Integrand size = 31, antiderivative size = 667 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {i f (e+f x)^2}{2 a d^2}-\frac {5 f^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {3 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{4 a d}+\frac {i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {5 i f^3 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^4}-\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{8 a d^2}-\frac {5 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^4}+\frac {9 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{8 a d^2}+\frac {i f^3 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}+\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{4 a d^3}-\frac {9 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{4 a d^4}+\frac {9 i f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{4 a d^4}-\frac {f^3 \text {sech}(c+d x)}{4 a d^4}+\frac {9 f (e+f x)^2 \text {sech}(c+d x)}{8 a d^2}-\frac {i f^2 (e+f x) \text {sech}^2(c+d x)}{4 a d^3}+\frac {f (e+f x)^2 \text {sech}^3(c+d x)}{4 a d^2}+\frac {i (e+f x)^3 \text {sech}^4(c+d x)}{4 a d}+\frac {i f^3 \tanh (c+d x)}{4 a d^4}-\frac {i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}-\frac {f^2 (e+f x) \text {sech}(c+d x) \tanh (c+d x)}{4 a d^3}+\frac {3 (e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{8 a d}-\frac {i f (e+f x)^2 \text {sech}^2(c+d x) \tanh (c+d x)}{4 a d^2}+\frac {(e+f x)^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 a d} \]
1/4*I*(f*x+e)^3*sech(d*x+c)^4/a/d-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+1/4*I*f^3*tanh(d*x+c)/a/d^4+9/8*I*f*(f *x+e)^2*polylog(2,I*exp(d*x+c))/a/d^2+5/2*I*f^3*polylog(2,-I*exp(d*x+c))/a /d^4+1/2*I*f^3*polylog(2,-exp(2*d*x+2*c))/a/d^4-5/2*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)*polylo g(3,-I*exp(d*x+c))/a/d^3-9/4*I*f^3*polylog(4,-I*exp(d*x+c))/a/d^4-1/4*I*f^ 2*(f*x+e)*sech(d*x+c)^2/a/d^3+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^2-1/2*I*f*(f*x+e)^2/ a/d^2+1/4*f*(f*x+e)^2*sech(d*x+c)^3/a/d^2+I*f^2*(f*x+e)*ln(1+exp(2*d*x+2*c ))/a/d^3-1/4*I*f*(f*x+e)^2*sech(d*x+c)^2*tanh(d*x+c)/a/d^2-9/8*I*f*(f*x+e) ^2*polylog(2,-I*exp(d*x+c))/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-9/4*I*f^2*(f*x+e)*polylog( 3,I*exp(d*x+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 = 9.27 (sec) , antiderivative size = 2008, normalized size of antiderivative = 3.01 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Result too large to show} \]
(-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}\) |
3.3.83.3.1 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[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_.)*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[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && 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_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (-Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x] + S imp[b^2*((n - 2)/(n - 1)) Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2]
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[((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Sech[a + b*x]^n/(b*n)) , x] + Simp[d*(m/(b*n)) Int[(c + d*x)^(m - 1)*Sech[a + b*x]^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_ .)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/a Int[(e + f*x)^m*Sech[ c + d*x]^(n + 2), x], x] + Simp[1/b Int[(e + f*x)^m*Sech[c + d*x]^(n + 1) *Tanh[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 + b^2, 0]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
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 = 188.52 (sec) , antiderivative size = 1909, normalized size of antiderivative = 2.86
-9/4*I*f^3*polylog(4,-I*exp(d*x+c))/a/d^4+9/8*I/a/d*e*f^2*ln(1-I*exp(d*x+c ))*x^2+9/4*I/a/d^2*e*f^2*polylog(2,I*exp(d*x+c))*x+9/4*I*f^3*polylog(4,I*e xp(d*x+c))/a/d^4+9/8*I/a/d^2*e^2*f*ln(1-I*exp(d*x+c))*c-9/8*I/a/d*e^2*f*ln (1+I*exp(d*x+c))*x-9/8*I/a/d^2*e^2*f*ln(1+I*exp(d*x+c))*c-9/4*I/a/d^2*e*f^ 2*polylog(2,-I*exp(d*x+c))*x-9/8*I/a/d*e*f^2*ln(1+I*exp(d*x+c))*x^2+1/4*(- 8*I*d^2*e*f^2*x-2*d*f^3*x*exp(d*x+c)-2*d*e*f^2*exp(d*x+c)+2*d^3*e^3*exp(3* d*x+3*c)+3*d^3*e^3*exp(5*d*x+5*c)+2*I*f^3*exp(4*d*x+4*c)+4*I*f^3*exp(2*d*x +2*c)-2*d^2*e*f^2*x*exp(d*x+c)+9*d^3*e*f^2*x^2*exp(d*x+c)-18*I*d^3*e*f^2*x ^2*exp(4*d*x+4*c)+2*I*f^3+3*d^3*f^3*x^3*exp(5*d*x+5*c)-2*d*f^3*x*exp(5*d*x +5*c)-2*d*e*f^2*exp(5*d*x+5*c)+2*d^3*f^3*x^3*exp(3*d*x+3*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*I*d^3*e^2*f*x*exp(4*d*x+4*c)- 18*I*d^2*e^2*f*exp(4*d*x+4*c)-4*I*d^2*e^2*f-4*I*d^2*f^3*x^2+6*I*d^3*f^3*x^ 3*exp(2*d*x+2*c)-18*I*d^2*f^3*x^2*exp(4*d*x+4*c)+18*d^2*e*f^2*x*exp(5*d*x+ 5*c)+6*d^3*e*f^2*x^2*exp(3*d*x+3*c)+6*d^3*e^2*f*x*exp(3*d*x+3*c)+9*d^3*e*f ^2*x^2*exp(5*d*x+5*c)+9*d^3*e^2*f*x*exp(5*d*x+5*c)+8*d^2*f^3*x^2*exp(3*d*x +3*c)+18*I*d^3*e*f^2*x^2*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)-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(4*d*x+4*c)+18*I*d^3*e^2*f*x*exp(2*d* x+2*c)+9*d^3*e^2*f*x*exp(d*x+c)-6*I*d^3*e^3*exp(4*d*x+4*c)+6*I*d^3*e^3*exp (2*d*x+2*c)-2*f^3*exp(d*x+c)-4*f^3*exp(3*d*x+3*c)-2*f^3*exp(5*d*x+5*c)+...
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 = 0.29 (sec) , antiderivative size = 3854, normalized size of antiderivative = 5.78 \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
1/8*(-8*I*d^2*e^2*f + 16*I*c*d*e*f^2 - 4*(2*I*c^2 - I)*f^3 - 3*(3*I*d^2*f^ 3*x^2 + 6*I*d^2*e*f^2*x + 3*I*d^2*e^2*f - 4*I*f^3 + (-3*I*d^2*f^3*x^2 - 6* I*d^2*e*f^2*x - 3*I*d^2*e^2*f + 4*I*f^3)*e^(6*d*x + 6*c) - 2*(3*d^2*f^3*x^ 2 + 6*d^2*e*f^2*x + 3*d^2*e^2*f - 4*f^3)*e^(5*d*x + 5*c) + (-3*I*d^2*f^3*x ^2 - 6*I*d^2*e*f^2*x - 3*I*d^2*e^2*f + 4*I*f^3)*e^(4*d*x + 4*c) - 4*(3*d^2 *f^3*x^2 + 6*d^2*e*f^2*x + 3*d^2*e^2*f - 4*f^3)*e^(3*d*x + 3*c) + (3*I*d^2 *f^3*x^2 + 6*I*d^2*e*f^2*x + 3*I*d^2*e^2*f - 4*I*f^3)*e^(2*d*x + 2*c) - 2* (3*d^2*f^3*x^2 + 6*d^2*e*f^2*x + 3*d^2*e^2*f - 4*f^3)*e^(d*x + c))*dilog(I *e^(d*x + c)) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2*e^2*f - 28*I *f^3 + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f + 28*I*f^3)*e^ (6*d*x + 6*c) - 2*(9*d^2*f^3*x^2 + 18*d^2*e*f^2*x + 9*d^2*e^2*f - 28*f^3)* e^(5*d*x + 5*c) + (-9*I*d^2*f^3*x^2 - 18*I*d^2*e*f^2*x - 9*I*d^2*e^2*f + 2 8*I*f^3)*e^(4*d*x + 4*c) - 4*(9*d^2*f^3*x^2 + 18*d^2*e*f^2*x + 9*d^2*e^2*f - 28*f^3)*e^(3*d*x + 3*c) + (9*I*d^2*f^3*x^2 + 18*I*d^2*e*f^2*x + 9*I*d^2 *e^2*f - 28*I*f^3)*e^(2*d*x + 2*c) - 2*(9*d^2*f^3*x^2 + 18*d^2*e*f^2*x + 9 *d^2*e^2*f - 28*f^3)*e^(d*x + c))*dilog(-I*e^(d*x + c)) - 8*(I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + 2*I*c*d*e*f^2 - I*c^2*f^3)*e^(6*d*x + 6*c) + 2*(3*d^3 *f^3*x^3 + 3*d^3*e^3 + 9*d^2*e^2*f - 2*(8*c + 1)*d*e*f^2 + 2*(4*c^2 - 1)*f ^3 + (9*d^3*e*f^2 + d^2*f^3)*x^2 + (9*d^3*e^2*f + 2*d^2*e*f^2 - 2*d*f^3)*x )*e^(5*d*x + 5*c) - 4*(3*I*d^3*f^3*x^3 + 3*I*d^3*e^3 + 9*I*d^2*e^2*f + ...
\[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{3} \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{3} x^{3} \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e f^{2} x^{2} \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e^{2} f x \operatorname {sech}^{3}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
-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. \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {sech}\left (d x + c\right )^{3}}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \text {sech}^3(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]