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 = 0.0457361, antiderivative size = 32, 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} \[ -2 a^2 \log (x)+2 a^2 \log (a x+1)-\frac{2 a}{x}+\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 77
Rubi steps
\begin{align*} \int \frac{e^{-2 \coth ^{-1}(a x)}}{x^3} \, dx &=-\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 = 0.0117426, 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 = 0.043, 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 = 1.02089, 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 = 1.53662, size = 88, normalized size = 2.75 \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 = 0.33974, 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 = 1.14377, size = 43, normalized size = 1.34 \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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