Optimal. Leaf size=36 \[ -2 a^2 \log (x)+2 a^2 \log (a x+i)-\frac {2 i a}{x}-\frac {1}{2 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 36, 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 = 36, 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 = 0.42, size = 39, normalized size = 1.08 \[ -\frac {4 \, a^{2} x^{2} \log \relax (x) - 4 \, a^{2} x^{2} \log \left (\frac {a x + i}{a}\right ) + 4 i \, a x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 32, normalized size = 0.89 \[ 2 \, a^{2} \log \left (a x + i\right ) - 2 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a i x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 45, normalized size = 1.25 \[ -\frac {1}{2 x^{2}}-\frac {2 i a}{x}-2 a^{2} \ln \relax (x )-2 i a^{2} \arctan \left (a x \right )+a^{2} \ln \left (a^{2} x^{2}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 42, normalized size = 1.17 \[ -2 i \, a^{2} \arctan \left (a x\right ) + a^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} \log \relax (x) - \frac {4 i \, a x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 27, normalized size = 0.75 \[ -a^2\,\mathrm {atan}\left (2\,a\,x+1{}\mathrm {i}\right )\,4{}\mathrm {i}-\frac {\frac {1}{2}+a\,x\,2{}\mathrm {i}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.19, size = 42, normalized size = 1.17 \[ - 2 a^{2} \left (\log {\left (4 a^{3} x \right )} - \log {\left (4 a^{3} x + 4 i a^{2} \right )}\right ) - \frac {4 i a x + 1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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