Integrand size = 12, antiderivative size = 34 \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,\frac {1}{6},-\frac {1}{6},-m,\frac {1}{x},-\frac {1}{x}\right )}{1+m} \]
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Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6308, 138} \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (-m-1,\frac {1}{6},-\frac {1}{6},-m,\frac {1}{x},-\frac {1}{x}\right )}{m+1} \]
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Rule 138
Rule 6308
Rubi steps \begin{align*} \text {integral}& = -\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (-1-m,\frac {1}{6},-\frac {1}{6},-m,\frac {1}{x},-\frac {1}{x}\right )}{1+m} \\ \end{align*}
\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx \]
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\[\int \frac {x^{m}}{\left (\frac {x -1}{1+x}\right )^{\frac {1}{6}}}d x\]
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\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\int { \frac {x^{m}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}} \,d x } \]
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\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\int \frac {x^{m}}{\sqrt [6]{\frac {x - 1}{x + 1}}}\, dx \]
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\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\int { \frac {x^{m}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}} \,d x } \]
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\[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\int { \frac {x^{m}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{6}}} \,d x } \]
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Timed out. \[ \int e^{\frac {1}{3} \coth ^{-1}(x)} x^m \, dx=\int \frac {x^m}{{\left (\frac {x-1}{x+1}\right )}^{1/6}} \,d x \]
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