Optimal. Leaf size=38 \[ \frac {x^{m+1} F_1\left (m+1;-\frac {i}{3},\frac {i}{3};m+2;i x,-i x\right )}{m+1} \]
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Rubi [A] time = 0.02, antiderivative size = 38, 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 = {5062, 133} \[ \frac {x^{m+1} F_1\left (m+1;-\frac {i}{3},\frac {i}{3};m+2;i x,-i x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 5062
Rubi steps
\begin {align*} \int e^{\frac {2}{3} \tan ^{-1}(x)} x^m \, dx &=\int (1-i x)^{\frac {i}{3}} (1+i x)^{-\frac {i}{3}} x^m \, dx\\ &=\frac {x^{1+m} F_1\left (1+m;-\frac {i}{3},\frac {i}{3};2+m;i x,-i x\right )}{1+m}\\ \end {align*}
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Mathematica [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int e^{\frac {2}{3} \tan ^{-1}(x)} x^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} e^{\left (\frac {2}{3} \, \arctan \relax (x)\right )}, x\right ) \]
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 \[ \int x^{m} e^{\left (\frac {2}{3} \, \arctan \relax (x)\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{\frac {2 \arctan \relax (x )}{3}} x^{m}\, dx \]
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 \[ \int x^{m} e^{\left (\frac {2}{3} \, \arctan \relax (x)\right )}\,{d x} \]
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 \[ \int x^m\,{\mathrm {e}}^{\frac {2\,\mathrm {atan}\relax (x)}{3}} \,d x \]
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 \[ \int x^{m} e^{\frac {2 \operatorname {atan}{\relax (x )}}{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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