Optimal. Leaf size=34 \[ \frac {x^{1+m} F_1\left (-1-m;\frac {1}{3},-\frac {1}{3};-m;\frac {1}{x},-\frac {1}{x}\right )}{1+m} \]
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Rubi [A]
time = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 = {6308, 138}
\begin {gather*} \frac {x^{m+1} F_1\left (-m-1;\frac {1}{3},-\frac {1}{3};-m;\frac {1}{x},-\frac {1}{x}\right )}{m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 6308
Rubi steps
\begin {align*} \int e^{\frac {2}{3} \coth ^{-1}(x)} x^m \, dx &=-\left (\left (\left (\frac {1}{x}\right )^m x^m\right ) \text {Subst}\left (\int \frac {x^{-2-m} \sqrt [3]{1+x}}{\sqrt [3]{1-x}} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {x^{1+m} F_1\left (-1-m;\frac {1}{3},-\frac {1}{3};-m;\frac {1}{x},-\frac {1}{x}\right )}{1+m}\\ \end {align*}
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Mathematica [F]
time = 0.56, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\frac {2}{3} \coth ^{-1}(x)} x^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\left (\frac {-1+x}{1+x}\right )^{\frac {1}{3}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m}}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x^m}{{\left (\frac {x-1}{x+1}\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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