Integrand size = 14, antiderivative size = 50 \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1+i a x}-4 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x) \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5170, 91, 81, 66} \[ \int e^{-4 i \arctan (a x)} x^m \, dx=-4 x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,-i a x)+\frac {4 x^{m+1}}{1+i a x}+\frac {x^{m+1}}{m+1} \]
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Rule 66
Rule 81
Rule 91
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m (1-i a x)^2}{(1+i a x)^2} \, dx \\ & = \frac {4 x^{1+m}}{1+i a x}+\frac {\int \frac {x^m \left (-a^2 (3+4 m)+i a^3 x\right )}{1+i a x} \, dx}{a^2} \\ & = \frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1+i a x}-(4 (1+m)) \int \frac {x^m}{1+i a x} \, dx \\ & = \frac {x^{1+m}}{1+m}+\frac {4 x^{1+m}}{1+i a x}-4 x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.16 \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\frac {x^{1+m} (-5 i-4 i m+a x-4 (1+m) (-i+a x) \operatorname {Hypergeometric2F1}(1,1+m,2+m,-i a x))}{(1+m) (-i+a x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.57 (sec) , antiderivative size = 428, normalized size of antiderivative = 8.56
method | result | size |
meijerg | \(-\frac {i \left (i a \right )^{-m} \left (\frac {x^{m} \left (i a \right )^{m} \left (6 a^{4} x^{4} m +6 i a^{3} x^{3} m +a^{2} x^{2} m^{4}+24 i a^{3} x^{3}+11 a^{2} x^{2} m^{3}-2 i a x \,m^{4}+46 a^{2} m^{2} x^{2}-21 i a x \,m^{3}+90 a^{2} m \,x^{2}-79 i a x \,m^{2}+72 a^{2} x^{2}-126 i a m x -m^{4}-72 i a x -10 m^{3}-35 m^{2}-50 m -24\right )}{\left (1+m \right ) m \left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{3}+9 m^{2}+26 m +24\right ) \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{6 a}+\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}-4 a^{2} m \,x^{2}+2 i a x \,m^{2}-6 a^{2} x^{2}+7 i a m x +6 i a x +m^{2}+3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} m \left (m^{2}+3 m +2\right ) \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{3 a}-\frac {i \left (i a \right )^{-m} \left (-\frac {x^{m} \left (i a \right )^{m} \left (-a^{2} m^{2} x^{2}+2 a^{2} m \,x^{2}+2 i a x \,m^{2}-5 i a m x +m^{2}-3 m +2\right )}{\left (i a x +1\right )^{3}}+x^{m} \left (i a \right )^{m} \left (m^{2}-3 m +2\right ) m \operatorname {LerchPhi}\left (-i a x , 1, m\right )\right )}{6 a}\) | \(428\) |
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\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \]
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\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int \frac {x^{m} \left (a^{2} x^{2} + 1\right )^{2}}{\left (a x - i\right )^{4}}\, dx \]
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\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \]
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\[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (a^{2} x^{2} + 1\right )}^{2} x^{m}}{{\left (i \, a x + 1\right )}^{4}} \,d x } \]
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Timed out. \[ \int e^{-4 i \arctan (a x)} x^m \, dx=\int \frac {x^m\,{\left (a^2\,x^2+1\right )}^2}{{\left (1+a\,x\,1{}\mathrm {i}\right )}^4} \,d x \]
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