Optimal. Leaf size=40 \[ \frac {x^{1+m} F_1\left (1+m;\frac {n}{2},-\frac {n}{2};2+m;i a x,-i a x\right )}{1+m} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {5170, 138}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 5170
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*}
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Mathematica [F]
time = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int e^{i n \text {ArcTan}(a x)} x^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{i n \arctan \left (a x \right )} x^{m}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{m} e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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