Integrand size = 12, antiderivative size = 179 \[ \int x^3 \coth ^3(a+b x) \, dx=-\frac {3 x^2}{2 b^2}+\frac {x^3}{2 b}-\frac {x^4}{4}-\frac {3 x^2 \coth (a+b x)}{2 b^2}-\frac {x^3 \coth ^2(a+b x)}{2 b}+\frac {3 x \log \left (1-e^{2 (a+b x)}\right )}{b^3}+\frac {x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {3 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^4}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac {3 x \operatorname {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {3 \operatorname {PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4} \] Output:
-3/2*x^2/b^2+1/2*x^3/b-1/4*x^4-3/2*x^2*coth(b*x+a)/b^2-1/2*x^3*coth(b*x+a) ^2/b+3*x*ln(1-exp(2*b*x+2*a))/b^3+x^3*ln(1-exp(2*b*x+2*a))/b+3/2*polylog(2 ,exp(2*b*x+2*a))/b^4+3/2*x^2*polylog(2,exp(2*b*x+2*a))/b^2-3/2*x*polylog(3 ,exp(2*b*x+2*a))/b^3+3/4*polylog(4,exp(2*b*x+2*a))/b^4
Leaf count is larger than twice the leaf count of optimal. \(422\) vs. \(2(179)=358\).
Time = 2.30 (sec) , antiderivative size = 422, normalized size of antiderivative = 2.36 \[ \int x^3 \coth ^3(a+b x) \, dx=\frac {1}{4} \left (x^4 \coth (a)-\frac {2 x^3 \text {csch}^2(a+b x)}{b}-\frac {2 e^{2 a} \left (6 b^2 e^{-2 a} x^2+b^4 e^{-2 a} x^4-6 b \left (1-e^{-2 a}\right ) x \log \left (1-e^{-a-b x}\right )-2 b^3 e^{-2 a} \left (-1+e^{2 a}\right ) x^3 \log \left (1-e^{-a-b x}\right )-6 b \left (1-e^{-2 a}\right ) x \log \left (1+e^{-a-b x}\right )-2 b^3 e^{-2 a} \left (-1+e^{2 a}\right ) x^3 \log \left (1+e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+6 b^2 \left (1-e^{-2 a}\right ) x^2 \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+6 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+6 b^2 \left (1-e^{-2 a}\right ) x^2 \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+12 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )+12 b \left (1-e^{-2 a}\right ) x \operatorname {PolyLog}\left (3,e^{-a-b x}\right )+12 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (4,-e^{-a-b x}\right )+12 \left (1-e^{-2 a}\right ) \operatorname {PolyLog}\left (4,e^{-a-b x}\right )\right )}{b^4 \left (-1+e^{2 a}\right )}+\frac {6 x^2 \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b^2}\right ) \] Input:
Integrate[x^3*Coth[a + b*x]^3,x]
Output:
(x^4*Coth[a] - (2*x^3*Csch[a + b*x]^2)/b - (2*E^(2*a)*((6*b^2*x^2)/E^(2*a) + (b^4*x^4)/E^(2*a) - 6*b*(1 - E^(-2*a))*x*Log[1 - E^(-a - b*x)] - (2*b^3 *(-1 + E^(2*a))*x^3*Log[1 - E^(-a - b*x)])/E^(2*a) - 6*b*(1 - E^(-2*a))*x* Log[1 + E^(-a - b*x)] - (2*b^3*(-1 + E^(2*a))*x^3*Log[1 + E^(-a - b*x)])/E ^(2*a) + 6*(1 - E^(-2*a))*PolyLog[2, -E^(-a - b*x)] + 6*b^2*(1 - E^(-2*a)) *x^2*PolyLog[2, -E^(-a - b*x)] + 6*(1 - E^(-2*a))*PolyLog[2, E^(-a - b*x)] + 6*b^2*(1 - E^(-2*a))*x^2*PolyLog[2, E^(-a - b*x)] + 12*b*(1 - E^(-2*a)) *x*PolyLog[3, -E^(-a - b*x)] + 12*b*(1 - E^(-2*a))*x*PolyLog[3, E^(-a - b* x)] + 12*(1 - E^(-2*a))*PolyLog[4, -E^(-a - b*x)] + 12*(1 - E^(-2*a))*Poly Log[4, E^(-a - b*x)]))/(b^4*(-1 + E^(2*a))) + (6*x^2*Csch[a]*Csch[a + b*x] *Sinh[b*x])/b^2)/4
Result contains complex when optimal does not.
Time = 1.49 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.49, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.917, Rules used = {3042, 26, 4203, 25, 26, 3042, 25, 26, 4201, 2620, 3011, 4203, 15, 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.
| \(\displaystyle \int x^3 \coth ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i x^3 \tan \left (i a+i b x+\frac {\pi }{2}\right )^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int x^3 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^3dx\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle i \left (-\int i x^3 \coth (a+b x)dx+\frac {3 i \int -x^2 \coth ^2(a+b x)dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\int i x^3 \coth (a+b x)dx-\frac {3 i \int x^2 \coth ^2(a+b x)dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-i \int x^3 \coth (a+b x)dx-\frac {3 i \int x^2 \coth ^2(a+b x)dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-i \int -i x^3 \tan \left (i a+i b x+\frac {\pi }{2}\right )dx-\frac {3 i \int -x^2 \tan \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-i \int -i x^3 \tan \left (i a+i b x+\frac {\pi }{2}\right )dx+\frac {3 i \int x^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\int x^3 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx+\frac {3 i \int x^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle i \left (-2 i \int \frac {e^{2 a+2 b x-i \pi } x^3}{1+e^{2 a+2 b x-i \pi }}dx+\frac {3 i \int x^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle i \left (\frac {3 i \int x^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx}{2 b}-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \int x^2 \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{2 b}\right )+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \int x^2 \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )^2dx}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 4203 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \int i x \coth (a+b x)dx}{b}-\int x^2dx+\frac {x^2 \coth (a+b x)}{b}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \int i x \coth (a+b x)dx}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (-\frac {2 \int x \coth (a+b x)dx}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (-\frac {2 \int -i x \tan \left (i a+i b x+\frac {\pi }{2}\right )dx}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \int x \tan \left (\frac {1}{2} (2 i a+\pi )+i b x\right )dx}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 4201 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \left (2 i \int \frac {e^{2 a+2 b x-i \pi } x}{1+e^{2 a+2 b x-i \pi }}dx-\frac {i x^2}{2}\right )}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \left (2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int \log \left (1+e^{2 a+2 b x-i \pi }\right )dx}{2 b}\right )-\frac {i x^2}{2}\right )}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle i \left (\frac {3 i \left (\frac {2 i \left (2 i \left (\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int e^{-2 a-2 b x+i \pi } \log \left (1+e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}\right )-\frac {i x^2}{2}\right )}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\int x \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )dx}{2 b}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\int e^{-2 a-2 b x+i \pi } \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )de^{2 a+2 b x-i \pi }}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle i \left (-2 i \left (\frac {x^3 \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {3 \left (\frac {\frac {x \operatorname {PolyLog}\left (3,-e^{2 a+2 b x-i \pi }\right )}{2 b}-\frac {\operatorname {PolyLog}\left (4,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{2 b}\right )}{2 b}\right )+\frac {3 i \left (\frac {2 i \left (2 i \left (\frac {\operatorname {PolyLog}\left (2,-e^{2 a+2 b x-i \pi }\right )}{4 b^2}+\frac {x \log \left (1+e^{2 a+2 b x-i \pi }\right )}{2 b}\right )-\frac {i x^2}{2}\right )}{b}+\frac {x^2 \coth (a+b x)}{b}-\frac {x^3}{3}\right )}{2 b}+\frac {i x^3 \coth ^2(a+b x)}{2 b}+\frac {i x^4}{4}\right )\) |
Input:
Int[x^3*Coth[a + b*x]^3,x]
Output:
I*((I/4)*x^4 + ((I/2)*x^3*Coth[a + b*x]^2)/b + (((3*I)/2)*(-1/3*x^3 + (x^2 *Coth[a + b*x])/b + ((2*I)*((-1/2*I)*x^2 + (2*I)*((x*Log[1 + E^(2*a - I*Pi + 2*b*x)])/(2*b) + PolyLog[2, -E^(2*a - I*Pi + 2*b*x)]/(4*b^2))))/b))/b - (2*I)*((x^3*Log[1 + E^(2*a - I*Pi + 2*b*x)])/(2*b) - (3*(-1/2*(x^2*PolyLo g[2, -E^(2*a - I*Pi + 2*b*x)])/b + ((x*PolyLog[3, -E^(2*a - I*Pi + 2*b*x)] )/(2*b) - PolyLog[4, -E^(2*a - I*Pi + 2*b*x)]/(4*b^2))/b))/(2*b)))
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[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[((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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(374\) vs. \(2(161)=322\).
Time = 0.74 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.09
| method | result | size |
| risch | \(-\frac {x^{4}}{4}-\frac {x^{2} \left (2 b x \,{\mathrm e}^{2 b x +2 a}+3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {6 a x}{b^{3}}-\frac {3 a^{4}}{2 b^{4}}-\frac {3 a^{2}}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{b^{4}}+\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{b}+\frac {6 a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{b}-\frac {2 a^{3} x}{b^{3}}+\frac {3 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {3 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{2}}-\frac {6 x \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {6 x \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {a^{3} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {2 a^{3} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {6 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 x^{2}}{b^{2}}\) | \(375\) |
Input:
int(x^3*coth(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-1/4*x^4-x^2*(2*b*x*exp(2*b*x+2*a)+3*exp(2*b*x+2*a)-3)/b^2/(exp(2*b*x+2*a) -1)^2-6*a*x/b^3-3/2/b^4*a^4-3/b^4*a^2+3/b^4*ln(1-exp(b*x+a))*a-3/b^4*a*ln( exp(b*x+a)-1)+3/b^3*ln(exp(b*x+a)+1)*x+3/b^3*ln(1-exp(b*x+a))*x+1/b^4*ln(1 -exp(b*x+a))*a^3+1/b*ln(exp(b*x+a)+1)*x^3+6/b^4*a*ln(exp(b*x+a))+1/b*ln(1- exp(b*x+a))*x^3-2/b^3*a^3*x+3*x^2*polylog(2,-exp(b*x+a))/b^2+3*x^2*polylog (2,exp(b*x+a))/b^2-6*x*polylog(3,-exp(b*x+a))/b^3-6*x*polylog(3,exp(b*x+a) )/b^3-1/b^4*a^3*ln(exp(b*x+a)-1)+6*polylog(4,-exp(b*x+a))/b^4+2/b^4*a^3*ln (exp(b*x+a))+6*polylog(4,exp(b*x+a))/b^4+3*polylog(2,-exp(b*x+a))/b^4+3*po lylog(2,exp(b*x+a))/b^4-3*x^2/b^2
Leaf count of result is larger than twice the leaf count of optimal. 1985 vs. \(2 (159) = 318\).
Time = 0.11 (sec) , antiderivative size = 1985, normalized size of antiderivative = 11.09 \[ \int x^3 \coth ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(x^3*coth(b*x+a)^3,x, algorithm="fricas")
Output:
-1/4*(b^4*x^4 + (b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2)*cosh(b*x + a)^4 + 4*(b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2)*sinh(b*x + a)^4 - 2*a^4 - 2*(b^4*x ^4 - 4*b^3*x^3 - 2*a^4 + 6*b^2*x^2 - 12*a^2)*cosh(b*x + a)^2 - 2*(b^4*x^4 - 4*b^3*x^3 - 2*a^4 + 6*b^2*x^2 - 3*(b^4*x^4 - 2*a^4 + 12*b^2*x^2 - 12*a^2 )*cosh(b*x + a)^2 - 12*a^2)*sinh(b*x + a)^2 - 12*a^2 - 12*((b^2*x^2 + 1)*c osh(b*x + a)^4 + 4*(b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 1)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x^2 + 1)*cosh(b*x + a)^2 - 2*(b^2* x^2 - 3*(b^2*x^2 + 1)*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 4*((b^2*x^2 + 1)*cosh(b*x + a)^3 - (b^2*x^2 + 1)*cosh(b*x + a))*sinh(b*x + a) + 1)*dilo g(cosh(b*x + a) + sinh(b*x + a)) - 12*((b^2*x^2 + 1)*cosh(b*x + a)^4 + 4*( b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 1)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x^2 + 1)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 1 )*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 4*((b^2*x^2 + 1)*cosh(b*x + a)^3 - (b^2*x^2 + 1)*cosh(b*x + a))*sinh(b*x + a) + 1)*dilog(-cosh(b*x + a) - s inh(b*x + a)) - 4*(b^3*x^3 + (b^3*x^3 + 3*b*x)*cosh(b*x + a)^4 + 4*(b^3*x^ 3 + 3*b*x)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x^3 + 3*b*x)*sinh(b*x + a) ^4 - 2*(b^3*x^3 + 3*b*x)*cosh(b*x + a)^2 - 2*(b^3*x^3 - 3*(b^3*x^3 + 3*b*x )*cosh(b*x + a)^2 + 3*b*x)*sinh(b*x + a)^2 + 3*b*x + 4*((b^3*x^3 + 3*b*x)* cosh(b*x + a)^3 - (b^3*x^3 + 3*b*x)*cosh(b*x + a))*sinh(b*x + a))*log(c...
\[ \int x^3 \coth ^3(a+b x) \, dx=\int x^{3} \coth ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate(x**3*coth(b*x+a)**3,x)
Output:
Integral(x**3*coth(a + b*x)**3, x)
Time = 0.13 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.69 \[ \int x^3 \coth ^3(a+b x) \, dx=\frac {b^{2} x^{4} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{4} + 12 \, x^{2} - 2 \, {\left (b^{2} x^{4} e^{\left (2 \, a\right )} + 4 \, b x^{3} e^{\left (2 \, a\right )} + 6 \, x^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{4 \, {\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} - \frac {b^{4} x^{4} + 6 \, b^{2} x^{2}}{2 \, b^{4}} + \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} + \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \] Input:
integrate(x^3*coth(b*x+a)^3,x, algorithm="maxima")
Output:
1/4*(b^2*x^4*e^(4*b*x + 4*a) + b^2*x^4 + 12*x^2 - 2*(b^2*x^4*e^(2*a) + 4*b *x^3*e^(2*a) + 6*x^2*e^(2*a))*e^(2*b*x))/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^(2 *b*x + 2*a) + b^2) - 1/2*(b^4*x^4 + 6*b^2*x^2)/b^4 + (b^3*x^3*log(e^(b*x + a) + 1) + 3*b^2*x^2*dilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x + a)))/b^4 + (b^3*x^3*log(-e^(b*x + a) + 1) + 3*b^2 *x^2*dilog(e^(b*x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4, e^( b*x + a)))/b^4 + 3*(b*x*log(e^(b*x + a) + 1) + dilog(-e^(b*x + a)))/b^4 + 3*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^4
\[ \int x^3 \coth ^3(a+b x) \, dx=\int { x^{3} \coth \left (b x + a\right )^{3} \,d x } \] Input:
integrate(x^3*coth(b*x+a)^3,x, algorithm="giac")
Output:
integrate(x^3*coth(b*x + a)^3, x)
Timed out. \[ \int x^3 \coth ^3(a+b x) \, dx=\int x^3\,{\mathrm {coth}\left (a+b\,x\right )}^3 \,d x \] Input:
int(x^3*coth(a + b*x)^3,x)
Output:
int(x^3*coth(a + b*x)^3, x)
\[ \int x^3 \coth ^3(a+b x) \, dx=\frac {-45-32 e^{2 b x +2 a} b^{4} x^{4}+234 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )+234 e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )+16 b^{4} x^{4}+16 e^{4 b x +4 a} b^{4} x^{4}+720 e^{4 b x +4 a} \left (\int \frac {x}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{2}+360 b^{2} x^{2}+234 \,\mathrm {log}\left (e^{b x +a}-1\right )+234 \,\mathrm {log}\left (e^{b x +a}+1\right )+45 e^{4 b x +4 a}-1440 e^{2 b x +2 a} \left (\int \frac {x}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{2}+96 b^{3} x^{3}-192 e^{2 b x +2 a} b^{3} x^{3}-288 e^{2 b x +2 a} b^{2} x^{2}+648 e^{2 b x +2 a} b x +128 \left (\int \frac {x^{3}}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{4}+288 \left (\int \frac {x^{2}}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{3}+720 \left (\int \frac {x}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{2}+128 e^{4 b x +4 a} \left (\int \frac {x^{3}}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{4}-468 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )-468 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )+288 e^{4 b x +4 a} \left (\int \frac {x^{2}}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{3}-256 e^{2 b x +2 a} \left (\int \frac {x^{3}}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{4}-576 e^{2 b x +2 a} \left (\int \frac {x^{2}}{e^{6 b x +6 a}-3 e^{4 b x +4 a}+3 e^{2 b x +2 a}-1}d x \right ) b^{3}-468 e^{4 b x +4 a} b x}{64 b^{4} \left (e^{4 b x +4 a}-2 e^{2 b x +2 a}+1\right )} \] Input:
int(x^3*coth(b*x+a)^3,x)
Output:
(128*e**(4*a + 4*b*x)*int(x**3/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3* e**(2*a + 2*b*x) - 1),x)*b**4 + 288*e**(4*a + 4*b*x)*int(x**2/(e**(6*a + 6 *b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**3 + 720*e**(4*a + 4*b*x)*int(x/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x ) - 1),x)*b**2 + 234*e**(4*a + 4*b*x)*log(e**(a + b*x) - 1) + 234*e**(4*a + 4*b*x)*log(e**(a + b*x) + 1) + 16*e**(4*a + 4*b*x)*b**4*x**4 - 468*e**(4 *a + 4*b*x)*b*x + 45*e**(4*a + 4*b*x) - 256*e**(2*a + 2*b*x)*int(x**3/(e** (6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**4 - 576 *e**(2*a + 2*b*x)*int(x**2/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**( 2*a + 2*b*x) - 1),x)*b**3 - 1440*e**(2*a + 2*b*x)*int(x/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**2 - 468*e**(2*a + 2*b *x)*log(e**(a + b*x) - 1) - 468*e**(2*a + 2*b*x)*log(e**(a + b*x) + 1) - 3 2*e**(2*a + 2*b*x)*b**4*x**4 - 192*e**(2*a + 2*b*x)*b**3*x**3 - 288*e**(2* a + 2*b*x)*b**2*x**2 + 648*e**(2*a + 2*b*x)*b*x + 128*int(x**3/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**4 + 288*int(x* *2/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b** 3 + 720*int(x/(e**(6*a + 6*b*x) - 3*e**(4*a + 4*b*x) + 3*e**(2*a + 2*b*x) - 1),x)*b**2 + 234*log(e**(a + b*x) - 1) + 234*log(e**(a + b*x) + 1) + 16* b**4*x**4 + 96*b**3*x**3 + 360*b**2*x**2 - 45)/(64*b**4*(e**(4*a + 4*b*x) - 2*e**(2*a + 2*b*x) + 1))