Optimal. Leaf size=45 \[ -\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2} \]
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Rubi [A] time = 0.0262917, antiderivative size = 45, 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} \[ -\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 77
Rubi steps
\begin{align*} \int e^{4 i \tan ^{-1}(a x)} x \, dx &=\int \frac{x (1+i a x)^2}{(1-i a x)^2} \, dx\\ &=\int \left (-\frac{4 i}{a}+x+\frac{4 i}{a (i+a x)^2}-\frac{8}{a (i+a x)}\right ) \, dx\\ &=-\frac{4 i x}{a}+\frac{x^2}{2}-\frac{4 i}{a^2 (i+a x)}-\frac{8 \log (i+a x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0240546, size = 45, normalized size = 1. \[ -\frac{4 i}{a^2 (a x+i)}-\frac{8 \log (a x+i)}{a^2}-\frac{4 i x}{a}+\frac{x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 53, normalized size = 1.2 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{4\,ix}{a}}-{\frac{4\,i}{{a}^{2} \left ( ax+i \right ) }}-4\,{\frac{\ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{2}}}+{\frac{8\,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 = 1.50673, size = 81, normalized size = 1.8 \begin{align*} -\frac{4 \,{\left (i \, a x + 1\right )}}{a^{4} x^{2} + a^{2}} + \frac{a x^{2} - 8 i \, x}{2 \, a} + \frac{8 i \, \arctan \left (a x\right )}{a^{2}} - \frac{4 \, \log \left (a^{2} x^{2} + 1\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57681, size = 127, normalized size = 2.82 \begin{align*} \frac{a^{3} x^{3} - 7 i \, a^{2} x^{2} + 8 \, a x -{\left (16 \, a x + 16 i\right )} \log \left (\frac{a x + i}{a}\right ) - 8 i}{2 \,{\left (a^{3} x + i \, a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.483821, size = 37, normalized size = 0.82 \begin{align*} - \frac{4 i a}{a^{4} x + i a^{3}} + \frac{x^{2}}{2} - \frac{4 i x}{a} - \frac{8 \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 [A] time = 1.11768, size = 61, normalized size = 1.36 \begin{align*} -\frac{8 \, \log \left (a x + i\right )}{a^{2}} - \frac{4 \, i}{{\left (a x + i\right )} a^{2}} + \frac{a^{4} x^{2} - 8 \, a^{3} i x}{2 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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