Optimal. Leaf size=16 \[ -\frac{i \log (-a x+i)}{a} \]
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Rubi [A] time = 0.0315238, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5073, 31} \[ -\frac{i \log (-a x+i)}{a} \]
Antiderivative was successfully verified.
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Rule 5073
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{-i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx &=\int \frac{1}{1+i a x} \, dx\\ &=-\frac{i \log (i-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0049329, size = 16, normalized size = 1. \[ -\frac{i \log (-a x+i)}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 26, normalized size = 1.6 \begin{align*}{\frac{-{\frac{i}{2}}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{a}}+{\frac{\arctan \left ( ax \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973474, size = 16, normalized size = 1. \begin{align*} -\frac{i \, \log \left (i \, a x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70131, size = 31, normalized size = 1.94 \begin{align*} -\frac{i \, \log \left (\frac{a x - i}{a}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070349, size = 12, normalized size = 0.75 \begin{align*} - \frac{i \log{\left (i a x + 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11537, size = 18, normalized size = 1.12 \begin{align*} -\frac{i \log \left (a i x + 1\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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