Optimal. Leaf size=34 \[ \frac {x^{m+1} F_1\left (-m-1;\frac {1}{3},-\frac {1}{3};-m;\frac {1}{x},-\frac {1}{x}\right )}{m+1} \]
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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 = {6173, 133} \[ \frac {x^{m+1} F_1\left (-m-1;\frac {1}{3},-\frac {1}{3};-m;\frac {1}{x},-\frac {1}{x}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 6173
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 ) \operatorname {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 = 0.85, size = 0, normalized size = 0.00 \[ \int e^{\frac {2}{3} \coth ^{-1}(x)} x^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x + 1\right )} x^{m} \left (\frac {x - 1}{x + 1}\right )^{\frac {2}{3}}}{x - 1}, x\right ) \]
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 \[ \int \frac {x^{m}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, 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 \[ \int \frac {x^{m}}{\left (\frac {x - 1}{x + 1}\right )^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {x^m}{{\left (\frac {x-1}{x+1}\right )}^{1/3}} \,d x \]
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 \[ \int \frac {x^{m}}{\sqrt [3]{\frac {x - 1}{x + 1}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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