Optimal. Leaf size=27 \[ -2 i a \log (x)+2 i a \log (-a x+i)-\frac {1}{x} \]
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Rubi [A] time = 0.03, antiderivative size = 27, 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} \[ -2 i a \log (x)+2 i a \log (-a x+i)-\frac {1}{x} \]
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^2} \, dx &=\int \frac {1-i a x}{x^2 (1+i a x)} \, dx\\ &=\int \left (\frac {1}{x^2}-\frac {2 i a}{x}+\frac {2 i a^2}{-i+a x}\right ) \, dx\\ &=-\frac {1}{x}-2 i a \log (x)+2 i a \log (i-a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ -2 i a \log (x)+2 i a \log (-a x+i)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 26, normalized size = 0.96 \[ \frac {-2 i \, a x \log \relax (x) + 2 i \, a x \log \left (\frac {a x - i}{a}\right ) - 1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 37, normalized size = 1.37 \[ -2 \, a i \log \left (-i + \frac {i}{a i x + 1}\right ) + \frac {a}{i - \frac {i}{a i x + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 34, normalized size = 1.26 \[ -\frac {1}{x}-2 i a \ln \relax (x )-2 a \arctan \left (a x \right )+i a \ln \left (a^{2} x^{2}+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 34, normalized size = 1.26 \[ 2 i \, a \log \left (i \, a x + 1\right ) - 2 i \, a \log \relax (x) - \frac {a x - i}{a x^{2} - i \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 17, normalized size = 0.63 \[ -4\,a\,\mathrm {atan}\left (2\,a\,x-\mathrm {i}\right )-\frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 32, normalized size = 1.19 \[ - 2 a \left (i \log {\left (4 a^{2} x \right )} - i \log {\left (4 a^{2} x - 4 i a \right )}\right ) - \frac {1}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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