Optimal. Leaf size=16 \[ \log (x)+\frac {4 i}{a x+i} \]
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Rubi [A] time = 0.02, antiderivative size = 16, 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, 88} \[ \log (x)+\frac {4 i}{a x+i} \]
Antiderivative was successfully verified.
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Rule 88
Rule 5062
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*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \[ \log (x)+\frac {4 i}{a x+i} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 18, normalized size = 1.12 \[ \frac {{\left (a x + i\right )} \log \relax (x) + 4 i}{a x + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 14, normalized size = 0.88 \[ \frac {4 \, i}{a x + i} + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 15, normalized size = 0.94 \[ \frac {4 i}{a x +i}+\ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 22, normalized size = 1.38 \[ -\frac {4 \, {\left (-i \, a x - 1\right )}}{a^{2} x^{2} + 1} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 14, normalized size = 0.88 \[ \ln \relax (x)+\frac {4{}\mathrm {i}}{a\,x+1{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 10, normalized size = 0.62 \[ \log {\relax (x )} + \frac {4 i}{a x + i} \]
Verification of antiderivative is not currently implemented for this CAS.
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