Integrand size = 18, antiderivative size = 201 \[ \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {6 x \text {arctanh}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^4}-\frac {3 x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {3 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^4}+\frac {3 x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}+\frac {3 x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {3 x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {3 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {3 \operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^4} \]
-6*x*arctanh(exp(b*x+a))/b^3-x^3*arctanh(exp(b*x+a))/b-3/2*x^2*csch(b*x+a) /b^2-1/2*x^3*coth(b*x+a)*csch(b*x+a)/b-3*polylog(2,-exp(b*x+a))/b^4-3/2*x^ 2*polylog(2,-exp(b*x+a))/b^2+3*polylog(2,exp(b*x+a))/b^4+3/2*x^2*polylog(2 ,exp(b*x+a))/b^2+3*x*polylog(3,-exp(b*x+a))/b^3-3*x*polylog(3,exp(b*x+a))/ b^3-3*polylog(4,-exp(b*x+a))/b^4+3*polylog(4,exp(b*x+a))/b^4
Time = 3.31 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.39 \[ \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {12 b^2 x^2 \text {csch}(a)+b^3 x^3 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )-24 b x \log \left (1-e^{a+b x}\right )-4 b^3 x^3 \log \left (1-e^{a+b x}\right )+24 b x \log \left (1+e^{a+b x}\right )+4 b^3 x^3 \log \left (1+e^{a+b x}\right )+12 \left (2+b^2 x^2\right ) \operatorname {PolyLog}\left (2,-e^{a+b x}\right )-12 \left (2+b^2 x^2\right ) \operatorname {PolyLog}\left (2,e^{a+b x}\right )-24 b x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )+24 b x \operatorname {PolyLog}\left (3,e^{a+b x}\right )+24 \operatorname {PolyLog}\left (4,-e^{a+b x}\right )-24 \operatorname {PolyLog}\left (4,e^{a+b x}\right )+b^3 x^3 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )-6 b^2 x^2 \text {csch}\left (\frac {a}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )-6 b^2 x^2 \text {sech}\left (\frac {a}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right ) \sinh \left (\frac {b x}{2}\right )}{8 b^4} \]
-1/8*(12*b^2*x^2*Csch[a] + b^3*x^3*Csch[(a + b*x)/2]^2 - 24*b*x*Log[1 - E^ (a + b*x)] - 4*b^3*x^3*Log[1 - E^(a + b*x)] + 24*b*x*Log[1 + E^(a + b*x)] + 4*b^3*x^3*Log[1 + E^(a + b*x)] + 12*(2 + b^2*x^2)*PolyLog[2, -E^(a + b*x )] - 12*(2 + b^2*x^2)*PolyLog[2, E^(a + b*x)] - 24*b*x*PolyLog[3, -E^(a + b*x)] + 24*b*x*PolyLog[3, E^(a + b*x)] + 24*PolyLog[4, -E^(a + b*x)] - 24* PolyLog[4, E^(a + b*x)] + b^3*x^3*Sech[(a + b*x)/2]^2 - 6*b^2*x^2*Csch[a/2 ]*Csch[(a + b*x)/2]*Sinh[(b*x)/2] - 6*b^2*x^2*Sech[a/2]*Sech[(a + b*x)/2]* Sinh[(b*x)/2])/b^4
Result contains complex when optimal does not.
Time = 1.71 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.98, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {5980, 3042, 26, 4670, 3011, 4674, 26, 3042, 26, 4670, 2715, 2838, 3011, 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.
| \(\displaystyle \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx\) |
| \(\Big \downarrow \) 5980 |
| \(\displaystyle \int x^3 \text {csch}^3(a+b x)dx+\int x^3 \text {csch}(a+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i x^3 \csc (i a+i b x)dx+\int -i x^3 \csc (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int x^3 \csc (i a+i b x)dx-i \int x^3 \csc (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle i \left (\frac {3 i \int x^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i \int x^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \int x^3 \csc (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \int x^3 \csc (i a+i b x)^3dx\) |
| \(\Big \downarrow \) 4674 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (-\frac {3 \int -i x \text {csch}(a+b x)dx}{b^2}+\frac {1}{2} \int -i x^3 \text {csch}(a+b x)dx-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {3 i \int x \text {csch}(a+b x)dx}{b^2}-\frac {1}{2} i \int x^3 \text {csch}(a+b x)dx-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {3 i \int i x \csc (i a+i b x)dx}{b^2}-\frac {1}{2} i \int i x^3 \csc (i a+i b x)dx-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (-\frac {3 \int x \csc (i a+i b x)dx}{b^2}+\frac {1}{2} \int x^3 \csc (i a+i b x)dx-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (-\frac {3 \left (\frac {i \int \log \left (1-e^{a+b x}\right )dx}{b}-\frac {i \int \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b^2}+\frac {1}{2} \left (\frac {3 i \int x^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i \int x^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (-\frac {3 \left (\frac {i \int e^{-a-b x} \log \left (1-e^{a+b x}\right )de^{a+b x}}{b^2}-\frac {i \int e^{-a-b x} \log \left (1+e^{a+b x}\right )de^{a+b x}}{b^2}+\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}\right )}{b^2}+\frac {1}{2} \left (\frac {3 i \int x^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i \int x^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {1}{2} \left (\frac {3 i \int x^2 \log \left (1-e^{a+b x}\right )dx}{b}-\frac {3 i \int x^2 \log \left (1+e^{a+b x}\right )dx}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 \left (\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {1}{2} \left (-\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,-e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \int x \operatorname {PolyLog}\left (2,e^{a+b x}\right )dx}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 \left (\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (3,-e^{a+b x}\right )dx}{b}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (3,e^{a+b x}\right )dx}{b}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {1}{2} \left (-\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (3,-e^{a+b x}\right )dx}{b}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\int \operatorname {PolyLog}\left (3,e^{a+b x}\right )dx}{b}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 \left (\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle i \left (-\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,-e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-i \left (\frac {1}{2} \left (-\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,-e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\int e^{-a-b x} \operatorname {PolyLog}\left (3,e^{a+b x}\right )de^{a+b x}}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}+\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}\right )-\frac {3 \left (\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle i \left (\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}\right )-i \left (\frac {1}{2} \left (\frac {2 i x^3 \text {arctanh}\left (e^{a+b x}\right )}{b}-\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,-e^{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (4,-e^{a+b x}\right )}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b}\right )}{b}+\frac {3 i \left (\frac {2 \left (\frac {x \operatorname {PolyLog}\left (3,e^{a+b x}\right )}{b}-\frac {\operatorname {PolyLog}\left (4,e^{a+b x}\right )}{b^2}\right )}{b}-\frac {x^2 \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b}\right )}{b}\right )-\frac {3 \left (\frac {2 i x \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {i \operatorname {PolyLog}\left (2,-e^{a+b x}\right )}{b^2}-\frac {i \operatorname {PolyLog}\left (2,e^{a+b x}\right )}{b^2}\right )}{b^2}-\frac {3 i x^2 \text {csch}(a+b x)}{2 b^2}-\frac {i x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
I*(((2*I)*x^3*ArcTanh[E^(a + b*x)])/b - ((3*I)*(-((x^2*PolyLog[2, -E^(a + b*x)])/b) + (2*((x*PolyLog[3, -E^(a + b*x)])/b - PolyLog[4, -E^(a + b*x)]/ b^2))/b))/b + ((3*I)*(-((x^2*PolyLog[2, E^(a + b*x)])/b) + (2*((x*PolyLog[ 3, E^(a + b*x)])/b - PolyLog[4, E^(a + b*x)]/b^2))/b))/b) - I*((((-3*I)/2) *x^2*Csch[a + b*x])/b^2 - ((I/2)*x^3*Coth[a + b*x]*Csch[a + b*x])/b - (3*( ((2*I)*x*ArcTanh[E^(a + b*x)])/b + (I*PolyLog[2, -E^(a + b*x)])/b^2 - (I*P olyLog[2, E^(a + b*x)])/b^2))/b^2 + (((2*I)*x^3*ArcTanh[E^(a + b*x)])/b - ((3*I)*(-((x^2*PolyLog[2, -E^(a + b*x)])/b) + (2*((x*PolyLog[3, -E^(a + b* x)])/b - PolyLog[4, -E^(a + b*x)]/b^2))/b))/b + ((3*I)*(-((x^2*PolyLog[2, E^(a + b*x)])/b) + (2*((x*PolyLog[3, E^(a + b*x)])/b - PolyLog[4, E^(a + b *x)]/b^2))/b))/b)/2)
3.5.53.3.1 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[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[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[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[Coth[(a_.) + (b_.)*(x_)]^(p_)*Csch[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*( x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2 ), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; FreeQ[ {a, b, c, d, m}, x] && IGtQ[p/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]
Time = 1.09 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.69
| method | result | size |
| risch | \(-\frac {x^{2} {\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a} b x +b x +3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {3 \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \operatorname {polylog}\left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \operatorname {polylog}\left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{2 b}+\frac {3 x^{2} \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {3 x \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) x^{3}}{2 b}-\frac {3 x^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {3 x \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) a}{b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{2 b^{4}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right ) a^{3}}{2 b^{4}}+\frac {6 a \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {a^{3} \operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b^{4}}\) | \(340\) |
-x^2*exp(b*x+a)*(exp(2*b*x+2*a)*b*x+b*x+3*exp(2*b*x+2*a)-3)/b^2/(exp(2*b*x +2*a)-1)^2+3*polylog(2,exp(b*x+a))/b^4+3*polylog(4,exp(b*x+a))/b^4-3*polyl og(2,-exp(b*x+a))/b^4-3*polylog(4,-exp(b*x+a))/b^4+3/b^3*ln(1-exp(b*x+a))* x-3/b^3*ln(exp(b*x+a)+1)*x+1/2/b*ln(1-exp(b*x+a))*x^3+3/2*x^2*polylog(2,ex p(b*x+a))/b^2-3*x*polylog(3,exp(b*x+a))/b^3-1/2/b*ln(exp(b*x+a)+1)*x^3-3/2 *x^2*polylog(2,-exp(b*x+a))/b^2+3*x*polylog(3,-exp(b*x+a))/b^3+3/b^4*ln(1- exp(b*x+a))*a-3/b^4*ln(exp(b*x+a)+1)*a+1/2/b^4*ln(1-exp(b*x+a))*a^3-1/2/b^ 4*ln(exp(b*x+a)+1)*a^3+6/b^4*a*arctanh(exp(b*x+a))+1/b^4*a^3*arctanh(exp(b *x+a))
Leaf count of result is larger than twice the leaf count of optimal. 1802 vs. \(2 (179) = 358\).
Time = 0.28 (sec) , antiderivative size = 1802, normalized size of antiderivative = 8.97 \[ \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx=\text {Too large to display} \]
-1/2*(2*(b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)^3 + 6*(b^3*x^3 + 3*b^2*x^2)*co sh(b*x + a)*sinh(b*x + a)^2 + 2*(b^3*x^3 + 3*b^2*x^2)*sinh(b*x + a)^3 + 2* (b^3*x^3 - 3*b^2*x^2)*cosh(b*x + a) - 3*((b^2*x^2 + 2)*cosh(b*x + a)^4 + 4 *(b^2*x^2 + 2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 2)*sinh(b*x + a) ^4 + b^2*x^2 - 2*(b^2*x^2 + 2)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 2)*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 4*((b^2*x^2 + 2)*cosh(b*x + a)^ 3 - (b^2*x^2 + 2)*cosh(b*x + a))*sinh(b*x + a) + 2)*dilog(cosh(b*x + a) + sinh(b*x + a)) + 3*((b^2*x^2 + 2)*cosh(b*x + a)^4 + 4*(b^2*x^2 + 2)*cosh(b *x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 2)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2 *x^2 + 2)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 2)*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 4*((b^2*x^2 + 2)*cosh(b*x + a)^3 - (b^2*x^2 + 2)*cos h(b*x + a))*sinh(b*x + a) + 2)*dilog(-cosh(b*x + a) - sinh(b*x + a)) + (b^ 3*x^3 + (b^3*x^3 + 6*b*x)*cosh(b*x + a)^4 + 4*(b^3*x^3 + 6*b*x)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x^3 + 6*b*x)*sinh(b*x + a)^4 - 2*(b^3*x^3 + 6*b *x)*cosh(b*x + a)^2 - 2*(b^3*x^3 - 3*(b^3*x^3 + 6*b*x)*cosh(b*x + a)^2 + 6 *b*x)*sinh(b*x + a)^2 + 6*b*x + 4*((b^3*x^3 + 6*b*x)*cosh(b*x + a)^3 - (b^ 3*x^3 + 6*b*x)*cosh(b*x + a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + ((a^3 + 6*a)*cosh(b*x + a)^4 + 4*(a^3 + 6*a)*cosh(b*x + a)*sin h(b*x + a)^3 + (a^3 + 6*a)*sinh(b*x + a)^4 + a^3 - 2*(a^3 + 6*a)*cosh(b*x + a)^2 - 2*(a^3 - 3*(a^3 + 6*a)*cosh(b*x + a)^2 + 6*a)*sinh(b*x + a)^2 ...
Timed out. \[ \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.30 \[ \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {{\left (b x^{3} e^{\left (3 \, a\right )} + 3 \, x^{2} e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x^{3} e^{a} - 3 \, x^{2} e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{2 \, b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{2 \, b^{4}} - \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \]
-((b*x^3*e^(3*a) + 3*x^2*e^(3*a))*e^(3*b*x) + (b*x^3*e^a - 3*x^2*e^a)*e^(b *x))/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^(2*b*x + 2*a) + b^2) - 1/2*(b^3*x^3*lo g(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 + 1/2*(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
\[ \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx=\int { x^{3} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx=\int \frac {x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\mathrm {sinh}\left (a+b\,x\right )}^3} \,d x \]