Optimal. Leaf size=125 \[ \frac {2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {1-i a x}{i a x+1}\right )}{n}-\frac {2^{\frac {n}{2}+1} (1-i a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{n} \]
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Rubi [A] time = 0.05, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5062, 105, 69, 131} \[ \frac {2 (1-i a x)^{-n/2} (1+i a x)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {1-i a x}{i a x+1}\right )}{n}-\frac {2^{\frac {n}{2}+1} (1-i a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 105
Rule 131
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{i n \tan ^{-1}(a x)}}{x} \, dx &=\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} \, _2F_1\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} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-i a x)\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 106, normalized size = 0.85 \[ \frac {2 (1-i a x)^{-n/2} \left ((1+i a x)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {a x+i}{i-a x}\right )-2^{n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-i a x)\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{i n \arctan \left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{i n \operatorname {atan}{\left (a x \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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