Integrand size = 12, antiderivative size = 402 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )+\sqrt {3} \arctan \left (\frac {1+\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}}{\sqrt {3}}\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+\arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \arctan \left (\frac {\sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}\right )+2 \text {arctanh}\left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )-\frac {1}{2} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \log \left (1+\frac {\sqrt [3]{1+\frac {1}{x}}}{\sqrt [3]{\frac {-1+x}{x}}}+\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{\frac {-1+x}{x}}}\right )+\frac {1}{2} \sqrt {3} \log \left (1-\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{2} \sqrt {3} \log \left (1+\frac {\sqrt {3} \sqrt [6]{\frac {-1+x}{x}}}{\sqrt [6]{1+\frac {1}{x}}}+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right ) \]
2*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))+arctan(2*((-1+x)/x)^(1/6)/(1+1/x) ^(1/6)-3^(1/2))+arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+3^(1/2))+2*arctanh ((1+1/x)^(1/6)/((-1+x)/x)^(1/6))-1/2*ln(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)-( 1+1/x)^(1/6)/((-1+x)/x)^(1/6))+1/2*ln(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)+(1+ 1/x)^(1/6)/((-1+x)/x)^(1/6))-arctan(1/3*(1-2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6 ))*3^(1/2))*3^(1/2)+arctan(1/3*(1+2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))*3^(1/2 ))*3^(1/2)+1/2*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*3^(1/2 )/(1+1/x)^(1/6))*3^(1/2)-1/2*ln(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)+((-1+x)/x )^(1/6)*3^(1/2)/(1+1/x)^(1/6))*3^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.06 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=\frac {12}{7} e^{\frac {7}{3} \coth ^{-1}(x)} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},1,\frac {19}{12},e^{4 \coth ^{-1}(x)}\right ) \]
Time = 0.53 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.92, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {6721, 140, 73, 104, 754, 27, 219, 854, 824, 27, 216, 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.
| \(\displaystyle \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx\) |
| \(\Big \downarrow \) 6721 |
| \(\displaystyle -\int \frac {\sqrt [6]{1+\frac {1}{x}} x}{\sqrt [6]{1-\frac {1}{x}}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 140 |
| \(\displaystyle -\int \frac {1}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}-\int \frac {x}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-\int \frac {x}{\sqrt [6]{1-\frac {1}{x}} \left (1+\frac {1}{x}\right )^{5/6}}d\frac {1}{x}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \int \frac {1}{\frac {1}{x^6}-1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\) |
| \(\Big \downarrow \) 754 |
| \(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{2 \left (-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{2 \left (\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \left (-\frac {1}{3} \int \frac {1}{1-\frac {1}{x^2}}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 6 \int \frac {1}{\left (2-\frac {1}{x^6}\right )^{5/6} x^4}d\sqrt [6]{1-\frac {1}{x}}-6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle 6 \int \frac {1}{\left (1+\frac {1}{x^6}\right ) x^4}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle 6 \left (\frac {1}{3} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \int -\frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{2 \left (-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \int -\frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{2 \left (\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )-6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 6 \left (\frac {1}{3} \int \frac {1}{1+\frac {1}{x^2}}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )-6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 6 \left (-\frac {1}{6} \int \frac {1-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{6} \int \frac {\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )-6 \left (-\frac {1}{6} \int \frac {2-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{6} \int \frac {\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+2}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle 6 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{2} \sqrt {3} \int -\frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )-6 \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 )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle 6 \left (\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \int \frac {1}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )-6 \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 )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle 6 \left (\frac {1}{6} \left (-\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (-\int \frac {1}{-1-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )-6 \left (\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-1\right )-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )+\frac {1}{6} \left (3 \int \frac {1}{-3-\frac {1}{x^2}}d\left (\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle 6 \left (\frac {1}{6} \left (-\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}}{-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \int \frac {\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}}{\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )-6 \left (\frac {1}{6} \left (-\frac {1}{2} \int \frac {1-\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}}{-\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\frac {1}{2} \int \frac {\frac {2 \sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}+\frac {1}{x^2}+1}d\frac {\sqrt [6]{1+\frac {1}{x}}}{\sqrt [6]{1-\frac {1}{x}}}-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle 6 \left (\frac {1}{3} \arctan \left (\frac {\sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )+\frac {1}{6} \left (\frac {1}{2} \sqrt {3} \log \left (-\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )-\arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}\right )\right )+\frac {1}{6} \left (\arctan \left (\frac {2 \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\sqrt {3}\right )-\frac {1}{2} \sqrt {3} \log \left (\frac {\sqrt {3} \sqrt [6]{1-\frac {1}{x}}}{\sqrt [6]{2-\frac {1}{x^6}}}+\frac {1}{x^2}+1\right )\right )\right )-6 \left (\frac {1}{6} \left (\frac {1}{2} \log \left (\frac {1}{x^2}-\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}-1}{\sqrt {3}}\right )\right )+\frac {1}{6} \left (-\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1}{\sqrt {3}}\right )-\frac {1}{2} \log \left (\frac {1}{x^2}+\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}+1\right )\right )-\frac {1}{3} \text {arctanh}\left (\frac {\sqrt [6]{\frac {1}{x}+1}}{\sqrt [6]{1-\frac {1}{x}}}\right )\right )\) |
6*(ArcTan[(1 - x^(-1))^(1/6)/(2 - x^(-6))^(1/6)]/3 + (-ArcTan[Sqrt[3] - (2 *(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6)] + (Sqrt[3]*Log[1 - (Sqrt[3]*(1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6 + (ArcTan[Sqrt[3] + (2*( 1 - x^(-1))^(1/6))/(2 - x^(-6))^(1/6)] - (Sqrt[3]*Log[1 + (Sqrt[3]*(1 - x^ (-1))^(1/6))/(2 - x^(-6))^(1/6) + x^(-2)])/2)/6) - 6*(-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)
3.2.16.3.1 Defintions of rubi rules used
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_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, 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[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
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[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
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 = 22.19 (sec) , antiderivative size = 2088, normalized size of antiderivative = 5.19
-3*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*ln(-18*RootOf(3*_Z* RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x-18*RootOf(3 *_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)-3*(-(1-x) /(1+x))^(5/6)*x+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*(-( 1-x)/(1+x))^(1/3)*x-3*(-(1-x)/(1+x))^(5/6)+3*(-(1-x)/(1+x))^(2/3)*x+18*Roo tOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+3*(- (1-x)/(1+x))^(2/3)+6*(-(1-x)/(1+x))^(1/2)*x+6*(-(1-x)/(1+x))^(1/2)-3*(-(1- x)/(1+x))^(1/3)*x-6*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)-3* (-(1-x)/(1+x))^(1/3)-3*(-(1-x)/(1+x))^(1/6)*x-3*(-(1-x)/(1+x))^(1/6)+1)-3* ln(-(18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)*x+3*(-(1 -x)/(1+x))^(5/6)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)*x+18*RootOf(3*_Z* RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)+3*(-(1-x)/(1+x))^(5/6)+3*Roo tOf(_Z^2+1)*(-(1-x)/(1+x))^(2/3)+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*( -(1-x)/(1+x))^(1/3)*x+6*(-(1-x)/(1+x))^(1/2)*x+3*RootOf(_Z^2+1)*(-(1-x)/(1 +x))^(1/3)*x+18*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(1/3)+ 6*(-(1-x)/(1+x))^(1/2)+3*RootOf(_Z^2+1)*(-(1-x)/(1+x))^(1/3)+6*RootOf(3*_Z *RootOf(_Z^2+1)+9*_Z^2-1)*x+3*(-(1-x)/(1+x))^(1/6)*x+RootOf(_Z^2+1)*x+3*(- (1-x)/(1+x))^(1/6))/x)*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)-RootOf(_Z^2+1) *ln((9*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*(-(1-x)/(1+x))^(2/3)*x-18*(-(1 -x)/(1+x))^(1/2)*RootOf(3*_Z*RootOf(_Z^2+1)+9*_Z^2-1)*RootOf(_Z^2+1)*x-...
Time = 0.25 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {-3} + 2} \log \left (\sqrt {2 \, \sqrt {-3} + 2} {\left (\sqrt {-3} - 1\right )} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {-3} + 2} \log \left (-\sqrt {2 \, \sqrt {-3} + 2} {\left (\sqrt {-3} - 1\right )} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) - \frac {1}{2} \, \sqrt {-2 \, \sqrt {-3} + 2} \log \left ({\left (\sqrt {-3} + 1\right )} \sqrt {-2 \, \sqrt {-3} + 2} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \sqrt {-2 \, \sqrt {-3} + 2} \log \left (-{\left (\sqrt {-3} + 1\right )} \sqrt {-2 \, \sqrt {-3} + 2} + 4 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right ) \]
-sqrt(3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) + 1/3*sqrt(3)) - sqrt( 3)*arctan(2/3*sqrt(3)*((x - 1)/(x + 1))^(1/6) - 1/3*sqrt(3)) + 1/2*sqrt(2* sqrt(-3) + 2)*log(sqrt(2*sqrt(-3) + 2)*(sqrt(-3) - 1) + 4*((x - 1)/(x + 1) )^(1/6)) - 1/2*sqrt(2*sqrt(-3) + 2)*log(-sqrt(2*sqrt(-3) + 2)*(sqrt(-3) - 1) + 4*((x - 1)/(x + 1))^(1/6)) - 1/2*sqrt(-2*sqrt(-3) + 2)*log((sqrt(-3) + 1)*sqrt(-2*sqrt(-3) + 2) + 4*((x - 1)/(x + 1))^(1/6)) + 1/2*sqrt(-2*sqrt (-3) + 2)*log(-(sqrt(-3) + 1)*sqrt(-2*sqrt(-3) + 2) + 4*((x - 1)/(x + 1))^ (1/6)) + 2*arctan(((x - 1)/(x + 1))^(1/6)) + 1/2*log(((x - 1)/(x + 1))^(1/ 3) + ((x - 1)/(x + 1))^(1/6) + 1) - 1/2*log(((x - 1)/(x + 1))^(1/3) - ((x - 1)/(x + 1))^(1/6) + 1) + log(((x - 1)/(x + 1))^(1/6) + 1) - log(((x - 1) /(x + 1))^(1/6) - 1)
\[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=\int \frac {1}{x \sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]
\[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=\int { \frac {1}{x \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right )}\right ) - \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1\right )}\right ) - \frac {1}{2} \, \sqrt {3} \log \left (\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \sqrt {3} \log \left (-\sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \arctan \left (\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \arctan \left (-\sqrt {3} + 2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + 2 \, \arctan \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) + \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} + 1\right ) - \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}} - 1 \right |}\right ) \]
-sqrt(3)*arctan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) + 1)) - sqrt(3)*arc tan(1/3*sqrt(3)*(2*((x - 1)/(x + 1))^(1/6) - 1)) - 1/2*sqrt(3)*log(sqrt(3) *((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + 1/2*sqrt(3)*log( -sqrt(3)*((x - 1)/(x + 1))^(1/6) + ((x - 1)/(x + 1))^(1/3) + 1) + arctan(s qrt(3) + 2*((x - 1)/(x + 1))^(1/6)) + arctan(-sqrt(3) + 2*((x - 1)/(x + 1) )^(1/6)) + 2*arctan(((x - 1)/(x + 1))^(1/6)) + 1/2*log(((x - 1)/(x + 1))^( 1/3) + ((x - 1)/(x + 1))^(1/6) + 1) - 1/2*log(((x - 1)/(x + 1))^(1/3) - (( x - 1)/(x + 1))^(1/6) + 1) + log(((x - 1)/(x + 1))^(1/6) + 1) - log(abs((( x - 1)/(x + 1))^(1/6) - 1))
Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.42 \[ \int \frac {e^{\frac {1}{3} \coth ^{-1}(x)}}{x} \, dx=2\,\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\right )-\mathrm {atan}\left ({\left (\frac {x-1}{x+1}\right )}^{1/6}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}-\mathrm {i}\right )-\mathrm {atan}\left (\frac {{\left (\frac {x-1}{x+1}\right )}^{1/6}\,1486016741376{}\mathrm {i}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (\sqrt {3}+1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{-743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {1486016741376\,{\left (\frac {x-1}{x+1}\right )}^{1/6}}{743008370688+\sqrt {3}\,743008370688{}\mathrm {i}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right ) \]
2*atan(((x - 1)/(x + 1))^(1/6)) - atan(((x - 1)/(x + 1))^(1/6)*1i)*2i - at an((((x - 1)/(x + 1))^(1/6)*1486016741376i)/(3^(1/2)*743008370688i - 74300 8370688))*(3^(1/2) - 1i) - atan((((x - 1)/(x + 1))^(1/6)*1486016741376i)/( 3^(1/2)*743008370688i + 743008370688))*(3^(1/2) + 1i) - atan((148601674137 6*((x - 1)/(x + 1))^(1/6))/(3^(1/2)*743008370688i - 743008370688))*(3^(1/2 )*1i + 1) - atan((1486016741376*((x - 1)/(x + 1))^(1/6))/(3^(1/2)*74300837 0688i + 743008370688))*(3^(1/2)*1i - 1)