Optimal. Leaf size=37 \[ -2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac {2 i a}{x}-\frac {1}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5062, 77} \[ -2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac {2 i a}{x}-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a x)}}{x^3} \, dx &=\int \frac {1-i a x}{x^3 (1+i a x)} \, dx\\ &=\int \left (\frac {1}{x^3}-\frac {2 i a}{x^2}-\frac {2 a^2}{x}+\frac {2 a^3}{-i+a x}\right ) \, dx\\ &=-\frac {1}{2 x^2}+\frac {2 i a}{x}-2 a^2 \log (x)+2 a^2 \log (i-a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ -2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac {2 i a}{x}-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 39, normalized size = 1.05 \[ -\frac {4 \, a^{2} x^{2} \log \relax (x) - 4 \, a^{2} x^{2} \log \left (\frac {a x - i}{a}\right ) - 4 i \, a x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 63, normalized size = 1.70 \[ 2 \, a^{2} i^{2} \log \left (\frac {i^{2}}{a i x + 1} + 1\right ) + \frac {\frac {6 \, a^{2} i^{2}}{a i x + 1} + 5 \, a^{2}}{2 \, {\left (i - \frac {i}{a i x + 1}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 1.22 \[ -\frac {1}{2 x^{2}}+\frac {2 i a}{x}-2 a^{2} \ln \relax (x )+2 i a^{2} \arctan \left (a x \right )+a^{2} \ln \left (a^{2} x^{2}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 50, normalized size = 1.35 \[ 2 \, a^{2} \log \left (i \, a x + 1\right ) - 2 \, a^{2} \log \relax (x) - \frac {4 \, a^{2} x^{2} - 3 i \, a x + 1}{2 i \, a x^{3} + 2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 26, normalized size = 0.70 \[ a^2\,\mathrm {atan}\left (2\,a\,x-\mathrm {i}\right )\,4{}\mathrm {i}+\frac {-\frac {1}{2}+a\,x\,2{}\mathrm {i}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 42, normalized size = 1.14 \[ - 2 a^{2} \left (\log {\left (4 a^{3} x \right )} - \log {\left (4 a^{3} x - 4 i a^{2} \right )}\right ) - \frac {- 4 i a x + 1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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