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 = 0.45 (sec) , antiderivative size = 417, normalized size of antiderivative = 8.34
method | result | size |
meijerg | \(\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {2 x^{1+m} \left (a^{2}\right )^{\frac {1}{2}+\frac {m}{2}}}{2 a^{2} x^{2}+2}+\frac {2 x^{1+m} \left (a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (-\frac {m^{2}}{4}+\frac {1}{4}\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}\right )}{2}+\frac {2 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (-2-m \right )}{\left (2+m \right ) \left (a^{2} x^{2}+1\right )}+\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} m \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{2}\right )}{a}-3 \left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (-3-m \right )}{\left (3+m \right ) a^{2} \left (a^{2} x^{2}+1\right )}+\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (1+m \right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{2 a^{2}}\right )-\frac {2 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (2 a^{2} x^{2}+m +2\right )}{\left (a^{2} x^{2}+1\right ) m}-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (2+m \right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{2}\right )}{a}+\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {m}{2}+\frac {5}{2}} \left (2 a^{2} x^{2}+m +3\right )}{\left (a^{2} x^{2}+1\right ) a^{4} \left (1+m \right )}-\frac {x^{1+m} \left (a^{2}\right )^{\frac {m}{2}+\frac {5}{2}} \left (3+m \right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{2 a^{4}}\right )}{2}\) | \(417\) |
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\[ \int e^{4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{4} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{2}} \,d x } \]
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\[ \int e^{4 i \arctan (a x)} x^m \, dx=\int \frac {x^{m} \left (a x - i\right )^{4}}{\left (a^{2} x^{2} + 1\right )^{2}}\, dx \]
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\[ \int e^{4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{4} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{2}} \,d x } \]
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\[ \int e^{4 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{4} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{2}} \,d x } \]
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Timed out. \[ \int e^{4 i \arctan (a x)} x^m \, dx=\int \frac {x^m\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^4}{{\left (a^2\,x^2+1\right )}^2} \,d x \]
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