Integrand size = 16, antiderivative size = 36 \[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {1}{8},-\frac {1}{8},2+m,i a x,-i a x\right )}{1+m} \]
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Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5170, 138} \[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\frac {x^{m+1} \operatorname {AppellF1}\left (m+1,\frac {1}{8},-\frac {1}{8},m+2,i a x,-i a x\right )}{m+1} \]
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Rule 138
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m \sqrt [8]{1+i a x}}{\sqrt [8]{1-i a x}} \, dx \\ & = \frac {x^{1+m} \operatorname {AppellF1}\left (1+m,\frac {1}{8},-\frac {1}{8},2+m,i a x,-i a x\right )}{1+m} \\ \end{align*}
\[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx \]
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\[\int {\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}^{\frac {1}{4}} x^{m}d x\]
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\[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\int { x^{m} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \]
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\[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\int x^{m} \sqrt [4]{\frac {i \left (a x - i\right )}{\sqrt {a^{2} x^{2} + 1}}}\, dx \]
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\[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\int { x^{m} \left (\frac {i \, a x + 1}{\sqrt {a^{2} x^{2} + 1}}\right )^{\frac {1}{4}} \,d x } \]
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Exception generated. \[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int e^{\frac {1}{4} i \arctan (a x)} x^m \, dx=\int x^m\,{\left (\frac {1+a\,x\,1{}\mathrm {i}}{\sqrt {a^2\,x^2+1}}\right )}^{1/4} \,d x \]
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