Integrand size = 15, antiderivative size = 40 \[ \int e^{i n \arctan (a x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {n}{2},-\frac {n}{2},2+m,i a x,-i a x\right )}{1+m} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {5170, 138} \[ \int e^{i n \arctan (a x)} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {n}{2},-\frac {n}{2},m+2,i a x,-i a x\right )}{m+1} \]
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Rule 138
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int x^m (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {n}{2},-\frac {n}{2},2+m,i a x,-i a x\right )}{1+m} \\ \end{align*}
\[ \int e^{i n \arctan (a x)} x^m \, dx=\int e^{i n \arctan (a x)} x^m \, dx \]
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\[\int {\mathrm e}^{i n \arctan \left (a x \right )} x^{m}d x\]
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\[ \int e^{i n \arctan (a x)} x^m \, dx=\int { x^{m} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{i n \arctan (a x)} x^m \, dx=\int x^{m} e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \]
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\[ \int e^{i n \arctan (a x)} x^m \, dx=\int { x^{m} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{i n \arctan (a x)} x^m \, dx=\int { x^{m} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{i n \arctan (a x)} x^m \, dx=\int x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \]
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