Integrand size = 14, antiderivative size = 39 \[ \int e^{-2 i \arctan (a x)} x^m \, dx=-\frac {x^{1+m}}{1+m}+\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x)}{1+m} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5170, 81, 66} \[ \int e^{-2 i \arctan (a x)} x^m \, dx=-\frac {x^{m+1}}{m+1}+\frac {2 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-i a x)}{m+1} \]
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Rule 66
Rule 81
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m (1-i a x)}{1+i a x} \, dx \\ & = -\frac {x^{1+m}}{1+m}+2 \int \frac {x^m}{1+i a x} \, dx \\ & = -\frac {x^{1+m}}{1+m}+\frac {2 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x)}{1+m} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.74 \[ \int e^{-2 i \arctan (a x)} x^m \, dx=\frac {x^{1+m} (-1+2 \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x))}{1+m} \]
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Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.30 (sec) , antiderivative size = 158, normalized size of antiderivative = 4.05
method | result | size |
meijerg | \(\frac {i \left (i a \right )^{-m} \left (\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m \,x^{2}-i a m x -2 i a x -m^{2}-3 m -2\right )}{\left (1+m \right ) m \left (i a x +1\right )}+x^{m} \left (i a \right )^{m} \left (2+m \right ) \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{a}-\frac {i \left (i a \right )^{-m} \left (\frac {x^{m} \left (i a \right )^{m} \left (-1-m \right )}{\left (1+m \right ) \left (i a x +1\right )}+x^{m} \left (i a \right )^{m} m \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{a}\) | \(158\) |
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\[ \int e^{-2 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )} x^{m}}{{\left (i \, a x + 1\right )}^{2}} \,d x } \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (29) = 58\).
Time = 2.40 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.41 \[ \int e^{-2 i \arctan (a x)} x^m \, dx=- \frac {i a m x^{m + 2} \Phi \left (a x e^{\frac {3 i \pi }{2}}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} - \frac {2 i a x^{m + 2} \Phi \left (a x e^{\frac {3 i \pi }{2}}, 1, m + 2\right ) \Gamma \left (m + 2\right )}{\Gamma \left (m + 3\right )} + \frac {m x^{m + 1} \Phi \left (a x e^{\frac {3 i \pi }{2}}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} + \frac {x^{m + 1} \Phi \left (a x e^{\frac {3 i \pi }{2}}, 1, m + 1\right ) \Gamma \left (m + 1\right )}{\Gamma \left (m + 2\right )} \]
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\[ \int e^{-2 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )} x^{m}}{{\left (i \, a x + 1\right )}^{2}} \,d x } \]
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\[ \int e^{-2 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )} x^{m}}{{\left (i \, a x + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int e^{-2 i \arctan (a x)} x^m \, dx=\int \frac {x^m\,\left (a^2\,x^2+1\right )}{{\left (1+a\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
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