Optimal. Leaf size=37 \[ -2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac{2 i a}{x}-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.0271134, antiderivative size = 37, 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} \[ -2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac{2 i a}{x}-\frac{1}{2 x^2} \]
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^3} \, dx &=\int \frac{1-i a x}{x^3 (1+i a x)} \, dx\\ &=\int \left (\frac{1}{x^3}-\frac{2 i a}{x^2}-\frac{2 a^2}{x}+\frac{2 a^3}{-i+a x}\right ) \, dx\\ &=-\frac{1}{2 x^2}+\frac{2 i a}{x}-2 a^2 \log (x)+2 a^2 \log (i-a x)\\ \end{align*}
Mathematica [A] time = 0.0107412, size = 37, normalized size = 1. \[ -2 a^2 \log (x)+2 a^2 \log (-a x+i)+\frac{2 i a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 45, normalized size = 1.2 \begin{align*} 2\,i{a}^{2}\arctan \left ( ax \right ) +{a}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) -{\frac{1}{2\,{x}^{2}}}+{\frac{2\,ia}{x}}-2\,{a}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00913, size = 68, normalized size = 1.84 \begin{align*} 2 \, a^{2} \log \left (i \, a x + 1\right ) - 2 \, a^{2} \log \left (x\right ) - \frac{4 \, a^{2} x^{2} - 3 i \, a x + 1}{2 i \, a x^{3} + 2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6601, size = 97, normalized size = 2.62 \begin{align*} -\frac{4 \, a^{2} x^{2} \log \left (x\right ) - 4 \, a^{2} x^{2} \log \left (\frac{a x - i}{a}\right ) - 4 i \, a x + 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.474097, size = 27, normalized size = 0.73 \begin{align*} - 2 a^{2} \left (\log{\left (x \right )} - \log{\left (x - \frac{i}{a} \right )}\right ) + \frac{4 i a x - 1}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.09567, size = 85, normalized size = 2.3 \begin{align*} 2 \, a^{2} i^{2} \log \left (\frac{i^{2}}{a i x + 1} + 1\right ) + \frac{\frac{6 \, a^{2} i^{2}}{a i x + 1} + 5 \, a^{2}}{2 \,{\left (i - \frac{i}{a i x + 1}\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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