Optimal. Leaf size=33 \[ 2 a^2 \log (x)-2 a^2 \log (1-a x)-\frac{2 a}{x}-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.0322523, antiderivative size = 33, 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} \[ 2 a^2 \log (x)-2 a^2 \log (1-a x)-\frac{2 a}{x}-\frac{1}{2 x^2} \]
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 = 0.0109985, size = 33, normalized size = 1. \[ 2 a^2 \log (x)-2 a^2 \log (1-a x)-\frac{2 a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 31, normalized size = 0.9 \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 = 0.94655, size = 41, normalized size = 1.24 \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 = 1.67456, size = 89, normalized size = 2.7 \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 = 0.489527, size = 27, normalized size = 0.82 \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 = 1.15007, size = 43, normalized size = 1.3 \begin{align*} -2 \, a^{2} \log \left ({\left | a x - 1 \right |}\right ) + 2 \, a^{2} \log \left ({\left | x \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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