Optimal. Leaf size=18 \[ 2 a \log (x)-2 a \log (a x+1)+\frac{1}{x} \]
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Rubi [A] time = 0.0432672, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 6126, 77} \[ 2 a \log (x)-2 a \log (a x+1)+\frac{1}{x} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{x^2} \, dx &=-\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 = 0.0092316, size = 18, 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 = 0.045, size = 19, 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 = 1.00079, size = 24, normalized size = 1.33 \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 = 1.52237, size = 59, normalized size = 3.28 \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 = 0.318615, size = 15, normalized size = 0.83 \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 = 1.14546, size = 27, normalized size = 1.5 \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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