Optimal. Leaf size=14 \[ \log (x)-2 \log (-a x+i) \]
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Rubi [A] time = 0.0200594, antiderivative size = 14, normalized size of antiderivative = 1., 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 5062
Rule 72
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*}
Mathematica [A] time = 0.0069279, size = 14, normalized size = 1. \[ \log (x)-2 \log (-a x+i) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 23, normalized size = 1.6 \begin{align*} -\ln \left ({a}^{2}{x}^{2}+1 \right ) -2\,i\arctan \left ( ax \right ) +\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02635, size = 16, normalized size = 1.14 \begin{align*} -2 \, \log \left (i \, a x + 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62398, size = 39, normalized size = 2.79 \begin{align*} \log \left (x\right ) - 2 \, \log \left (\frac{a x - i}{a}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.418123, size = 10, normalized size = 0.71 \begin{align*} \log{\left (x \right )} - 2 \log{\left (x - \frac{i}{a} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09633, size = 63, normalized size = 4.5 \begin{align*} -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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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