Optimal. Leaf size=40 \[ \frac{x^{m+1} F_1\left (m+1;\frac{n}{2},-\frac{n}{2};m+2;i a x,-i a x\right )}{m+1} \]
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Rubi [A] time = 0.0280571, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {5062, 133} \[ \frac{x^{m+1} F_1\left (m+1;\frac{n}{2},-\frac{n}{2};m+2;i a x,-i a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 133
Rubi steps
\begin{align*} \int e^{i n \tan ^{-1}(a x)} x^m \, dx &=\int x^m (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx\\ &=\frac{x^{1+m} F_1\left (1+m;\frac{n}{2},-\frac{n}{2};2+m;i a x,-i a x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.220206, size = 0, normalized size = 0. \[ \int e^{i n \tan ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{in\arctan \left ( ax \right ) }}{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 (i \, n \arctan \left (a 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 (\frac{x^{m}}{\left (-\frac{a x + i}{a x - i}\right )^{\frac{1}{2} \, n}}, 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} e^{\left (i \, n \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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