Optimal. Leaf size=14 \[ \log (x)-2 \log (-a x+i) \]
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Rubi [A] time = 0.02, antiderivative size = 14, 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, 72} \[ \log (x)-2 \log (-a x+i) \]
Antiderivative was successfully verified.
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Rule 72
Rule 5062
Rubi steps
\begin {align*} \int \frac {e^{-2 i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac {1-i a x}{x (1+i a x)} \, dx\\ &=\int \left (\frac {1}{x}-\frac {2 a}{-i+a x}\right ) \, dx\\ &=\log (x)-2 \log (i-a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.00 \[ \log (x)-2 \log (-a x+i) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 15, normalized size = 1.07 \[ \log \relax (x) - 2 \, \log \left (\frac {a x - i}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.13, size = 47, normalized size = 3.36 \[ -a i {\left (\frac {i \log \left (-i + \frac {i}{a i x + 1}\right )}{a} + \frac {i \log \left (\frac {1}{\sqrt {a^{2} x^{2} + 1} {\left | a \right |}}\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 23, normalized size = 1.64 \[ \ln \relax (x )-\ln \left (a^{2} x^{2}+1\right )-2 i \arctan \left (a x \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 12, normalized size = 0.86 \[ -2 \, \log \left (i \, a x + 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 14, normalized size = 1.00 \[ \ln \relax (x)-2\,\ln \left (x-\frac {1{}\mathrm {i}}{a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 17, normalized size = 1.21 \[ \log {\left (3 a x \right )} - 2 \log {\left (3 a x - 3 i \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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