Integrand size = 15, antiderivative size = 125 \[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\frac {2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {1-i a x}{1+i a x}\right )}{n}-\frac {2^{1+\frac {n}{2}} (1-i a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{n} \]
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Time = 0.04 (sec) , antiderivative size = 125, 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, 132, 71, 133} \[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\frac {2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {1-i a x}{i a x+1}\right )}{n}-\frac {2^{\frac {n}{2}+1} (1-i a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{n} \]
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Rule 71
Rule 132
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} \, dx \\ & = -\left ((i a) \int (1-i a x)^{-1-\frac {n}{2}} (1+i a x)^{n/2} \, dx\right )+\int \frac {(1-i a x)^{-1-\frac {n}{2}} (1+i a x)^{n/2}}{x} \, dx \\ & = \frac {2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {1-i a x}{1+i a x}\right )}{n}-\frac {2^{1+\frac {n}{2}} (1-i a x)^{-n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-i a x)\right )}{n} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85 \[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\frac {2 (1-i a x)^{-n/2} \left ((1+i a x)^{n/2} \operatorname {Hypergeometric2F1}\left (1,-\frac {n}{2},1-\frac {n}{2},\frac {i+a x}{i-a x}\right )-2^{n/2} \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},-\frac {n}{2},1-\frac {n}{2},\frac {1}{2} (1-i a x)\right )\right )}{n} \]
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\[\int \frac {{\mathrm e}^{i n \arctan \left (a x \right )}}{x}d x\]
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\[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x} \,d x } \]
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\[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\int \frac {e^{i n \operatorname {atan}{\left (a x \right )}}}{x}\, dx \]
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\[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x} \,d x } \]
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\[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\int { \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {e^{i n \arctan (a x)}}{x} \, dx=\int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{x} \,d x \]
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