Integrand size = 12, antiderivative size = 237 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {11}{27} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{1+\frac {1}{x}} x+\frac {7}{18} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{1+\frac {1}{x}} x^2+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{1+\frac {1}{x}} x^3-\frac {19 \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19 \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{\sqrt {3}}\right )}{54 \sqrt {3}}+\frac {19}{81} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {19}{162} \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{1-\frac {1}{x}}}\right ) \sqrt [6]{1-\frac {1}{x}}}\right ) \] Output:
11/27*(1-1/x)^(5/6)*(1+1/x)^(1/6)*x+7/18*(1-1/x)^(5/6)*(1+1/x)^(1/6)*x^2+1 /3*(1-1/x)^(5/6)*(1+1/x)^(1/6)*x^3-19/162*arctan(1/3*(1-2*(1+1/x)^(1/6)/(1 -1/x)^(1/6))*3^(1/2))*3^(1/2)+19/162*arctan(1/3*(1+2*(1+1/x)^(1/6)/(1-1/x) ^(1/6))*3^(1/2))*3^(1/2)+19/81*arctanh((1+1/x)^(1/6)/(1-1/x)^(1/6))+19/162 *arctanh((1+1/x)^(1/6)/(1+(1+1/x)^(1/3)/(1-1/x)^(1/3))/(1-1/x)^(1/6))
Time = 5.29 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.80 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{324} \left (\frac {864 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^3}+\frac {1368 e^{\frac {1}{3} \coth ^{-1}(x)}}{\left (-1+e^{2 \coth ^{-1}(x)}\right )^2}+\frac {732 e^{\frac {1}{3} \coth ^{-1}(x)}}{-1+e^{2 \coth ^{-1}(x)}}+38 \sqrt {3} \arctan \left (\frac {-1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )+38 \sqrt {3} \arctan \left (\frac {1+2 e^{\frac {1}{3} \coth ^{-1}(x)}}{\sqrt {3}}\right )-38 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}\right )+38 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}\right )-19 \log \left (1-e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )+19 \log \left (1+e^{\frac {1}{3} \coth ^{-1}(x)}+e^{\frac {2}{3} \coth ^{-1}(x)}\right )\right ) \] Input:
Integrate[E^(ArcCoth[x]/3)*x^2,x]
Output:
((864*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x]))^3 + (1368*E^(ArcCoth[x]/3) )/(-1 + E^(2*ArcCoth[x]))^2 + (732*E^(ArcCoth[x]/3))/(-1 + E^(2*ArcCoth[x] )) + 38*Sqrt[3]*ArcTan[(-1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] + 38*Sqrt[3]*Arc Tan[(1 + 2*E^(ArcCoth[x]/3))/Sqrt[3]] - 38*Log[1 - E^(ArcCoth[x]/3)] + 38* Log[1 + E^(ArcCoth[x]/3)] - 19*Log[1 - E^(ArcCoth[x]/3) + E^((2*ArcCoth[x] )/3)] + 19*Log[1 + E^(ArcCoth[x]/3) + E^((2*ArcCoth[x])/3)])/324
Time = 0.69 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.12, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6721, 110, 27, 168, 27, 168, 27, 104, 754, 27, 219, 1142, 25, 1083, 217, 1103}
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.
| \(\displaystyle \int x^2 e^{\frac {1}{3} \coth ^{-1}(x)} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [6]{1+\frac {1}{x}} x^4}{\sqrt [6]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3-\frac {1}{3} \int \frac {\left (7+\frac {6}{x}\right ) x^3}{3 \sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3-\frac {1}{9} \int \frac {\left (7+\frac {6}{x}\right ) x^3}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{2} \int -\frac {\left (22+\frac {21}{x}\right ) x^2}{3 \sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2-\frac {1}{6} \int \frac {\left (22+\frac {21}{x}\right ) x^2}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (\int -\frac {19 x}{3 \sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}+22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-\frac {19}{3} \int \frac {x}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \int \frac {1}{\frac {1}{x^6}-1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{2 \left (-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{2 \left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (\frac {1}{6} \left (\frac {1}{2} \int -\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {3}{2} \int \frac {1}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (-\frac {3}{2} \int \frac {1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-1\right )-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (\frac {1}{6} \left (-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{9} \left (\frac {1}{6} \left (22 \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x-38 \left (\frac {1}{6} \left (\frac {1}{2} \log \left (\frac {1}{x^2}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {1}{x^2}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\right )+\frac {7}{2} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^2\right )+\frac {1}{3} \left (1-\frac {1}{x}\right )^{5/6} \sqrt [6]{\frac {1}{x}+1} x^3\) |
Input:
Int[E^(ArcCoth[x]/3)*x^2,x]
Output:
((1 - x^(-1))^(5/6)*(1 + x^(-1))^(1/6)*x^3)/3 + ((7*(1 - x^(-1))^(5/6)*(1 + x^(-1))^(1/6)*x^2)/2 + (22*(1 - x^(-1))^(5/6)*(1 + x^(-1))^(1/6)*x - 38* (-1/3*ArcTanh[(1 + x^(-1))^(1/6)/(1 - x^(-1))^(1/6)] + (-(Sqrt[3]*ArcTan[( -1 + (2*(1 + x^(-1))^(1/6))/(1 - x^(-1))^(1/6))/Sqrt[3]]) + Log[1 - (1 + x ^(-1))^(1/6)/(1 - x^(-1))^(1/6) + x^(-2)]/2)/6 + (-(Sqrt[3]*ArcTan[(1 + (2 *(1 + x^(-1))^(1/6))/(1 - x^(-1))^(1/6))/Sqrt[3]]) - Log[1 + (1 + x^(-1))^ (1/6)/(1 - x^(-1))^(1/6) + x^(-2)]/2)/6))/6)/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[-a /b, n]], s = Denominator[Rt[-a/b, n]], k, u}, Simp[u = Int[(r - s*Cos[(2*k* Pi)/n]*x)/(r^2 - 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x] + Int[(r + s*Cos[(2 *k*Pi)/n]*x)/(r^2 + 2*r*s*Cos[(2*k*Pi)/n]*x + s^2*x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 - s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] / ; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && NegQ[a/b]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(x_)^(m_.), x_Symbol] :> -Subst[Int[(1 + x /a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x] /; FreeQ[{a, n}, x] && !IntegerQ[n] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.39 (sec) , antiderivative size = 1163, normalized size of antiderivative = 4.91
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1163\) |
| risch | \(\text {Expression too large to display}\) | \(2895\) |
Input:
int(1/((x-1)/(1+x))^(1/6)*x^2,x,method=_RETURNVERBOSE)
Output:
1/54*(1+x)*(18*x^2+21*x+22)*(-(1-x)/(1+x))^(5/6)+19/162*ln(9*RootOf(9*_Z^2 -3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x+9*RootOf(9*_Z^2-3 *_Z+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/2)* x+3*(-(1-x)/(1+x))^(5/6)+3*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(9*_Z^2-3*_Z+1) *(-(1-x)/(1+x))^(1/2)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)*x+3*(- (1-x)/(1+x))^(2/3)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)+9*RootOf( 9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/3)*x+9*RootOf(9* _Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)-3*(-(1-x)/(1+x))^(1/3)-3*(-(1-x)/(1+x))^ (1/6)*x+3*RootOf(9*_Z^2-3*_Z+1)-3*(-(1-x)/(1+x))^(1/6)-2)-19/54*ln(9*RootO f(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x+9*RootOf( 9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x) )^(1/2)*x+3*(-(1-x)/(1+x))^(5/6)+3*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(9*_Z^2 -3*_Z+1)*(-(1-x)/(1+x))^(1/2)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3 )*x+3*(-(1-x)/(1+x))^(2/3)+18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/3)+9 *RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/3)*x+9*R ootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/6)-3*(-(1-x)/(1+x))^(1/3)-3*(-(1-x) /(1+x))^(1/6)*x+3*RootOf(9*_Z^2-3*_Z+1)-3*(-(1-x)/(1+x))^(1/6)-2)*RootOf(9 *_Z^2-3*_Z+1)+19/54*RootOf(9*_Z^2-3*_Z+1)*ln(-9*RootOf(9*_Z^2-3*_Z+1)*(-(1 -x)/(1+x))^(2/3)*x+3*(-(1-x)/(1+x))^(5/6)*x-9*RootOf(9*_Z^2-3*_Z+1)*(-(1-x )/(1+x))^(2/3)-18*RootOf(9*_Z^2-3*_Z+1)*(-(1-x)/(1+x))^(1/2)*x+3*(-(1-x...
Time = 0.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.73 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\frac {1}{54} \, {\left (18 \, x^{3} + 39 \, x^{2} + 43 \, x + 22\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}} - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/6)*x^2,x, algorithm="fricas")
Output:
1/54*(18*x^3 + 39*x^2 + 43*x + 22)*((x - 1)/(x + 1))^(5/6) - 19/162*sqrt(3 )*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 1/3*sqrt(3)) - 19/162*sqrt( 3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) - 1/3*sqrt(3)) + 19/324*log( ((x - 1)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 19/324*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + 19/162*log(((x - 1)/(x + 1))^(1/6) + 1) - 19/162*log(((x - 1)/(x + 1))^(1/6) - 1)
\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\int \frac {x^{2}}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \] Input:
integrate(1/((x-1)/(1+x))**(1/6)*x**2,x)
Output:
Integral(x**2/((x - 1)/(x + 1))**(1/6), x)
Time = 0.11 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.93 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {19 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {17}{6}} - 8 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {11}{6}} + 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {3 \, {\left (x - 1\right )}}{x + 1} - \frac {3 \, {\left (x - 1\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {{\left (x - 1\right )}^{3}}{{\left (x + 1\right )}^{3}} - 1\right )}} + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/6)*x^2,x, algorithm="maxima")
Output:
-19/162*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 19/1 62*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) - 1/27*(19* ((x - 1)/(x + 1))^(17/6) - 8*((x - 1)/(x + 1))^(11/6) + 61*((x - 1)/(x + 1 ))^(5/6))/(3*(x - 1)/(x + 1) - 3*(x - 1)^2/(x + 1)^2 + (x - 1)^3/(x + 1)^3 - 1) + 19/324*log(((x - 1)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 19/324*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + 19/1 62*log(((x - 1)/(x + 1))^(1/6) + 1) - 19/162*log(((x - 1)/(x + 1))^(1/6) - 1)
Time = 0.14 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.91 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \frac {19}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) + \frac {\frac {8 \, {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{x + 1} - \frac {19 \, {\left (x - 1\right )}^{2} \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{{\left (x + 1\right )}^{2}} - 61 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {5}{6}}}{27 \, {\left (\frac {x - 1}{x + 1} - 1\right )}^{3}} + \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{324} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \frac {19}{162} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {19}{162} \, \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \] Input:
integrate(1/((x-1)/(1+x))^(1/6)*x^2,x, algorithm="giac")
Output:
-19/162*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - 19/1 62*sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) + 1/27*(8*( x - 1)*((x - 1)/(x + 1))^(5/6)/(x + 1) - 19*(x - 1)^2*((x - 1)/(x + 1))^(5 /6)/(x + 1)^2 - 61*((x - 1)/(x + 1))^(5/6))/((x - 1)/(x + 1) - 1)^3 + 19/3 24*log(((x - 1)/(x + 1))^(1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 19/324*log (((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + 19/162*log(((x - 1)/(x + 1))^(1/6) + 1) - 19/162*log(abs(((x - 1)/(x + 1))^(1/6) - 1))
Time = 0.15 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.71 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=-\frac {\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,19{}\mathrm {i}}{81}-\frac {\frac {61\,{\left (\frac {x-1}{x+1}\right )}^{5/6}}{27}-\frac {8\,{\left (\frac {x-1}{x+1}\right )}^{11/6}}{27}+\frac {19\,{\left (\frac {x-1}{x+1}\right )}^{17/6}}{27}}{\frac {3\,\left (x-1\right )}{x+1}-\frac {3\,{\left (x-1\right )}^2}{{\left (x+1\right )}^2}+\frac {{\left (x-1\right )}^3}{{\left (x+1\right )}^3}-1}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,4952198{}\mathrm {i}}{14348907\,\left (-\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19\,\sqrt {3}}{162}-\frac {19}{162}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,4952198{}\mathrm {i}}{14348907\,\left (\frac {2476099}{14348907}+\frac {\sqrt {3}\,2476099{}\mathrm {i}}{14348907}\right )}\right )\,\left (\frac {19\,\sqrt {3}}{162}+\frac {19}{162}{}\mathrm {i}\right ) \] Input:
int(x^2/((x - 1)/(x + 1))^(1/6),x)
Output:
- (atan(((x - 1)/(x + 1))^(1/6)*1i)*19i)/81 - ((61*((x - 1)/(x + 1))^(5/6) )/27 - (8*((x - 1)/(x + 1))^(11/6))/27 + (19*((x - 1)/(x + 1))^(17/6))/27) /((3*(x - 1))/(x + 1) - (3*(x - 1)^2)/(x + 1)^2 + (x - 1)^3/(x + 1)^3 - 1) - atan((((x - 1)/(x + 1))^(1/6)*4952198i)/(14348907*((3^(1/2)*2476099i)/1 4348907 - 2476099/14348907)))*((19*3^(1/2))/162 - 19i/162) - atan((((x - 1 )/(x + 1))^(1/6)*4952198i)/(14348907*((3^(1/2)*2476099i)/14348907 + 247609 9/14348907)))*((19*3^(1/2))/162 + 19i/162)
\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^2 \, dx=\int \frac {\left (x +1\right )^{\frac {1}{6}} x^{2}}{\left (x -1\right )^{\frac {1}{6}}}d x \] Input:
int(1/((x-1)/(1+x))^(1/6)*x^2,x)
Output:
int(((x + 1)**(1/6)*x**2)/(x - 1)**(1/6),x)