Optimal. Leaf size=36 \[ \frac{x^{m+1} F_1\left (m+1;\frac{1}{4},-\frac{1}{4};m+2;i a x,-i a x\right )}{m+1} \]
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Rubi [A] time = 0.0260548, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {5062, 133} \[ \frac{x^{m+1} F_1\left (m+1;\frac{1}{4},-\frac{1}{4};m+2;i a x,-i a x\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 133
Rubi steps
\begin{align*} \int e^{\frac{1}{2} i \tan ^{-1}(a x)} x^m \, dx &=\int \frac{x^m \sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx\\ &=\frac{x^{1+m} F_1\left (1+m;\frac{1}{4},-\frac{1}{4};2+m;i a x,-i a x\right )}{1+m}\\ \end{align*}
Mathematica [F] time = 0.17397, size = 0, normalized size = 0. \[ \int e^{\frac{1}{2} i \tan ^{-1}(a x)} x^m \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.126, size = 0, normalized size = 0. \begin{align*} \int \sqrt{{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}}{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} \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}\,{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} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}, 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} \sqrt{\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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