Optimal. Leaf size=120 \[ \frac {2 a^2 n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n-2}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-i a x}{i a x+1}\right )}{2-n}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{2 x^2} \]
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Rubi [A] time = 0.05, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5062, 96, 131} \[ \frac {2 a^2 n (1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n-2}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-i a x}{i a x+1}\right )}{2-n}-\frac {(1-i a x)^{1-\frac {n}{2}} (1+i a x)^{\frac {n+2}{2}}}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 96
Rule 131
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{i n \tan ^{-1}(a x)}}{x^3} \, dx &=\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)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-i a x}{1+i a x}\right )}{2-n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 114, normalized size = 0.95 \[ \frac {(a x+i) (1-i a x)^{-n/2} (1+i a x)^{n/2} \left (4 a^2 n x^2 \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {a x+i}{i-a x}\right )-(n-2) (a x-i)^2\right )}{2 (n-2) x^2 (a x-i)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x^{3} \left (-\frac {a x + i}{a x - i}\right )^{\frac {1}{2} \, n}}, x\right ) \]
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 = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{i n \arctan \left (a x \right )}}{x^{3}}\, dx \]
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 \[ \int \frac {e^{\left (i \, n \arctan \left (a x\right )\right )}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )\,1{}\mathrm {i}}}{x^3} \,d x \]
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 \[ \int \frac {e^{i n \operatorname {atan}{\left (a x \right )}}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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