Integrand size = 15, antiderivative size = 120 \[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 x^2}+\frac {2 a^2 n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-i a x}{1+i a x}\right )}{2-n} \]
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Time = 0.04 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5170, 98, 133} \[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\frac {2 a^2 n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n-2}{2}} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-i a x}{i a x+1}\right )}{2-n}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{2 x^2} \]
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Rule 98
Rule 133
Rule 5170
Rubi steps \begin{align*} \text {integral}& = \int \frac {(1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^3} \, dx \\ & = -\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 x^2}+\frac {1}{2} (i a n) \int \frac {(1-i a x)^{-n/2} (1+i a x)^{n/2}}{x^2} \, dx \\ & = -\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {2+n}{2}}}{2 x^2}+\frac {2 a^2 n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {1}{2} (-2+n)} \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {1-i a x}{1+i a x}\right )}{2-n} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.95 \[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\frac {(1-i a x)^{-n/2} (1+i a x)^{n/2} (i+a x) \left (-\left ((-2+n) (-i+a x)^2\right )+4 a^2 n x^2 \operatorname {Hypergeometric2F1}\left (2,1-\frac {n}{2},2-\frac {n}{2},\frac {i+a x}{i-a x}\right )\right )}{2 (-2+n) x^2 (-i+a x)} \]
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\[\int \frac {{\mathrm e}^{i n \arctan \left (a x \right )}}{x^{3}}d x\]
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\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int \frac {e^{i n \operatorname {atan}{\left (a x \right )}}}{x^{3}}\, dx \]
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\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {e^{i n \arctan (a x)}}{x^3} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{x^3} \,d x \]
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