Optimal. Leaf size=21 \[ 2 a \log (x)-2 a \log (1-a x)-\frac{1}{x} \]
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Rubi [A] time = 0.0283413, antiderivative size = 21, 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} \[ 2 a \log (x)-2 a \log (1-a x)-\frac{1}{x} \]
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 = 0.0098314, size = 21, normalized size = 1. \[ 2 a \log (x)-2 a \log (1-a x)-\frac{1}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 21, normalized size = 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 = 0.950452, size = 27, normalized size = 1.29 \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 = 1.7354, size = 59, normalized size = 2.81 \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 = 0.553719, size = 17, normalized size = 0.81 \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 = 1.16747, size = 30, normalized size = 1.43 \begin{align*} -2 \, a \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a \log \left ({\left | x \right |}\right ) - \frac{1}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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