Integrand size = 15, antiderivative size = 171 \[ \int e^{i n \arctan (a x)} x^3 \, dx=\frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}-\frac {2^{-2+\frac {n}{2}} n \left (8+n^2\right ) (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)} \]
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Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5170, 102, 152, 71} \[ \int e^{i n \arctan (a x)} x^3 \, dx=-\frac {2^{\frac {n}{2}-2} n \left (n^2+8\right ) (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)}-\frac {(1+i a x)^{\frac {n+2}{2}} \left (2 i a n x+n^2+6\right ) (1-i a x)^{1-\frac {n}{2}}}{24 a^4}+\frac {x^2 (1+i a x)^{\frac {n+2}{2}} (1-i a x)^{1-\frac {n}{2}}}{4 a^2} \]
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Rule 71
Rule 102
Rule 152
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int x^3 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx \\ & = \frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}+\frac {\int x (1-i a x)^{-n/2} (1+i a x)^{n/2} (-2-i a n x) \, dx}{4 a^2} \\ & = \frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}+\frac {\left (i n \left (8+n^2\right )\right ) \int (1-i a x)^{-n/2} (1+i a x)^{n/2} \, dx}{24 a^3} \\ & = \frac {x^2 (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{4 a^2}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}} \left (6+n^2+2 i a n x\right )}{24 a^4}-\frac {2^{-2+\frac {n}{2}} n \left (8+n^2\right ) (1-i a x)^{1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{3 a^4 (2-n)} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.23 \[ \int e^{i n \arctan (a x)} x^3 \, dx=\frac {(1-i a x)^{-n/2} (i+a x) \left (-i 2^{3+\frac {n}{2}} n \operatorname {Hypergeometric2F1}\left (-2-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )+i 2^{3+\frac {n}{2}} (-1+n) \operatorname {Hypergeometric2F1}\left (-1-\frac {n}{2},1-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )+(-2+n) \left (a^2 x^2 (1+i a x)^{n/2} (-i+a x)-i 2^{1+\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},-\frac {n}{2},2-\frac {n}{2},\frac {1}{2} (1-i a x)\right )\right )\right )}{4 a^4 (-2+n)} \]
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\[\int {\mathrm e}^{i n \arctan \left (a x \right )} x^{3}d x\]
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\[ \int e^{i n \arctan (a x)} x^3 \, dx=\int { x^{3} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{i n \arctan (a x)} x^3 \, dx=\int x^{3} e^{i n \operatorname {atan}{\left (a x \right )}}\, dx \]
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\[ \int e^{i n \arctan (a x)} x^3 \, dx=\int { x^{3} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]
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\[ \int e^{i n \arctan (a x)} x^3 \, dx=\int { x^{3} e^{\left (i \, n \arctan \left (a x\right )\right )} \,d x } \]
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Timed out. \[ \int e^{i n \arctan (a x)} x^3 \, dx=\int x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}} \,d x \]
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