Optimal. Leaf size=32 \[ 2 a^2 \log (x)-2 a^2 \log (a x+1)+\frac{2 a}{x}-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.0297747, antiderivative size = 32, 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} \[ 2 a^2 \log (x)-2 a^2 \log (a x+1)+\frac{2 a}{x}-\frac{1}{2 x^2} \]
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^3} \, dx &=\int \frac{1-a x}{x^3 (1+a x)} \, dx\\ &=\int \left (\frac{1}{x^3}-\frac{2 a}{x^2}+\frac{2 a^2}{x}-\frac{2 a^3}{1+a x}\right ) \, dx\\ &=-\frac{1}{2 x^2}+\frac{2 a}{x}+2 a^2 \log (x)-2 a^2 \log (1+a x)\\ \end{align*}
Mathematica [A] time = 0.0101006, size = 32, normalized size = 1. \[ 2 a^2 \log (x)-2 a^2 \log (a x+1)+\frac{2 a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 31, normalized size = 1. \begin{align*} -{\frac{1}{2\,{x}^{2}}}+2\,{\frac{a}{x}}+2\,{a}^{2}\ln \left ( x \right ) -2\,{a}^{2}\ln \left ( ax+1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.991817, size = 41, normalized size = 1.28 \begin{align*} -2 \, a^{2} \log \left (a x + 1\right ) + 2 \, a^{2} \log \left (x\right ) + \frac{4 \, a x - 1}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8204, size = 89, normalized size = 2.78 \begin{align*} -\frac{4 \, a^{2} x^{2} \log \left (a x + 1\right ) - 4 \, a^{2} x^{2} \log \left (x\right ) - 4 \, 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 = 0.669226, size = 26, normalized size = 0.81 \begin{align*} - 2 a^{2} \left (- \log{\left (x \right )} + \log{\left (x + \frac{1}{a} \right )}\right ) + \frac{4 a x - 1}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15121, size = 68, normalized size = 2.12 \begin{align*} 2 \, a^{2} \log \left ({\left | -\frac{1}{a x + 1} + 1 \right |}\right ) + \frac{5 \, a^{2} - \frac{6 \, a^{2}}{a x + 1}}{2 \,{\left (\frac{1}{a x + 1} - 1\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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