Optimal. Leaf size=49 \[ \frac{x^2}{a^2}+\frac{2 i x}{a^3}-\frac{2 \log (-a x+i)}{a^4}-\frac{2 i x^3}{3 a}-\frac{x^4}{4} \]
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Rubi [A] time = 0.0352147, antiderivative size = 49, 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} \[ \frac{x^2}{a^2}+\frac{2 i x}{a^3}-\frac{2 \log (-a x+i)}{a^4}-\frac{2 i x^3}{3 a}-\frac{x^4}{4} \]
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^3 \, dx &=\int \frac{x^3 (1-i a x)}{1+i a x} \, dx\\ &=\int \left (\frac{2 i}{a^3}+\frac{2 x}{a^2}-\frac{2 i x^2}{a}-x^3-\frac{2}{a^3 (-i+a x)}\right ) \, dx\\ &=\frac{2 i x}{a^3}+\frac{x^2}{a^2}-\frac{2 i x^3}{3 a}-\frac{x^4}{4}-\frac{2 \log (i-a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0182823, size = 49, normalized size = 1. \[ \frac{x^2}{a^2}+\frac{2 i x}{a^3}-\frac{2 \log (-a x+i)}{a^4}-\frac{2 i x^3}{3 a}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 55, normalized size = 1.1 \begin{align*} -{\frac{{x}^{4}}{4}}-{\frac{{\frac{2\,i}{3}}{x}^{3}}{a}}+{\frac{{x}^{2}}{{a}^{2}}}+{\frac{2\,ix}{{a}^{3}}}-{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{4}}}-{\frac{2\,i\arctan \left ( ax \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00334, size = 59, normalized size = 1.2 \begin{align*} -\frac{i \,{\left (-3 i \, a^{3} x^{4} + 8 \, a^{2} x^{3} + 12 i \, a x^{2} - 24 \, x\right )}}{12 \, a^{3}} - \frac{2 \, \log \left (i \, a x + 1\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63158, size = 112, normalized size = 2.29 \begin{align*} -\frac{3 \, a^{4} x^{4} + 8 i \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 24 i \, a x + 24 \, \log \left (\frac{a x - i}{a}\right )}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.360324, size = 41, normalized size = 0.84 \begin{align*} - \frac{x^{4}}{4} - \frac{2 i x^{3}}{3 a} + \frac{x^{2}}{a^{2}} + \frac{2 i x}{a^{3}} - \frac{2 \log{\left (a x - i \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.13513, size = 108, normalized size = 2.2 \begin{align*} -\frac{{\left (a i x + 1\right )}^{4}{\left (\frac{20 \, i^{2}}{a i x + 1} - \frac{84 \, i^{4}}{{\left (a i x + 1\right )}^{3}} - \frac{54 \, i^{2}}{{\left (a i x + 1\right )}^{2}} + 3\right )}}{12 \, a^{4} i^{4}} + \frac{2 \, \log \left (\frac{1}{\sqrt{a^{2} x^{2} + 1}{\left | a \right |}}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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