Optimal. Leaf size=38 \[ \frac{x^{m+1} F_1\left (m+1;-\frac{i}{6},\frac{i}{6};m+2;i x,-i x\right )}{m+1} \]
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Rubi [A] time = 0.023389, antiderivative size = 38, 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 = {5062, 133} \[ \frac{x^{m+1} F_1\left (m+1;-\frac{i}{6},\frac{i}{6};m+2;i x,-i x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 133
Rubi steps
\begin{align*} \int e^{\frac{1}{3} \tan ^{-1}(x)} x^m \, dx &=\int (1-i x)^{\frac{i}{6}} (1+i x)^{-\frac{i}{6}} x^m \, dx\\ &=\frac{x^{1+m} F_1\left (1+m;-\frac{i}{6},\frac{i}{6};2+m;i x,-i x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.198047, size = 0, normalized size = 0. \[ \int e^{\frac{1}{3} \tan ^{-1}(x)} x^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\frac{\arctan \left ( x \right ) }{3}}}}{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} e^{\left (\frac{1}{3} \, \arctan \left (x\right )\right )}\,{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} e^{\left (\frac{1}{3} \, \arctan \left (x\right )\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} e^{\frac{\operatorname{atan}{\left (x \right )}}{3}}\, dx \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} e^{\left (\frac{1}{3} \, \arctan \left (x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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