Optimal. Leaf size=40 \[ \frac{2 x}{a^2}+\frac{2 i \log (-a x+i)}{a^3}-\frac{i x^2}{a}-\frac{x^3}{3} \]
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Rubi [A] time = 0.0297327, antiderivative size = 40, 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, 77} \[ \frac{2 x}{a^2}+\frac{2 i \log (-a x+i)}{a^3}-\frac{i x^2}{a}-\frac{x^3}{3} \]
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^2 \, dx &=\int \frac{x^2 (1-i a x)}{1+i a x} \, dx\\ &=\int \left (\frac{2}{a^2}-\frac{2 i x}{a}-x^2+\frac{2 i}{a^2 (-i+a x)}\right ) \, dx\\ &=\frac{2 x}{a^2}-\frac{i x^2}{a}-\frac{x^3}{3}+\frac{2 i \log (i-a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0125777, size = 40, normalized size = 1. \[ \frac{2 x}{a^2}+\frac{2 i \log (-a x+i)}{a^3}-\frac{i x^2}{a}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 47, normalized size = 1.2 \begin{align*} -{\frac{{x}^{3}}{3}}-{\frac{i{x}^{2}}{a}}+2\,{\frac{x}{{a}^{2}}}+{\frac{i\ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}}}-2\,{\frac{\arctan \left ( ax \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03743, size = 47, normalized size = 1.18 \begin{align*} -\frac{a^{2} x^{3} + 3 i \, a x^{2} - 6 \, x}{3 \, a^{2}} + \frac{2 i \, \log \left (i \, a x + 1\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53685, size = 88, normalized size = 2.2 \begin{align*} -\frac{a^{3} x^{3} + 3 i \, a^{2} x^{2} - 6 \, a x - 6 i \, \log \left (\frac{a x - i}{a}\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.360423, size = 31, normalized size = 0.78 \begin{align*} - \frac{x^{3}}{3} - \frac{i x^{2}}{a} + \frac{2 x}{a^{2}} + \frac{2 i \log{\left (a x - i \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09594, size = 92, normalized size = 2.3 \begin{align*} -\frac{2 \, i \log \left (\frac{1}{\sqrt{a^{2} x^{2} + 1}{\left | a \right |}}\right )}{a^{3}} - \frac{{\left (a i x + 1\right )}^{3}{\left (\frac{6 \, i^{2}}{a i x + 1} - \frac{15 \, i^{2}}{{\left (a i x + 1\right )}^{2}} + 1\right )}}{3 \, a^{3} i^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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