Integrand size = 12, antiderivative size = 158 \[ \int x^3 \cosh ^2(x) \coth ^3(x) \, dx=\frac {3 x}{8}-\frac {3 x^2}{2}+\frac {3 x^3}{4}-\frac {x^4}{2}-\frac {3}{2} x^2 \coth (x)-\frac {1}{2} x^3 \coth ^2(x)+3 x \log \left (1-e^{2 x}\right )+2 x^3 \log \left (1-e^{2 x}\right )+\frac {3 \operatorname {PolyLog}\left (2,e^{2 x}\right )}{2}+3 x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )-3 x \operatorname {PolyLog}\left (3,e^{2 x}\right )+\frac {3 \operatorname {PolyLog}\left (4,e^{2 x}\right )}{2}-\frac {3}{8} \cosh (x) \sinh (x)-\frac {3}{4} x^2 \cosh (x) \sinh (x)+\frac {3}{4} x \sinh ^2(x)+\frac {1}{2} x^3 \sinh ^2(x) \]
3/8*x-3/2*x^2+3/4*x^3-1/2*x^4-3/2*x^2*coth(x)-1/2*x^3*coth(x)^2+3*x*ln(1-e xp(2*x))+2*x^3*ln(1-exp(2*x))+3/2*polylog(2,exp(2*x))+3*x^2*polylog(2,exp( 2*x))-3*x*polylog(3,exp(2*x))+3/2*polylog(4,exp(2*x))-3/8*cosh(x)*sinh(x)- 3/4*x^2*cosh(x)*sinh(x)+3/4*x*sinh(x)^2+1/2*x^3*sinh(x)^2
Time = 0.64 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.21 \[ \int x^3 \cosh ^2(x) \coth ^3(x) \, dx=-3 \left (1+2 x^2\right ) \operatorname {PolyLog}\left (2,-e^{-x}\right )+\frac {1}{8} \left (12 x^2+4 x^4+3 x \cosh (2 x)+2 x^3 \cosh (2 x)-12 x^2 \coth (x)-4 x^3 \text {csch}^2(x)+24 x \log \left (1-e^{-x}\right )+16 x^3 \log \left (1-e^{-x}\right )+24 x \log \left (1+e^{-x}\right )+16 x^3 \log \left (1+e^{-x}\right )-24 \left (1+2 x^2\right ) \operatorname {PolyLog}\left (2,e^{-x}\right )-96 x \operatorname {PolyLog}\left (3,-e^{-x}\right )-96 x \operatorname {PolyLog}\left (3,e^{-x}\right )-96 \operatorname {PolyLog}\left (4,-e^{-x}\right )-96 \operatorname {PolyLog}\left (4,e^{-x}\right )-3 \cosh (x) \sinh (x)-6 x^2 \cosh (x) \sinh (x)\right ) \]
-3*(1 + 2*x^2)*PolyLog[2, -E^(-x)] + (12*x^2 + 4*x^4 + 3*x*Cosh[2*x] + 2*x ^3*Cosh[2*x] - 12*x^2*Coth[x] - 4*x^3*Csch[x]^2 + 24*x*Log[1 - E^(-x)] + 1 6*x^3*Log[1 - E^(-x)] + 24*x*Log[1 + E^(-x)] + 16*x^3*Log[1 + E^(-x)] - 24 *(1 + 2*x^2)*PolyLog[2, E^(-x)] - 96*x*PolyLog[3, -E^(-x)] - 96*x*PolyLog[ 3, E^(-x)] - 96*PolyLog[4, -E^(-x)] - 96*PolyLog[4, E^(-x)] - 3*Cosh[x]*Si nh[x] - 6*x^2*Cosh[x]*Sinh[x])/8
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.
| \(\displaystyle \int x^3 \cosh ^2(x) \coth ^3(x) \, dx\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle \int x^3 \coth ^3(x)dx+\int x^3 \cosh ^2(x) \coth (x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+\int i x^3 \tan \left (i x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \int x^3 \tan \left (i x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (-\int i x^3 \coth (x)dx+\frac {3}{2} i \int -x^2 \coth ^2(x)dx+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (-\int i x^3 \coth (x)dx-\frac {3}{2} i \int x^2 \coth ^2(x)dx+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (-i \int x^3 \coth (x)dx-\frac {3}{2} i \int x^2 \coth ^2(x)dx+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (-i \int -i x^3 \tan \left (i x+\frac {\pi }{2}\right )dx-\frac {3}{2} i \int -x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (-i \int -i x^3 \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {3}{2} i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (-\int x^3 \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {3}{2} i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (-2 i \int -\frac {e^{2 x} x^3}{1-e^{2 x}}dx+\frac {3}{2} i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \int \frac {e^{2 x} x^3}{1-e^{2 x}}dx+\frac {3}{2} i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (\frac {3}{2} i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+2 i \left (\frac {3}{2} \int x^2 \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (-\int x^2dx+2 i \int i x \coth (x)dx+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \int i x \coth (x)dx-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (-2 \int x \coth (x)dx-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (-2 \int -i x \tan \left (i x+\frac {\pi }{2}\right )dx-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \int x \tan \left (i x+\frac {\pi }{2}\right )dx-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (2 i \int -\frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \int \frac {e^{2 x} x}{1-e^{2 x}}dx-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \log \left (1-e^{2 x}\right )de^{2 x}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \int x^3 \cosh ^2(x) \coth (x)dx+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )\) |
| \(\Big \downarrow \) 5973 |
| \(\displaystyle i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )+\int x^3 \coth (x)dx+\int x^3 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )+\int -i x^3 \tan \left (i x+\frac {\pi }{2}\right )dx+\int x^3 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )-i \int x^3 \tan \left (i x+\frac {\pi }{2}\right )dx+\int x^3 \cosh (x) \sinh (x)dx\) |
| \(\Big \downarrow \) 4199 |
| \(\displaystyle i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )+\int x^3 \cosh (x) \sinh (x)dx-i \left (2 i \int -\frac {e^{2 x} x^3}{1-e^{2 x}}dx-\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )+\int x^3 \cosh (x) \sinh (x)dx-i \left (-2 i \int \frac {e^{2 x} x^3}{1-e^{2 x}}dx-\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )+\int x^3 \cosh (x) \sinh (x)dx-i \left (-2 i \left (\frac {3}{2} \int x^2 \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )-\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -i \left (-2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )-\frac {i x^4}{4}\right )+i \left (2 i \left (\frac {3}{2} \left (\int x \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x^2 \operatorname {PolyLog}\left (2,e^{2 x}\right )\right )-\frac {1}{2} x^3 \log \left (1-e^{2 x}\right )\right )+\frac {3}{2} i \left (2 i \left (-2 i \left (-\frac {\operatorname {PolyLog}\left (2,e^{2 x}\right )}{4}-\frac {1}{2} x \log \left (1-e^{2 x}\right )\right )-\frac {i x^2}{2}\right )-\frac {x^3}{3}+x^2 \coth (x)\right )+\frac {i x^4}{4}+\frac {1}{2} i x^3 \coth ^2(x)\right )+\int x^3 \cosh (x) \sinh (x)dx\) |
3.5.23.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_.) + Pi*(k_.) + (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 ))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In tegerQ[4*k] && IGtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si mp[b*d*(m/(f*(n - 1))) Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] , x] - Simp[b^2 Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Time = 0.45 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(-\frac {x^{4}}{2}+\left (-\frac {3}{32}+\frac {3}{16} x -\frac {3}{16} x^{2}+\frac {1}{8} x^{3}\right ) {\mathrm e}^{2 x}+\left (\frac {3}{32}+\frac {3}{16} x +\frac {3}{16} x^{2}+\frac {1}{8} x^{3}\right ) {\mathrm e}^{-2 x}-\frac {x^{2} \left (2 \,{\mathrm e}^{2 x} x +3 \,{\mathrm e}^{2 x}-3\right )}{\left ({\mathrm e}^{2 x}-1\right )^{2}}-3 x^{2}+3 x \ln \left (1-{\mathrm e}^{x}\right )+3 \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )+3 x \ln \left ({\mathrm e}^{x}+1\right )+3 \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )+2 x^{3} \ln \left (1-{\mathrm e}^{x}\right )+6 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-12 x \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )+12 \operatorname {polylog}\left (4, {\mathrm e}^{x}\right )+2 x^{3} \ln \left ({\mathrm e}^{x}+1\right )+6 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )-12 x \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )+12 \operatorname {polylog}\left (4, -{\mathrm e}^{x}\right )\) | \(184\) |
-1/2*x^4+(-3/32+3/16*x-3/16*x^2+1/8*x^3)*exp(x)^2+(3/32+3/16*x+3/16*x^2+1/ 8*x^3)/exp(x)^2-x^2*(2*x*exp(x)^2+3*exp(x)^2-3)/(exp(x)^2-1)^2-3*x^2+3*x*l n(1-exp(x))+3*polylog(2,exp(x))+3*x*ln(exp(x)+1)+3*polylog(2,-exp(x))+2*x^ 3*ln(1-exp(x))+6*x^2*polylog(2,exp(x))-12*x*polylog(3,exp(x))+12*polylog(4 ,exp(x))+2*x^3*ln(exp(x)+1)+6*x^2*polylog(2,-exp(x))-12*x*polylog(3,-exp(x ))+12*polylog(4,-exp(x))
Leaf count of result is larger than twice the leaf count of optimal. 2067 vs. \(2 (126) = 252\).
Time = 0.27 (sec) , antiderivative size = 2067, normalized size of antiderivative = 13.08 \[ \int x^3 \cosh ^2(x) \coth ^3(x) \, dx=\text {Too large to display} \]
1/32*((4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^8 + 8*(4*x^3 - 6*x^2 + 6*x - 3)*co sh(x)*sinh(x)^7 + (4*x^3 - 6*x^2 + 6*x - 3)*sinh(x)^8 - 2*(8*x^4 + 4*x^3 + 42*x^2 + 6*x - 3)*cosh(x)^6 - 2*(8*x^4 + 4*x^3 - 14*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^2 + 42*x^2 + 6*x - 3)*sinh(x)^6 + 4*(14*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^3 - 3*(8*x^4 + 4*x^3 + 42*x^2 + 6*x - 3)*cosh(x))*sinh(x)^5 + 4*(8*x^4 - 14*x^3 + 24*x^2 + 3*x)*cosh(x)^4 + 2*(35*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^4 + 16*x^4 - 28*x^3 - 15*(8*x^4 + 4*x^3 + 42*x^2 + 6*x - 3)*c osh(x)^2 + 48*x^2 + 6*x)*sinh(x)^4 + 8*(7*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x )^5 - 5*(8*x^4 + 4*x^3 + 42*x^2 + 6*x - 3)*cosh(x)^3 + 2*(8*x^4 - 14*x^3 + 24*x^2 + 3*x)*cosh(x))*sinh(x)^3 + 4*x^3 - 2*(8*x^4 + 4*x^3 + 6*x^2 + 6*x + 3)*cosh(x)^2 + 2*(14*(4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^6 - 15*(8*x^4 + 4*x^3 + 42*x^2 + 6*x - 3)*cosh(x)^4 - 8*x^4 - 4*x^3 + 12*(8*x^4 - 14*x^3 + 24*x^2 + 3*x)*cosh(x)^2 - 6*x^2 - 6*x - 3)*sinh(x)^2 + 6*x^2 + 96*((2*x^2 + 1)*cosh(x)^6 + 6*(2*x^2 + 1)*cosh(x)*sinh(x)^5 + (2*x^2 + 1)*sinh(x)^6 - 2*(2*x^2 + 1)*cosh(x)^4 + (15*(2*x^2 + 1)*cosh(x)^2 - 4*x^2 - 2)*sinh(x) ^4 + 4*(5*(2*x^2 + 1)*cosh(x)^3 - 2*(2*x^2 + 1)*cosh(x))*sinh(x)^3 + (2*x^ 2 + 1)*cosh(x)^2 + (15*(2*x^2 + 1)*cosh(x)^4 - 12*(2*x^2 + 1)*cosh(x)^2 + 2*x^2 + 1)*sinh(x)^2 + 2*(3*(2*x^2 + 1)*cosh(x)^5 - 4*(2*x^2 + 1)*cosh(x)^ 3 + (2*x^2 + 1)*cosh(x))*sinh(x))*dilog(cosh(x) + sinh(x)) + 96*((2*x^2 + 1)*cosh(x)^6 + 6*(2*x^2 + 1)*cosh(x)*sinh(x)^5 + (2*x^2 + 1)*sinh(x)^6 ...
\[ \int x^3 \cosh ^2(x) \coth ^3(x) \, dx=\int x^{3} \cosh ^{2}{\left (x \right )} \coth ^{3}{\left (x \right )}\, dx \]
Time = 0.25 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.51 \[ \int x^3 \cosh ^2(x) \coth ^3(x) \, dx=-x^{4} + 2 \, x^{3} \log \left (e^{x} + 1\right ) + 2 \, x^{3} \log \left (-e^{x} + 1\right ) + 6 \, x^{2} {\rm Li}_2\left (-e^{x}\right ) + 6 \, x^{2} {\rm Li}_2\left (e^{x}\right ) - 3 \, x^{2} + 3 \, x \log \left (e^{x} + 1\right ) + 3 \, x \log \left (-e^{x} + 1\right ) - 12 \, x {\rm Li}_{3}(-e^{x}) - 12 \, x {\rm Li}_{3}(e^{x}) + \frac {16 \, x^{4} - 8 \, x^{3} + 84 \, x^{2} + {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (6 \, x\right )} + 2 \, {\left (8 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (4 \, x\right )} - 4 \, {\left (8 \, x^{4} + 14 \, x^{3} + 24 \, x^{2} - 3 \, x\right )} e^{\left (2 \, x\right )} + {\left (4 \, x^{3} + 6 \, x^{2} + 6 \, x + 3\right )} e^{\left (-2 \, x\right )} - 12 \, x - 6}{32 \, {\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )}} + 3 \, {\rm Li}_2\left (-e^{x}\right ) + 3 \, {\rm Li}_2\left (e^{x}\right ) + 12 \, {\rm Li}_{4}(-e^{x}) + 12 \, {\rm Li}_{4}(e^{x}) \]
-x^4 + 2*x^3*log(e^x + 1) + 2*x^3*log(-e^x + 1) + 6*x^2*dilog(-e^x) + 6*x^ 2*dilog(e^x) - 3*x^2 + 3*x*log(e^x + 1) + 3*x*log(-e^x + 1) - 12*x*polylog (3, -e^x) - 12*x*polylog(3, e^x) + 1/32*(16*x^4 - 8*x^3 + 84*x^2 + (4*x^3 - 6*x^2 + 6*x - 3)*e^(6*x) + 2*(8*x^4 - 4*x^3 + 6*x^2 - 6*x + 3)*e^(4*x) - 4*(8*x^4 + 14*x^3 + 24*x^2 - 3*x)*e^(2*x) + (4*x^3 + 6*x^2 + 6*x + 3)*e^( -2*x) - 12*x - 6)/(e^(4*x) - 2*e^(2*x) + 1) + 3*dilog(-e^x) + 3*dilog(e^x) + 12*polylog(4, -e^x) + 12*polylog(4, e^x)
\[ \int x^3 \cosh ^2(x) \coth ^3(x) \, dx=\int { x^{3} \cosh \left (x\right )^{2} \coth \left (x\right )^{3} \,d x } \]
Timed out. \[ \int x^3 \cosh ^2(x) \coth ^3(x) \, dx=\int x^3\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {coth}\left (x\right )}^3 \,d x \]