Optimal. Leaf size=16 \[ \log (x)+\frac{4 i}{a x+i} \]
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Rubi [A] time = 0.0215648, antiderivative size = 16, 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, 88} \[ \log (x)+\frac{4 i}{a x+i} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 i \tan ^{-1}(a x)}}{x} \, dx &=\int \frac{(1+i a x)^2}{x (1-i a x)^2} \, dx\\ &=\int \left (\frac{1}{x}-\frac{4 i a}{(i+a x)^2}\right ) \, dx\\ &=\frac{4 i}{i+a x}+\log (x)\\ \end{align*}
Mathematica [A] time = 0.0096094, size = 16, normalized size = 1. \[ \log (x)+\frac{4 i}{a x+i} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 15, normalized size = 0.9 \begin{align*}{\frac{4\,i}{ax+i}}+\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52583, size = 30, normalized size = 1.88 \begin{align*} -\frac{4 \,{\left (-i \, a x - 1\right )}}{a^{2} x^{2} + 1} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60104, size = 49, normalized size = 3.06 \begin{align*} \frac{{\left (a x + i\right )} \log \left (x\right ) + 4 i}{a x + i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.475003, size = 15, normalized size = 0.94 \begin{align*} \frac{4 i a}{a^{2} x + i a} + \log{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10788, size = 19, normalized size = 1.19 \begin{align*} \frac{4 \, i}{a x + i} + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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