Optimal. Leaf size=47 \[ \frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (3 x+\sqrt{6}\right ) \]
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Rubi [A] time = 0.0227613, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {633, 31} \[ \frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (3 x+\sqrt{6}\right ) \]
Antiderivative was successfully verified.
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Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{-5+2 x}{-2+3 x^2} \, dx &=\frac{1}{4} \left (4-5 \sqrt{6}\right ) \int \frac{1}{-\sqrt{6}+3 x} \, dx+\frac{1}{4} \left (4+5 \sqrt{6}\right ) \int \frac{1}{\sqrt{6}+3 x} \, dx\\ &=\frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (\sqrt{6}+3 x\right )\\ \end{align*}
Mathematica [A] time = 0.027503, size = 47, normalized size = 1. \[ \frac{1}{12} \left (4-5 \sqrt{6}\right ) \log \left (\sqrt{6}-3 x\right )+\frac{1}{12} \left (4+5 \sqrt{6}\right ) \log \left (3 x+\sqrt{6}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 24, normalized size = 0.5 \begin{align*}{\frac{\ln \left ( 3\,{x}^{2}-2 \right ) }{3}}+{\frac{5\,\sqrt{6}}{6}{\it Artanh} \left ({\frac{x\sqrt{6}}{2}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40894, size = 49, normalized size = 1.04 \begin{align*} -\frac{5}{12} \, \sqrt{6} \log \left (\frac{3 \, x - \sqrt{6}}{3 \, x + \sqrt{6}}\right ) + \frac{1}{3} \, \log \left (3 \, x^{2} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.03127, size = 105, normalized size = 2.23 \begin{align*} \frac{5}{12} \, \sqrt{6} \log \left (\frac{3 \, x^{2} + 2 \, \sqrt{6} x + 2}{3 \, x^{2} - 2}\right ) + \frac{1}{3} \, \log \left (3 \, x^{2} - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.106597, size = 42, normalized size = 0.89 \begin{align*} \left (\frac{1}{3} - \frac{5 \sqrt{6}}{12}\right ) \log{\left (x - \frac{\sqrt{6}}{3} \right )} + \left (\frac{1}{3} + \frac{5 \sqrt{6}}{12}\right ) \log{\left (x + \frac{\sqrt{6}}{3} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07114, size = 50, normalized size = 1.06 \begin{align*} \frac{1}{12} \,{\left (5 \, \sqrt{6} + 4\right )} \log \left ({\left | x + \frac{1}{3} \, \sqrt{6} \right |}\right ) - \frac{1}{12} \,{\left (5 \, \sqrt{6} - 4\right )} \log \left ({\left | x - \frac{1}{3} \, \sqrt{6} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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