Integrand size = 12, antiderivative size = 155 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )-\frac {3}{2} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {3}{2} \log \left (1+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{2} \log \left (1+\frac {1}{x}\right )-\frac {\log (x)}{2} \]
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Time = 0.04 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6306, 132, 62, 93} \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}\right )-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{\frac {x-1}{x}}}{\sqrt {3} \sqrt [3]{\frac {1}{x}+1}}+\frac {1}{\sqrt {3}}\right )-\frac {3}{2} \log \left (\sqrt [3]{\frac {1}{x}+1}-\sqrt [3]{\frac {x-1}{x}}\right )-\frac {3}{2} \log \left (\frac {\sqrt [3]{\frac {x-1}{x}}}{\sqrt [3]{\frac {1}{x}+1}}+1\right )-\frac {1}{2} \log \left (\frac {1}{x}+1\right )-\frac {\log (x)}{2} \]
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Rule 62
Rule 93
Rule 132
Rule 6306
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {\sqrt [3]{1+x}}{\sqrt [3]{1-x} x} \, dx,x,\frac {1}{x}\right ) \\ & = -\text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right )-\text {Subst}\left (\int \frac {1}{\sqrt [3]{1-x} x (1+x)^{2/3}} \, dx,x,\frac {1}{x}\right ) \\ & = -\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )-\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{\frac {-1+x}{x}}}{\sqrt {3} \sqrt [3]{1+\frac {1}{x}}}\right )-\frac {3}{2} \log \left (\sqrt [3]{1+\frac {1}{x}}-\sqrt [3]{\frac {-1+x}{x}}\right )-\frac {3}{2} \log \left (1+\frac {\sqrt [3]{\frac {-1+x}{x}}}{\sqrt [3]{1+\frac {1}{x}}}\right )-\frac {1}{2} \log \left (1+\frac {1}{x}\right )-\frac {\log (x)}{2} \\ \end{align*}
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 = 0.17 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\frac {3}{2} e^{\frac {8}{3} \coth ^{-1}(x)} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},1,\frac {5}{3},e^{4 \coth ^{-1}(x)}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.66 (sec) , antiderivative size = 1026, normalized size of antiderivative = 6.62
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Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - 1\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x + 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + {\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + x + 1}{x + 1}\right ) \]
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\[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\int \frac {1}{x \sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=-\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right )}\right ) + \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right )}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} + 1\right ) - \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}} - 1\right ) \]
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Time = 0.29 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.51 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=\sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + 1\right )}\right ) + \frac {1}{2} \, \log \left (\left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} + \frac {{\left (x - 1\right )} \left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}}}{x + 1} + 1\right ) - \log \left ({\left | \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}} - 1 \right |}\right ) \]
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Time = 4.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\frac {2}{3} \coth ^{-1}(x)}}{x} \, dx=-\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}-1296\right )-\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}+648-\sqrt {3}\,648{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )+\ln \left (1296\,{\left (\frac {x-1}{x+1}\right )}^{2/3}+648+\sqrt {3}\,648{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ) \]
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