Integrand size = 14, antiderivative size = 114 \[ \int e^{6 i \arctan (a x)} x^m \, dx=-\frac {x^{1+m} (1+i a x)^2}{(1+m) (1-i a x)^2}+\frac {4 i x^{1+m} \left (i (1+m)^2+a \left (3+3 m+m^2\right ) x\right )}{(1+m) (1-i a x)^2}+\frac {2 \left (3+4 m+2 m^2\right ) x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,i a x)}{1+m} \]
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Time = 0.07 (sec) , antiderivative size = 114, 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, 102, 150, 66} \[ \int e^{6 i \arctan (a x)} x^m \, dx=\frac {2 \left (2 m^2+4 m+3\right ) x^{m+1} \operatorname {Hypergeometric2F1}(1,m+1,m+2,i a x)}{m+1}+\frac {4 i x^{m+1} \left (a \left (m^2+3 m+3\right ) x+i (m+1)^2\right )}{(m+1) (1-i a x)^2}-\frac {(1+i a x)^2 x^{m+1}}{(m+1) (1-i a x)^2} \]
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Rule 66
Rule 102
Rule 150
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {x^m (1+i a x)^3}{(1-i a x)^3} \, dx \\ & = -\frac {x^{1+m} (1+i a x)^2}{(1+m) (1-i a x)^2}+\frac {i \int \frac {x^m (1+i a x) \left (-2 i a (1+m)+2 a^2 (3+m) x\right )}{(1-i a x)^3} \, dx}{a (1+m)} \\ & = -\frac {x^{1+m} (1+i a x)^2}{(1+m) (1-i a x)^2}+\frac {4 i x^{1+m} \left (i (1+m)^2+a \left (3+3 m+m^2\right ) x\right )}{(1+m) (1-i a x)^2}+\left (2 \left (3+4 m+2 m^2\right )\right ) \int \frac {x^m}{1-i a x} \, dx \\ & = -\frac {x^{1+m} (1+i a x)^2}{(1+m) (1-i a x)^2}+\frac {4 i x^{1+m} \left (i (1+m)^2+a \left (3+3 m+m^2\right ) x\right )}{(1+m) (1-i a x)^2}+\frac {2 \left (3+4 m+2 m^2\right ) x^{1+m} \operatorname {Hypergeometric2F1}(1,1+m,2+m,i a x)}{1+m} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.82 \[ \int e^{6 i \arctan (a x)} x^m \, dx=\frac {x^{1+m} \left (5-10 i a x-a^2 x^2+4 m (2-3 i a x)+m^2 (4-4 i a x)+2 \left (3+4 m+2 m^2\right ) (i+a x)^2 \operatorname {Hypergeometric2F1}(1,1+m,2+m,i a x)\right )}{(1+m) (i+a x)^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.76 (sec) , antiderivative size = 748, normalized size of antiderivative = 6.56
method | result | size |
meijerg | \(\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (-a^{2} m^{2} x^{2}+2 a^{2} m \,x^{2}+3 a^{2} x^{2}-m^{2}+4 m +5\right )}{2 \left (1+m \right ) \left (a^{2} x^{2}+1\right )^{2}}+\frac {4 x^{1+m} \left (a^{2}\right )^{\frac {1}{2}+\frac {m}{2}} \left (\frac {1}{16} m^{3}-\frac {3}{16} m^{2}-\frac {1}{16} m +\frac {3}{16}\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{1+m}\right )}{4}+\frac {3 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (a^{2} m \,x^{2}+m -2\right )}{2 \left (a^{2} x^{2}+1\right )^{2}}-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (m -2\right ) m \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{4}\right )}{2 a}-\frac {15 \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 (a^{2} m \,x^{2}+a^{2} x^{2}+m -1\right )}{2 \left (a^{2} x^{2}+1\right )^{2} a^{2}}-\frac {x^{1+m} \left (a^{2}\right )^{\frac {3}{2}+\frac {m}{2}} \left (1+m \right ) \left (m -1\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{4 a^{2}}\right )}{4}-\frac {5 i \left (a^{2}\right )^{-\frac {m}{2}} \left (-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (a^{2} m \,x^{2}+4 a^{2} x^{2}+m +2\right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} m \left (2+m \right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{4}\right )}{a}+\frac {15 \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 (a^{2} m \,x^{2}+5 a^{2} x^{2}+m +3\right )}{2 a^{4} \left (a^{2} x^{2}+1\right )^{2}}+\frac {x^{1+m} \left (a^{2}\right )^{\frac {m}{2}+\frac {5}{2}} \left (m^{2}+4 m +3\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{4 a^{4}}\right )}{4}+\frac {3 i \left (a^{2}\right )^{-\frac {m}{2}} \left (\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (8 a^{4} x^{4}+a^{2} m^{2} x^{2}+8 a^{2} m \,x^{2}+16 a^{2} x^{2}+m^{2}+6 m +8\right )}{2 \left (a^{2} x^{2}+1\right )^{2} m}-\frac {x^{m} \left (a^{2}\right )^{\frac {m}{2}} \left (m^{2}+6 m +8\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {m}{2}\right )}{4}\right )}{2 a}-\frac {\left (a^{2}\right )^{-\frac {1}{2}-\frac {m}{2}} \left (\frac {x^{1+m} \left (a^{2}\right )^{\frac {7}{2}+\frac {m}{2}} \left (8 a^{4} x^{4}+a^{2} m^{2} x^{2}+10 a^{2} m \,x^{2}+25 a^{2} x^{2}+m^{2}+8 m +15\right )}{2 \left (a^{2} x^{2}+1\right )^{2} \left (1+m \right ) a^{6}}-\frac {x^{1+m} \left (a^{2}\right )^{\frac {7}{2}+\frac {m}{2}} \left (m^{2}+8 m +15\right ) \operatorname {LerchPhi}\left (-a^{2} x^{2}, 1, \frac {1}{2}+\frac {m}{2}\right )}{4 a^{6}}\right )}{4}\) | \(748\) |
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\[ \int e^{6 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{6} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{3}} \,d x } \]
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\[ \int e^{6 i \arctan (a x)} x^m \, dx=- \int \left (- \frac {x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx - \int \frac {15 a^{2} x^{2} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx - \int \left (- \frac {15 a^{4} x^{4} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx - \int \frac {a^{6} x^{6} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx - \int \left (- \frac {6 i a x x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx - \int \frac {20 i a^{3} x^{3} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx - \int \left (- \frac {6 i a^{5} x^{5} x^{m}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\right )\, dx \]
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\[ \int e^{6 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{6} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{3}} \,d x } \]
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\[ \int e^{6 i \arctan (a x)} x^m \, dx=\int { \frac {{\left (i \, a x + 1\right )}^{6} x^{m}}{{\left (a^{2} x^{2} + 1\right )}^{3}} \,d x } \]
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Timed out. \[ \int e^{6 i \arctan (a x)} x^m \, dx=\int \frac {x^m\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^6}{{\left (a^2\,x^2+1\right )}^3} \,d x \]
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