Optimal. Leaf size=20 \[ -2 a \log (x)+2 a \log (a x+1)-\frac{1}{x} \]
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Rubi [A] time = 0.0270994, antiderivative size = 20, 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 = {6126, 77} \[ -2 a \log (x)+2 a \log (a x+1)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{-2 \tanh ^{-1}(a x)}}{x^2} \, dx &=\int \frac{1-a x}{x^2 (1+a x)} \, dx\\ &=\int \left (\frac{1}{x^2}-\frac{2 a}{x}+\frac{2 a^2}{1+a x}\right ) \, dx\\ &=-\frac{1}{x}-2 a \log (x)+2 a \log (1+a x)\\ \end{align*}
Mathematica [A] time = 0.0086829, size = 20, normalized size = 1. \[ -2 a \log (x)+2 a \log (a x+1)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 21, normalized size = 1.1 \begin{align*} -{x}^{-1}-2\,a\ln \left ( x \right ) +2\,a\ln \left ( ax+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.946294, size = 27, normalized size = 1.35 \begin{align*} 2 \, a \log \left (a x + 1\right ) - 2 \, a \log \left (x\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96922, size = 58, normalized size = 2.9 \begin{align*} \frac{2 \, a x \log \left (a x + 1\right ) - 2 \, a x \log \left (x\right ) - 1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07266, size = 17, normalized size = 0.85 \begin{align*} - 2 a \left (\log{\left (x \right )} - \log{\left (x + \frac{1}{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 = 1.19509, size = 41, normalized size = 2.05 \begin{align*} -2 \, a \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right ) + \frac{a}{\frac{1}{a x + 1} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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