Optimal. Leaf size=28 \[ \frac{x^{m+1} F_1\left (m+1;\frac{1}{6},-\frac{1}{6};m+2;x,-x\right )}{m+1} \]
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Rubi [A] time = 0.0238578, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 133} \[ \frac{x^{m+1} F_1\left (m+1;\frac{1}{6},-\frac{1}{6};m+2;x,-x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 133
Rubi steps
\begin{align*} \int e^{\frac{1}{3} \tanh ^{-1}(x)} x^m \, dx &=\int \frac{x^m \sqrt [6]{1+x}}{\sqrt [6]{1-x}} \, dx\\ &=\frac{x^{1+m} F_1\left (1+m;\frac{1}{6},-\frac{1}{6};2+m;x,-x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.231273, size = 0, normalized size = 0. \[ \int e^{\frac{1}{3} \tanh ^{-1}(x)} x^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{{(1+x){\frac{1}{\sqrt{-{x}^{2}+1}}}}}{x}^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \left (-\frac{\sqrt{-x^{2} + 1}}{x - 1}\right )^{\frac{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \left (\frac{x + 1}{\sqrt{-x^{2} + 1}}\right )^{\frac{1}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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