Integrand size = 29, antiderivative size = 463 \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {3 i f (e+f x)^2}{2 a d^2}-\frac {6 f^2 (e+f x) \arctan \left (e^{c+d x}\right )}{a d^3}+\frac {(e+f x)^3 \arctan \left (e^{c+d x}\right )}{a d}+\frac {3 i f^2 (e+f x) \log \left (1+e^{2 (c+d x)}\right )}{a d^3}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{a d^4}-\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{2 a d^2}-\frac {3 i f^3 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{a d^4}+\frac {3 i f (e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{2 a d^2}+\frac {3 i f^3 \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 a d^4}+\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{a d^3}-\frac {3 i f^2 (e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{a d^3}-\frac {3 i f^3 \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{a d^4}+\frac {3 i f^3 \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{a d^4}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 a d^2}+\frac {i (e+f x)^3 \text {sech}^2(c+d x)}{2 a d}-\frac {3 i f (e+f x)^2 \tanh (c+d x)}{2 a d^2}+\frac {(e+f x)^3 \text {sech}(c+d x) \tanh (c+d x)}{2 a d} \]
3/2*I*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/a/d^2-6*f^2*(f*x+e)*arctan(exp(d *x+c))/a/d^3+(f*x+e)^3*arctan(exp(d*x+c))/a/d+3*I*f^3*polylog(2,-I*exp(d*x +c))/a/d^4+3/2*I*f^3*polylog(2,-exp(2*d*x+2*c))/a/d^4+1/2*I*(f*x+e)^3*sech (d*x+c)^2/a/d-3*I*f^3*polylog(2,I*exp(d*x+c))/a/d^4-3/2*I*f*(f*x+e)^2*tanh (d*x+c)/a/d^2+3*I*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/a/d^3-3*I*f^3*polyl og(4,-I*exp(d*x+c))/a/d^4+3*I*f^3*polylog(4,I*exp(d*x+c))/a/d^4-3/2*I*f*(f *x+e)^2/a/d^2+3*I*f^2*(f*x+e)*ln(1+exp(2*d*x+2*c))/a/d^3+3/2*f*(f*x+e)^2*s ech(d*x+c)/a/d^2-3/2*I*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/a/d^2-3*I*f^2* (f*x+e)*polylog(3,I*exp(d*x+c))/a/d^3+1/2*(f*x+e)^3*sech(d*x+c)*tanh(d*x+c )/a/d
Time = 8.07 (sec) , antiderivative size = 828, normalized size of antiderivative = 1.79 \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {\frac {(e+f x)^4}{f}+\frac {4 \left (1-i e^c\right ) (e+f x)^3 \log \left (1+i e^{-c-d x}\right )}{d}+\frac {12 i \left (i+e^c\right ) f \left (d^2 (e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{-c-d x}\right )+2 f \left (d (e+f x) \operatorname {PolyLog}\left (3,-i e^{-c-d x}\right )+f \operatorname {PolyLog}\left (4,-i e^{-c-d x}\right )\right )\right )}{d^4}}{8 a \left (i+e^c\right )}-\frac {-4 d^2 e \left (1+i e^c\right ) f \left (d^2 e^2-12 f^2\right ) x+\left (-12 f^2+d^2 (e+f x)^2\right )^2+12 d \left (1+i e^c\right ) f^2 \left (d^2 e^2-4 f^2\right ) x \log \left (1-i e^{-c-d x}\right )+12 d^3 e \left (1+i e^c\right ) f^3 x^2 \log \left (1-i e^{-c-d x}\right )+4 d^3 \left (1+i e^c\right ) f^4 x^3 \log \left (1-i e^{-c-d x}\right )+4 d e \left (1+i e^c\right ) f \left (d^2 e^2-12 f^2\right ) \log \left (i-e^{c+d x}\right )+12 \left (1+i e^c\right ) f^2 \left (-d^2 e^2+4 f^2\right ) \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )-24 d^2 e \left (1+i e^c\right ) f^3 x \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )-12 d^2 \left (1+i e^c\right ) f^4 x^2 \operatorname {PolyLog}\left (2,i e^{-c-d x}\right )-24 d e \left (1+i e^c\right ) f^3 \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )-24 d \left (1+i e^c\right ) f^4 x \operatorname {PolyLog}\left (3,i e^{-c-d x}\right )-24 \left (1+i e^c\right ) f^4 \operatorname {PolyLog}\left (4,i e^{-c-d x}\right )}{8 a d^4 \left (-i+e^c\right ) f}+\frac {x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )}{8 a \left (\cosh \left (\frac {c}{2}\right )-i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right )}+\frac {i (e+f x)^3}{2 a d \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}-\frac {3 i \left (e^2 f \sinh \left (\frac {d x}{2}\right )+2 e f^2 x \sinh \left (\frac {d x}{2}\right )+f^3 x^2 \sinh \left (\frac {d x}{2}\right )\right )}{a d^2 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \left (\cosh \left (\frac {c}{2}+\frac {d x}{2}\right )+i \sinh \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]
-1/8*((e + f*x)^4/f + (4*(1 - I*E^c)*(e + f*x)^3*Log[1 + I*E^(-c - d*x)])/ d + ((12*I)*(I + E^c)*f*(d^2*(e + f*x)^2*PolyLog[2, (-I)*E^(-c - d*x)] + 2 *f*(d*(e + f*x)*PolyLog[3, (-I)*E^(-c - d*x)] + f*PolyLog[4, (-I)*E^(-c - d*x)])))/d^4)/(a*(I + E^c)) - (-4*d^2*e*(1 + I*E^c)*f*(d^2*e^2 - 12*f^2)*x + (-12*f^2 + d^2*(e + f*x)^2)^2 + 12*d*(1 + I*E^c)*f^2*(d^2*e^2 - 4*f^2)* x*Log[1 - I*E^(-c - d*x)] + 12*d^3*e*(1 + I*E^c)*f^3*x^2*Log[1 - I*E^(-c - d*x)] + 4*d^3*(1 + I*E^c)*f^4*x^3*Log[1 - I*E^(-c - d*x)] + 4*d*e*(1 + I* E^c)*f*(d^2*e^2 - 12*f^2)*Log[I - E^(c + d*x)] + 12*(1 + I*E^c)*f^2*(-(d^2 *e^2) + 4*f^2)*PolyLog[2, I*E^(-c - d*x)] - 24*d^2*e*(1 + I*E^c)*f^3*x*Pol yLog[2, I*E^(-c - d*x)] - 12*d^2*(1 + I*E^c)*f^4*x^2*PolyLog[2, I*E^(-c - d*x)] - 24*d*e*(1 + I*E^c)*f^3*PolyLog[3, I*E^(-c - d*x)] - 24*d*(1 + I*E^ c)*f^4*x*PolyLog[3, I*E^(-c - d*x)] - 24*(1 + I*E^c)*f^4*PolyLog[4, I*E^(- c - d*x)])/(8*a*d^4*(-I + E^c)*f) + (x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3))/(8*a*(Cosh[c/2] - I*Sinh[c/2])*(Cosh[c/2] + I*Sinh[c/2])) + ((I/ 2)*(e + f*x)^3)/(a*d*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + (d*x)/2])^2) - (( 3*I)*(e^2*f*Sinh[(d*x)/2] + 2*e*f^2*x*Sinh[(d*x)/2] + f^3*x^2*Sinh[(d*x)/2 ]))/(a*d^2*(Cosh[c/2] + I*Sinh[c/2])*(Cosh[c/2 + (d*x)/2] + I*Sinh[c/2 + ( d*x)/2]))
Time = 2.90 (sec) , antiderivative size = 442, normalized size of antiderivative = 0.95, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {6105, 3042, 4674, 3042, 4668, 2715, 2838, 3011, 5974, 3042, 4672, 26, 3042, 26, 4201, 2620, 2715, 2838, 7163, 2720, 7143}
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.
\(\displaystyle \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx\) |
\(\Big \downarrow \) 6105 |
\(\displaystyle \frac {\int (e+f x)^3 \text {sech}^3(c+d x)dx}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )^3dx}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4674 |
\(\displaystyle \frac {-\frac {3 f^2 \int (e+f x) \text {sech}(c+d x)dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \text {sech}(c+d x)dx+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {3 f^2 \int (e+f x) \csc \left (i c+i d x+\frac {\pi }{2}\right )dx}{d^2}+\frac {1}{2} \int (e+f x)^3 \csc \left (i c+i d x+\frac {\pi }{2}\right )dx+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 4668 |
\(\displaystyle \frac {-\frac {3 f^2 \left (-\frac {i f \int \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {i f \int \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {-\frac {3 f^2 \left (-\frac {i f \int e^{-c-d x} \log \left (1-i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {i f \int e^{-c-d x} \log \left (1+i e^{c+d x}\right )de^{c+d x}}{d^2}+\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}\right )}{d^2}+\frac {1}{2} \left (-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {1}{2} \left (-\frac {3 i f \int (e+f x)^2 \log \left (1-i e^{c+d x}\right )dx}{d}+\frac {3 i f \int (e+f x)^2 \log \left (1+i e^{c+d x}\right )dx}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \int (e+f x)^3 \text {sech}^2(c+d x) \tanh (c+d x)dx}{a}\) |
\(\Big \downarrow \) 5974 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (\frac {3 f \int (e+f x)^2 \text {sech}^2(c+d x)dx}{2 d}-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \int (e+f x)^2 \csc \left (i c+i d x+\frac {\pi }{2}\right )^2dx}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 4672 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 i f \int -i (e+f x) \tanh (c+d x)dx}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 f \int (e+f x) \tanh (c+d x)dx}{d}\right )}{2 d}-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}-\frac {2 f \int -i (e+f x) \tan (i c+i d x)dx}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \int (e+f x) \tan (i c+i d x)dx}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \int \frac {e^{2 (c+d x)} (e+f x)}{1+e^{2 (c+d x)}}dx-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int \log \left (1+e^{2 (c+d x)}\right )dx}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}-\frac {f \int e^{-2 (c+d x)} \log \left (1+e^{2 (c+d x)}\right )de^{2 (c+d x)}}{4 d^2}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \int (e+f x) \operatorname {PolyLog}\left (2,i e^{c+d x}\right )dx}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 7163 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \int \operatorname {PolyLog}\left (3,i e^{c+d x}\right )dx}{d}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {\frac {1}{2} \left (\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \int e^{-c-d x} \operatorname {PolyLog}\left (3,i e^{c+d x}\right )de^{c+d x}}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}+\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}\right )-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}\right )}{a}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {2 (e+f x) \arctan \left (e^{c+d x}\right )}{d}-\frac {i f \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d^2}+\frac {i f \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d^2}\right )}{d^2}+\frac {1}{2} \left (\frac {2 (e+f x)^3 \arctan \left (e^{c+d x}\right )}{d}+\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,-i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,-i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right )}{d}\right )}{d}-\frac {3 i f \left (\frac {2 f \left (\frac {(e+f x) \operatorname {PolyLog}\left (3,i e^{c+d x}\right )}{d}-\frac {f \operatorname {PolyLog}\left (4,i e^{c+d x}\right )}{d^2}\right )}{d}-\frac {(e+f x)^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right )}{d}\right )}{d}\right )+\frac {3 f (e+f x)^2 \text {sech}(c+d x)}{2 d^2}+\frac {(e+f x)^3 \tanh (c+d x) \text {sech}(c+d x)}{2 d}}{a}-\frac {i \left (-\frac {(e+f x)^3 \text {sech}^2(c+d x)}{2 d}+\frac {3 f \left (\frac {(e+f x)^2 \tanh (c+d x)}{d}+\frac {2 i f \left (2 i \left (\frac {f \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{4 d^2}+\frac {(e+f x) \log \left (e^{2 (c+d x)}+1\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}\right )}{d}\right )}{2 d}\right )}{a}\) |
((-3*f^2*((2*(e + f*x)*ArcTan[E^(c + d*x)])/d - (I*f*PolyLog[2, (-I)*E^(c + d*x)])/d^2 + (I*f*PolyLog[2, I*E^(c + d*x)])/d^2))/d^2 + ((2*(e + f*x)^3 *ArcTan[E^(c + d*x)])/d + ((3*I)*f*(-(((e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/d) + (2*f*(((e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/d - (f*PolyLog[ 4, (-I)*E^(c + d*x)])/d^2))/d))/d - ((3*I)*f*(-(((e + f*x)^2*PolyLog[2, I* E^(c + d*x)])/d) + (2*f*(((e + f*x)*PolyLog[3, I*E^(c + d*x)])/d - (f*Poly Log[4, I*E^(c + d*x)])/d^2))/d))/d)/2 + (3*f*(e + f*x)^2*Sech[c + d*x])/(2 *d^2) + ((e + f*x)^3*Sech[c + d*x]*Tanh[c + d*x])/(2*d))/a - (I*(-1/2*((e + f*x)^3*Sech[c + d*x]^2)/d + (3*f*(((2*I)*f*(((-1/2*I)*(e + f*x)^2)/f + ( 2*I)*(((e + f*x)*Log[1 + E^(2*(c + d*x))])/(2*d) + (f*PolyLog[2, -E^(2*(c + d*x))])/(4*d^2))))/d + ((e + f*x)^2*Tanh[c + d*x])/d))/(2*d)))/a
3.3.71.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x _Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I Int[ (c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[csc[(e_.) + 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_))^(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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
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. 1079 vs. \(2 (416 ) = 832\).
Time = 22.22 (sec) , antiderivative size = 1080, normalized size of antiderivative = 2.33
(d*f^3*x^3*exp(d*x+c)+3*d*e*f^2*x^2*exp(d*x+c)+3*d*e^2*f*x*exp(d*x+c)+d*e^ 3*exp(d*x+c)+3*f^3*x^2*exp(d*x+c)-3*I*f^3*x^2+6*e*f^2*x*exp(d*x+c)-6*I*e*f ^2*x+3*e^2*f*exp(d*x+c)-3*I*e^2*f)/(exp(d*x+c)-I)^2/d^2/a-3*I*f^3*polylog( 4,-I*exp(d*x+c))/a/d^4-3/2*I/a/d*e*f^2*ln(1+I*exp(d*x+c))*x^2-3*I/a/d^2*e* f^2*polylog(2,-I*exp(d*x+c))*x+3*I*f^3*polylog(4,I*exp(d*x+c))/a/d^4-3*I/a /d^4*f^3*c*ln(1+exp(2*d*x+2*c))+6*I/a/d^4*f^3*c*ln(exp(d*x+c))+3/2*I/a/d^2 *e^2*f*polylog(2,I*exp(d*x+c))-3/2*I/a/d^2*e^2*f*polylog(2,-I*exp(d*x+c))+ 1/2*I/a/d^4*f^3*ln(1-I*exp(d*x+c))*c^3-1/2*I/a/d^4*f^3*ln(1+I*exp(d*x+c))* c^3+1/2*I/a/d*f^3*ln(1-I*exp(d*x+c))*x^3+3/2*I/a/d^2*f^3*polylog(2,I*exp(d *x+c))*x^2-3*I/a/d^3*f^3*polylog(3,I*exp(d*x+c))*x-1/2*I/a/d*f^3*ln(1+I*ex p(d*x+c))*x^3-3/2*I/a/d^2*f^3*polylog(2,-I*exp(d*x+c))*x^2+3*I/a/d^3*f^3*p olylog(3,-I*exp(d*x+c))*x+1/a/d*e^3*arctan(exp(d*x+c))-3/a/d^2*e^2*f*c*arc tan(exp(d*x+c))-3/2*I/a/d^3*e*f^2*ln(1-I*exp(d*x+c))*c^2+3/2*I/a/d^3*e*f^2 *ln(1+I*exp(d*x+c))*c^2+3/2*I/a/d*e^2*f*ln(1-I*exp(d*x+c))*x+3/2*I/a/d^2*e ^2*f*ln(1-I*exp(d*x+c))*c-3/2*I/a/d*e^2*f*ln(1+I*exp(d*x+c))*x-3/2*I/a/d^2 *e^2*f*ln(1+I*exp(d*x+c))*c+3/2*I/a/d*e*f^2*ln(1-I*exp(d*x+c))*x^2+3*I/a/d ^2*e*f^2*polylog(2,I*exp(d*x+c))*x+3/a/d^3*f^2*c^2*e*arctan(exp(d*x+c))+3* I/a/d^3*e*f^2*polylog(3,-I*exp(d*x+c))+3*I/a/d^3*e*f^2*ln(1+exp(2*d*x+2*c) )-6*I/a/d^3*e*f^2*ln(exp(d*x+c))-3*I/a/d^3*e*f^2*polylog(3,I*exp(d*x+c))-6 *I/a/d^3*f^3*c*x+6*I/a/d^3*f^3*ln(1+I*exp(d*x+c))*x+6*I/a/d^4*f^3*ln(1+...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1461 vs. \(2 (390) = 780\).
Time = 0.26 (sec) , antiderivative size = 1461, normalized size of antiderivative = 3.16 \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx=\text {Too large to display} \]
1/2*(-6*I*d^2*e^2*f + 12*I*c*d*e*f^2 - 6*I*c^2*f^3 - 3*(I*d^2*f^3*x^2 + 2* I*d^2*e*f^2*x + I*d^2*e^2*f + (-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e^ 2*f)*e^(2*d*x + 2*c) - 2*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f)*e^(d*x + c))*dilog(I*e^(d*x + c)) - 3*(-I*d^2*f^3*x^2 - 2*I*d^2*e*f^2*x - I*d^2*e ^2*f + 4*I*f^3 + (I*d^2*f^3*x^2 + 2*I*d^2*e*f^2*x + I*d^2*e^2*f - 4*I*f^3) *e^(2*d*x + 2*c) + 2*(d^2*f^3*x^2 + 2*d^2*e*f^2*x + d^2*e^2*f - 4*f^3)*e^( d*x + c))*dilog(-I*e^(d*x + c)) - 6*(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^(2*d*x + 2*c) + 2*(d^3*f^3*x^3 + d^3*e^3 + 3*d^2 *e^2*f - 12*c*d*e*f^2 + 6*c^2*f^3 + 3*(d^3*e*f^2 - d^2*f^3)*x^2 + 3*(d^3*e ^2*f - 2*d^2*e*f^2)*x)*e^(d*x + c) + (-I*d^3*e^3 + 3*I*c*d^2*e^2*f - 3*I*c ^2*d*e*f^2 + I*c^3*f^3 + (I*d^3*e^3 - 3*I*c*d^2*e^2*f + 3*I*c^2*d*e*f^2 - I*c^3*f^3)*e^(2*d*x + 2*c) + 2*(d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*e^(d*x + c))*log(e^(d*x + c) + I) + (I*d^3*e^3 - 3*I*c*d^2*e^2*f - 3*(-I*c^2 + 4*I)*d*e*f^2 + (-I*c^3 + 12*I*c)*f^3 + (-I*d^3*e^3 + 3*I*c*d ^2*e^2*f - 3*(I*c^2 - 4*I)*d*e*f^2 + (I*c^3 - 12*I*c)*f^3)*e^(2*d*x + 2*c) - 2*(d^3*e^3 - 3*c*d^2*e^2*f + 3*(c^2 - 4)*d*e*f^2 - (c^3 - 12*c)*f^3)*e^ (d*x + c))*log(e^(d*x + c) - I) + (I*d^3*f^3*x^3 + 3*I*d^3*e*f^2*x^2 + 3*I *c*d^2*e^2*f - 3*I*c^2*d*e*f^2 + (I*c^3 - 12*I*c)*f^3 - 3*(-I*d^3*e^2*f + 4*I*d*f^3)*x + (-I*d^3*f^3*x^3 - 3*I*d^3*e*f^2*x^2 - 3*I*c*d^2*e^2*f + 3*I *c^2*d*e*f^2 + (-I*c^3 + 12*I*c)*f^3 - 3*(I*d^3*e^2*f - 4*I*d*f^3)*x)*e...
\[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx=- \frac {i \left (\int \frac {e^{3} \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {f^{3} x^{3} \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e f^{2} x^{2} \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx + \int \frac {3 e^{2} f x \operatorname {sech}{\left (c + d x \right )}}{\sinh {\left (c + d x \right )} - i}\, dx\right )}{a} \]
-I*(Integral(e**3*sech(c + d*x)/(sinh(c + d*x) - I), x) + Integral(f**3*x* *3*sech(c + d*x)/(sinh(c + d*x) - I), x) + Integral(3*e*f**2*x**2*sech(c + d*x)/(sinh(c + d*x) - I), x) + Integral(3*e**2*f*x*sech(c + d*x)/(sinh(c + d*x) - I), x))/a
Time = 0.41 (sec) , antiderivative size = 685, normalized size of antiderivative = 1.48 \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx=-\frac {1}{2} \, e^{3} {\left (\frac {4 \, e^{\left (-d x - c\right )}}{-2 \, {\left (-2 i \, a e^{\left (-d x - c\right )} - a e^{\left (-2 \, d x - 2 \, c\right )} + a\right )} d} + \frac {i \, \log \left (e^{\left (-d x - c\right )} + i\right )}{a d} - \frac {i \, \log \left (i \, e^{\left (-d x - c\right )} + 1\right )}{a d}\right )} + \frac {3 i \, {\left (d x \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right )\right )} e^{2} f}{2 \, a d^{2}} - \frac {6 i \, e f^{2} x}{a d^{2}} + \frac {-3 i \, f^{3} x^{2} - 6 i \, e f^{2} x - 3 i \, e^{2} f + {\left (d f^{3} x^{3} e^{c} + 3 \, e^{2} f e^{c} + 3 \, {\left (d e f^{2} + f^{3}\right )} x^{2} e^{c} + 3 \, {\left (d e^{2} f + 2 \, e f^{2}\right )} x e^{c}\right )} e^{\left (d x\right )}}{a d^{2} e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d^{2} e^{\left (d x + c\right )} - a d^{2}} - \frac {3 i \, {\left (d^{2} x^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(-i \, e^{\left (d x + c\right )})\right )} e f^{2}}{2 \, a d^{3}} + \frac {3 i \, {\left (d^{2} x^{2} \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) + 2 \, d x {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right ) - 2 \, {\rm Li}_{3}(i \, e^{\left (d x + c\right )})\right )} e f^{2}}{2 \, a d^{3}} + \frac {6 i \, e f^{2} \log \left (i \, e^{\left (d x + c\right )} + 1\right )}{a d^{3}} - \frac {i \, {\left (d^{3} x^{3} \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(-i \, e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(-i \, e^{\left (d x + c\right )})\right )} f^{3}}{2 \, a d^{4}} + \frac {i \, {\left (d^{3} x^{3} \log \left (-i \, e^{\left (d x + c\right )} + 1\right ) + 3 \, d^{2} x^{2} {\rm Li}_2\left (i \, e^{\left (d x + c\right )}\right ) - 6 \, d x {\rm Li}_{3}(i \, e^{\left (d x + c\right )}) + 6 \, {\rm Li}_{4}(i \, e^{\left (d x + c\right )})\right )} f^{3}}{2 \, a d^{4}} - \frac {3 i \, {\left (d^{2} e^{2} f - 4 \, f^{3}\right )} {\left (d x \log \left (i \, e^{\left (d x + c\right )} + 1\right ) + {\rm Li}_2\left (-i \, e^{\left (d x + c\right )}\right )\right )}}{2 \, a d^{4}} - \frac {i \, d^{4} f^{3} x^{4} + 4 i \, d^{4} e f^{2} x^{3} + 6 i \, d^{4} e^{2} f x^{2}}{8 \, a d^{4}} + \frac {i \, d^{4} f^{3} x^{4} + 4 i \, d^{4} e f^{2} x^{3} - 6 \, {\left (-i \, d^{2} e^{2} f + 4 i \, f^{3}\right )} d^{2} x^{2}}{8 \, a d^{4}} \]
-1/2*e^3*(4*e^(-d*x - c)/((4*I*a*e^(-d*x - c) + 2*a*e^(-2*d*x - 2*c) - 2*a )*d) + I*log(e^(-d*x - c) + I)/(a*d) - I*log(I*e^(-d*x - c) + 1)/(a*d)) + 3/2*I*(d*x*log(-I*e^(d*x + c) + 1) + dilog(I*e^(d*x + c)))*e^2*f/(a*d^2) - 6*I*e*f^2*x/(a*d^2) + (-3*I*f^3*x^2 - 6*I*e*f^2*x - 3*I*e^2*f + (d*f^3*x^ 3*e^c + 3*e^2*f*e^c + 3*(d*e*f^2 + f^3)*x^2*e^c + 3*(d*e^2*f + 2*e*f^2)*x* e^c)*e^(d*x))/(a*d^2*e^(2*d*x + 2*c) - 2*I*a*d^2*e^(d*x + c) - a*d^2) - 3/ 2*I*(d^2*x^2*log(I*e^(d*x + c) + 1) + 2*d*x*dilog(-I*e^(d*x + c)) - 2*poly log(3, -I*e^(d*x + c)))*e*f^2/(a*d^3) + 3/2*I*(d^2*x^2*log(-I*e^(d*x + c) + 1) + 2*d*x*dilog(I*e^(d*x + c)) - 2*polylog(3, I*e^(d*x + c)))*e*f^2/(a* d^3) + 6*I*e*f^2*log(I*e^(d*x + c) + 1)/(a*d^3) - 1/2*I*(d^3*x^3*log(I*e^( d*x + c) + 1) + 3*d^2*x^2*dilog(-I*e^(d*x + c)) - 6*d*x*polylog(3, -I*e^(d *x + c)) + 6*polylog(4, -I*e^(d*x + c)))*f^3/(a*d^4) + 1/2*I*(d^3*x^3*log( -I*e^(d*x + c) + 1) + 3*d^2*x^2*dilog(I*e^(d*x + c)) - 6*d*x*polylog(3, I* e^(d*x + c)) + 6*polylog(4, I*e^(d*x + c)))*f^3/(a*d^4) - 3/2*I*(d^2*e^2*f - 4*f^3)*(d*x*log(I*e^(d*x + c) + 1) + dilog(-I*e^(d*x + c)))/(a*d^4) - 1 /8*(I*d^4*f^3*x^4 + 4*I*d^4*e*f^2*x^3 + 6*I*d^4*e^2*f*x^2)/(a*d^4) + 1/8*( I*d^4*f^3*x^4 + 4*I*d^4*e*f^2*x^3 - 6*(-I*d^2*e^2*f + 4*I*f^3)*d^2*x^2)/(a *d^4)
\[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \operatorname {sech}\left (d x + c\right )}{i \, a \sinh \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+i a \sinh (c+d x)} \, dx=\int \frac {{\left (e+f\,x\right )}^3}{\mathrm {cosh}\left (c+d\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]