3/8*(f*x+e)^3*sech(d*x+c)*tanh(d*x+c)/a/d+I*f^2*(f*x+e)*ln(1+ exp(2*d*x+2*c))/a/d^3+1/4*(f*x+e)^3*sech(d*x+c)^3*tanh(d*x+c)/a/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2008\) vs. \(2(667)=1334\).
Time = 8.83 (sec) , antiderivative size = 2008, normalized size of antiderivative = 3.01 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \] Input:
Integrate[((e + f*x)^3*Sech[c + d*x]^3)/(a + I*a*Sinh[c + d*x]),x]
Output:
(-3*E^c*((d^2*e^3*x)/E^c - (4*e*f^2*x)/E^c - (e*(1 - I*E^c)*(d^2*e^2 - 4*f ^2)*x)/E^c + (3*d^2*e^2*f*x^2)/(2*E^c) - (2*f^3*x^2)/E^c + (d^2*e*f^2*x^3) /E^c + (d^2*f^3*x^4)/(4*E^c) + ((1 - I*E^c)*f*(3*d^2*e^2 - 4*f^2)*x*Log[1 + I*E^(-c - d*x)])/(d*E^c) + (3*d*e*(1 - I*E^c)*f^2*x^2*Log[1 + I*E^(-c - d*x)])/E^c + (d*(1 - I*E^c)*f^3*x^3*Log[1 + I*E^(-c - d*x)])/E^c + (e*(1 - I*E^c)*(d^2*e^2 - 4*f^2)*Log[I + E^(c + d*x)])/(d*E^c) - ((1 - I*E^c)*f*( 3*d^2*e^2 - 4*f^2)*PolyLog[2, (-I)*E^(-c - d*x)])/(d^2*E^c) - (6*e*(1 - I* E^c)*f^2*x*PolyLog[2, (-I)*E^(-c - d*x)])/E^c - (3*(1 - I*E^c)*f^3*x^2*Pol yLog[2, (-I)*E^(-c - d*x)])/E^c - (6*e*(1 - I*E^c)*f^2*PolyLog[3, (-I)*E^( -c - d*x)])/(d*E^c) - (6*(1 - I*E^c)*f^3*x*PolyLog[3, (-I)*E^(-c - d*x)])/ (d*E^c) - (6*(1 - I*E^c)*f^3*PolyLog[4, (-I)*E^(-c - d*x)])/(d^2*E^c)))/(8 *a*d^2*(I + E^c)) - (-12*d^2*e*(1 + I*E^c)*f*(3*d^2*e^2 - 28*f^2)*x + (28* f^2 - 3*d^2*(e + f*x)^2)^2 + 12*d*(1 + I*E^c)*f^2*(9*d^2*e^2 - 28*f^2)*x*L og[1 - I*E^(-c - d*x)] + 108*d^3*e*(1 + I*E^c)*f^3*x^2*Log[1 - I*E^(-c - d *x)] + 36*d^3*(1 + I*E^c)*f^4*x^3*Log[1 - I*E^(-c - d*x)] + 12*d*e*(1 + I* E^c)*f*(3*d^2*e^2 - 28*f^2)*Log[I - E^(c + d*x)] + 12*(1 + I*E^c)*f^2*(-9* d^2*e^2 + 28*f^2)*PolyLog[2, I*E^(-c - d*x)] - 216*d^2*e*(1 + I*E^c)*f^3*x *PolyLog[2, I*E^(-c - d*x)] - 108*d^2*(1 + I*E^c)*f^4*x^2*PolyLog[2, I*E^( -c - d*x)] - 216*d*e*(1 + I*E^c)*f^3*PolyLog[3, I*E^(-c - d*x)] - 216*d*(1 + I*E^c)*f^4*x*PolyLog[3, I*E^(-c - d*x)] - 216*(1 + I*E^c)*f^4*PolyLo...
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.
| \(\displaystyle \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 6105 |
| \(\displaystyle \frac {\int (e+f x)^3 \text {sech}^5(c+d x)dx}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^5dx}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {-\frac {f^2 \int (e+f x) \text {sech}^3(c+d x)dx}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \text {sech}^3(c+d x)dx+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 4673 |
| \(\displaystyle \frac {-\frac {f^2 \left (\frac {1}{2} \int (e+f x) \text {sech}(c+d x)dx+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {f^2 \left (\frac {1}{2} \int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {-\frac {f^2 \left (\frac {1}{2} \left (-\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {-\frac {f^2 \left (\frac {1}{2} \left (-\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {3}{4} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {3}{4} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3 f^2 \int (e+f x) \text {sech}(c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \text {sech}(c+d x)dx+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3 f^2 \int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3 f^2 \left (-\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {3}{4} \left (-\frac {3 f^2 \left (-\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^4(c+d x) \tanh (c+d x)dx}{a}\) |
| \(\Big \downarrow \) 5974 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (\frac {3 f \int (e+f x)^2 \text {sech}^4(c+d x)dx}{4 d}-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )^4dx}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (\frac {3 f \left (-\frac {f^2 \int \text {sech}^2(c+d x)dx}{3 d^2}+\frac {2}{3} \int (e+f x)^2 \text {sech}^2(c+d x)dx+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (-\frac {f^2 \int \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{3 d^2}+\frac {2}{3} \int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (-\frac {i f^2 \int 1d(-i \tanh (c+d x))}{3 d^3}+\frac {2}{3} \int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (\frac {2}{3} \int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 4672 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \tanh (c+d x)dx}{d}\right )-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 f \int (e+f x) \tanh (c+d x)dx}{d}\right )-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \tan (i c+i d x)dx}{d}\right )-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \int (e+f x) \tan (i c+i d x)dx}{d}\right )-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int e^{-2 (c+d x)} \log \left (1+e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle \frac {\frac {3}{4} \left (\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )-\frac {f^2 \left (\frac {1}{2} \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )+\frac {f \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x) \tanh (c+d x) \text {sech}(c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^4(c+d x)}{4 d}+\frac {3 f \left (-\frac {f^2 \tanh (c+d x)}{3 d^3}+\frac {2}{3} \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )+\frac {f (e+f x) \text {sech}^2(c+d x)}{3 d^2}+\frac {(e+f x)^2 \tanh (c+d x) \text {sech}^2(c+d x)}{3 d}\right )}{4 d}\right )}{a}\) |
Input:
Int[((e + f*x)^3*Sech[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. 1908 vs. \(2 (589 ) = 1178\).
Time = 1.41 (sec) , antiderivative size = 1909, normalized size of antiderivative = 2.86
\[\text {Expression too large to display}\]
Input:
int((f*x+e)^3*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)
Output:
1/4*(18*I*d^3*e^2*f*x*exp(2*d*x+2*c)-36*I*d^2*e*f^2*x*exp(4*d*x+4*c)-44*I* d^2*e*f^2*x*exp(2*d*x+2*c)-18*I*d^3*e^2*f*x*exp(4*d*x+4*c)-18*I*d^3*e*f^2* x^2*exp(4*d*x+4*c)+18*I*d^3*e*f^2*x^2*exp(2*d*x+2*c)+9*d^3*e*f^2*x^2*exp(5 *d*x+5*c)-6*I*d^3*e^3*exp(4*d*x+4*c)+3*d^3*f^3*x^3*exp(5*d*x+5*c)+2*d^3*f^ 3*x^3*exp(3*d*x+3*c)+6*I*d^3*e^3*exp(2*d*x+2*c)-2*d*e*f^2*exp(5*d*x+5*c)-2 *d*f^3*x*exp(5*d*x+5*c)+9*d^2*f^3*x^2*exp(5*d*x+5*c)+9*d^2*e^2*f*exp(5*d*x +5*c)+18*d^2*e*f^2*x*exp(5*d*x+5*c)+3*d^3*e^3*exp(5*d*x+5*c)+4*I*f^3*exp(2 *d*x+2*c)+2*d^3*e^3*exp(3*d*x+3*c)+2*I*f^3*exp(4*d*x+4*c)-8*I*d^2*e*f^2*x+ 6*d^3*e^2*f*x*exp(3*d*x+3*c)+6*d^3*e*f^2*x^2*exp(3*d*x+3*c)+9*d^3*e^2*f*x* exp(5*d*x+5*c)-22*I*d^2*f^3*x^2*exp(2*d*x+2*c)-22*I*d^2*e^2*f*exp(2*d*x+2* c)+6*I*d^3*f^3*x^3*exp(2*d*x+2*c)-6*I*d^3*f^3*x^3*exp(4*d*x+4*c)-18*I*d^2* f^3*x^2*exp(4*d*x+4*c)-18*I*d^2*e^2*f*exp(4*d*x+4*c)-2*f^3*exp(5*d*x+5*c)- 4*I*d^2*f^3*x^2-4*I*d^2*e^2*f+2*I*f^3-4*d*e*f^2*exp(3*d*x+3*c)+16*d^2*e*f^ 2*x*exp(3*d*x+3*c)-d^2*e^2*f*exp(d*x+c)-2*d*f^3*x*exp(d*x+c)-2*d*e*f^2*exp (d*x+c)-2*d^2*e*f^2*x*exp(d*x+c)+9*d^3*e*f^2*x^2*exp(d*x+c)+8*d^2*f^3*x^2* exp(3*d*x+3*c)+8*d^2*e^2*f*exp(3*d*x+3*c)+3*d^3*e^3*exp(d*x+c)-4*f^3*exp(3 *d*x+3*c)-2*f^3*exp(d*x+c)-d^2*f^3*x^2*exp(d*x+c)+3*d^3*f^3*x^3*exp(d*x+c) -4*d*f^3*x*exp(3*d*x+3*c)+9*d^3*e^2*f*x*exp(d*x+c))/(exp(d*x+c)+I)^2/(exp( d*x+c)-I)^4/d^4/a+5/a/d^4*f^3*c*arctan(exp(d*x+c))-3/4/a/d^4*f^3*c^3*arcta n(exp(d*x+c))-5/a/d^3*e*f^2*arctan(exp(d*x+c))+3/4/a/d*e^3*arctan(exp(d...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3854 vs. \(2 (560) = 1120\).
Time = 0.14 (sec) , antiderivative size = 3854, normalized size of antiderivative = 5.78 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="fricas ")
Output:
Too large to include
\[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{3} \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{3} x^{3} \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e f^{2} x^{2} \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e^{2} f x \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \] Input:
integrate((f*x+e)**3*sech(d*x+c)**3/(a+I*a*sinh(d*x+c)),x)
Output:
-I*(Integral(e**3*sech(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(f**3 *x**3*sech(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(3*e*f**2*x**2*se ch(c + d*x)**3/(sinh(c + d*x) - I), x) + Integral(3*e**2*f*x*sech(c + d*x) **3/(sinh(c + d*x) - I), x))/a
Exception generated. \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((f*x+e)^3*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="maxima ")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
\[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {sech}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \] Input:
integrate((f*x+e)^3*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x, algorithm="giac")
Output:
integrate((f*x + e)^3*sech(d*x + c)^3/(I*a*sinh(d*x + c) + a), x)
Timed out. \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \] Input:
int((e + f*x)^3/(cosh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)),x)
Output:
int((e + f*x)^3/(cosh(c + d*x)^3*(a + a*sinh(c + d*x)*1i)), x)
\[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\frac {\left (\int \frac {\mathrm {sech}\left (d x +c \right )^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{3}+\left (\int \frac {\mathrm {sech}\left (d x +c \right )^{3} x^{3}}{\sinh \left (d x +c \right ) i +1}d x \right ) f^{3}+3 \left (\int \frac {\mathrm {sech}\left (d x +c \right )^{3} x^{2}}{\sinh \left (d x +c \right ) i +1}d x \right ) e \,f^{2}+3 \left (\int \frac {\mathrm {sech}\left (d x +c \right )^{3} x}{\sinh \left (d x +c \right ) i +1}d x \right ) e^{2} f}{a} \] Input:
int((f*x+e)^3*sech(d*x+c)^3/(a+I*a*sinh(d*x+c)),x)
Output:
(int(sech(c + d*x)**3/(sinh(c + d*x)*i + 1),x)*e**3 + int((sech(c + d*x)** 3*x**3)/(sinh(c + d*x)*i + 1),x)*f**3 + 3*int((sech(c + d*x)**3*x**2)/(sin h(c + d*x)*i + 1),x)*e*f**2 + 3*int((sech(c + d*x)**3*x)/(sinh(c + d*x)*i + 1),x)*e**2*f)/a