Optimal. Leaf size=20 \[ -x-\frac{2 i \log (-a x+i)}{a} \]
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Rubi [A] time = 0.0101968, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5061, 43} \[ -x-\frac{2 i \log (-a x+i)}{a} \]
Antiderivative was successfully verified.
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Rule 5061
Rule 43
Rubi steps
\begin{align*} \int e^{-2 i \tan ^{-1}(a x)} \, dx &=\int \frac{1-i a x}{1+i a x} \, dx\\ &=\int \left (-1-\frac{2 i}{-i+a x}\right ) \, dx\\ &=-x-\frac{2 i \log (i-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0088313, size = 30, normalized size = 1.5 \[ -\frac{i \log \left (a^2 x^2+1\right )}{a}+\frac{2 \tan ^{-1}(a x)}{a}-x \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.041, size = 30, normalized size = 1.5 \begin{align*} -x-{\frac{i\ln \left ({a}^{2}{x}^{2}+1 \right ) }{a}}+2\,{\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 = 1.01925, size = 22, normalized size = 1.1 \begin{align*} -x - \frac{2 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 = 1.52253, size = 45, normalized size = 2.25 \begin{align*} -\frac{a x + 2 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 = 0.330293, size = 14, normalized size = 0.7 \begin{align*} - x - \frac{2 i \log{\left (a x - i \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1094, size = 90, normalized size = 4.5 \begin{align*} a^{2}{\left (\frac{{\left (a i x + 1\right )} i}{a^{3}} + \frac{2 \, i \log \left (\frac{1}{\sqrt{a^{2} x^{2} + 1}{\left | a \right |}}\right )}{a^{3}} - \frac{i}{{\left (a i x + 1\right )} a^{3}}\right )} + \frac{i}{{\left (a i x + 1\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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