Optimal. Leaf size=13 \[ \log (x)-2 \log (a x+i) \]
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Rubi [A] time = 0.0194988, antiderivative size = 13, 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 = 0.0058213, size = 13, normalized size = 1. \[ \log (x)-2 \log (a x+i) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 23, normalized size = 1.8 \begin{align*} 2\,i\arctan \left ( ax \right ) -\ln \left ({a}^{2}{x}^{2}+1 \right ) +\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50648, size = 28, normalized size = 2.15 \begin{align*} 2 i \, \arctan \left (a x\right ) - \log \left (a^{2} x^{2} + 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 = 1.68503, size = 39, normalized size = 3. \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 = 0.407838, size = 10, normalized size = 0.77 \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 [A] time = 1.13246, size = 20, normalized size = 1.54 \begin{align*} 2 \, i^{2} \log \left (a x + i\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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