Optimal. Leaf size=49 \[ \frac{2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{i a}{x^2}-\frac{1}{3 x^3} \]
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Rubi [A] time = 0.0317706, 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} \[ \frac{2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{i a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{-2 i \tan ^{-1}(a x)}}{x^4} \, dx &=\int \frac{1-i a x}{x^4 (1+i a x)} \, dx\\ &=\int \left (\frac{1}{x^4}-\frac{2 i a}{x^3}-\frac{2 a^2}{x^2}+\frac{2 i a^3}{x}-\frac{2 i a^4}{-i+a x}\right ) \, dx\\ &=-\frac{1}{3 x^3}+\frac{i a}{x^2}+\frac{2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (i-a x)\\ \end{align*}
Mathematica [A] time = 0.0148524, size = 49, normalized size = 1. \[ \frac{2 a^2}{x}+2 i a^3 \log (x)-2 i a^3 \log (-a x+i)+\frac{i a}{x^2}-\frac{1}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 55, normalized size = 1.1 \begin{align*} 2\,{a}^{3}\arctan \left ( ax \right ) -i{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) -{\frac{1}{3\,{x}^{3}}}+{\frac{ia}{{x}^{2}}}+2\,i{a}^{3}\ln \left ( x \right ) +2\,{\frac{{a}^{2}}{x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04681, size = 77, normalized size = 1.57 \begin{align*} -2 i \, a^{3} \log \left (i \, a x + 1\right ) + 2 i \, a^{3} \log \left (x\right ) + \frac{6 i \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 i \, a x - 1}{3 i \, a x^{4} + 3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64331, size = 117, normalized size = 2.39 \begin{align*} \frac{6 i \, a^{3} x^{3} \log \left (x\right ) - 6 i \, a^{3} x^{3} \log \left (\frac{a x - i}{a}\right ) + 6 \, a^{2} x^{2} + 3 i \, a x - 1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.520236, size = 39, normalized size = 0.8 \begin{align*} - 2 a^{3} \left (- i \log{\left (x \right )} + i \log{\left (x - \frac{i}{a} \right )}\right ) + \frac{6 a^{2} x^{2} + 3 i a x - 1}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10177, size = 104, normalized size = 2.12 \begin{align*} 2 \, a^{3} i \log \left (-i + \frac{i}{a i x + 1}\right ) + \frac{\frac{24 \, a^{3} i^{2}}{a i x + 1} + 10 \, a^{3} - \frac{15 \, a^{3} i^{2}}{{\left (a i x + 1\right )}^{2}}}{3 \,{\left (i - \frac{i}{a i x + 1}\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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