Optimal. Leaf size=30 \[ \frac{2 \log (-a x+i)}{a^2}-\frac{2 i x}{a}-\frac{x^2}{2} \]
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Rubi [A] time = 0.0210554, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5062, 77} \[ \frac{2 \log (-a x+i)}{a^2}-\frac{2 i x}{a}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 77
Rubi steps
\begin{align*} \int e^{-2 i \tan ^{-1}(a x)} x \, dx &=\int \frac{x (1-i a x)}{1+i a x} \, dx\\ &=\int \left (-\frac{2 i}{a}-x+\frac{2}{a (-i+a x)}\right ) \, dx\\ &=-\frac{2 i x}{a}-\frac{x^2}{2}+\frac{2 \log (i-a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0101914, size = 30, normalized size = 1. \[ \frac{2 \log (-a x+i)}{a^2}-\frac{2 i x}{a}-\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 38, normalized size = 1.3 \begin{align*} -{\frac{{x}^{2}}{2}}-{\frac{2\,ix}{a}}+{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}}+{\frac{2\,i\arctan \left ( ax \right ) }{{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08784, size = 38, normalized size = 1.27 \begin{align*} \frac{i \,{\left (i \, a x^{2} - 4 \, x\right )}}{2 \, a} + \frac{2 \, \log \left (i \, a x + 1\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.51843, size = 69, normalized size = 2.3 \begin{align*} -\frac{a^{2} x^{2} + 4 i \, a x - 4 \, \log \left (\frac{a x - i}{a}\right )}{2 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.175426, size = 22, normalized size = 0.73 \begin{align*} - \frac{x^{2}}{2} - \frac{2 i x}{a} + \frac{2 \log{\left (a x - i \right )}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11053, size = 78, normalized size = 2.6 \begin{align*} \frac{i{\left (\frac{4 \, i \log \left (\frac{1}{\sqrt{a^{2} x^{2} + 1}{\left | a \right |}}\right )}{a} + \frac{{\left (a i x + 1\right )}^{2}{\left (i - \frac{6 \, i}{a i x + 1}\right )}}{a i^{2}}\right )}}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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