Optimal. Leaf size=27 \[ -2 i a \log (x)+2 i a \log (-a x+i)-\frac{1}{x} \]
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Rubi [A] time = 0.0255769, antiderivative size = 27, 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} \[ -2 i a \log (x)+2 i a \log (-a x+i)-\frac{1}{x} \]
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^2} \, dx &=\int \frac{1-i a x}{x^2 (1+i a x)} \, dx\\ &=\int \left (\frac{1}{x^2}-\frac{2 i a}{x}+\frac{2 i a^2}{-i+a x}\right ) \, dx\\ &=-\frac{1}{x}-2 i a \log (x)+2 i a \log (i-a x)\\ \end{align*}
Mathematica [A] time = 0.0092211, size = 27, normalized size = 1. \[ -2 i a \log (x)+2 i a \log (-a x+i)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 34, normalized size = 1.3 \begin{align*} -2\,a\arctan \left ( ax \right ) +ia\ln \left ({a}^{2}{x}^{2}+1 \right ) -{x}^{-1}-2\,ia\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05493, size = 46, normalized size = 1.7 \begin{align*} 2 i \, a \log \left (i \, a x + 1\right ) - 2 i \, a \log \left (x\right ) - \frac{a x - i}{a x^{2} - i \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74535, size = 70, normalized size = 2.59 \begin{align*} \frac{-2 i \, a x \log \left (x\right ) + 2 i \, a x \log \left (\frac{a x - i}{a}\right ) - 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.420873, size = 20, normalized size = 0.74 \begin{align*} - 2 a \left (i \log{\left (x \right )} - i \log{\left (x - \frac{i}{a} \right )}\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09886, size = 50, normalized size = 1.85 \begin{align*} -2 \, a i \log \left (-i + \frac{i}{a i x + 1}\right ) + \frac{a}{i - \frac{i}{a i x + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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