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 = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, 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 31
Rule 633
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*}
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Mathematica [A] time = 0.03, size = 47, normalized size = 1.00 \[ \frac {1}{12} \left (4-5 \sqrt {6}\right ) \log \left (\sqrt {6}-3 x\right )+\frac {1}{12} \left (4+5 \sqrt {6}\right ) \log \left (3 x+\sqrt {6}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 47, normalized size = 1.00 \[ \frac {1}{12} \left (4-5 \sqrt {6}\right ) \log \left (\sqrt {6}-3 x\right )+\frac {1}{12} \left (4+5 \sqrt {6}\right ) \log \left (3 x+\sqrt {6}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 40, normalized size = 0.85 \[ \frac {5}{12} \, \sqrt {6} \log \left (\frac {3 \, x^{2} + 2 \, \sqrt {6} x + 2}{3 \, x^{2} - 2}\right ) + \frac {1}{3} \, \log \left (3 \, x^{2} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 37, normalized size = 0.79 \[ \frac {1}{12} \, {\left (5 \, \sqrt {6} + 4\right )} \log \left ({\left | x + \frac {1}{3} \, \sqrt {6} \right |}\right ) - \frac {1}{12} \, {\left (5 \, \sqrt {6} - 4\right )} \log \left ({\left | x - \frac {1}{3} \, \sqrt {6} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 24, normalized size = 0.51
method | result | size |
default | \(\frac {\ln \left (3 x^{2}-2\right )}{3}+\frac {5 \sqrt {6}\, \arctanh \left (\frac {x \sqrt {6}}{2}\right )}{6}\) | \(24\) |
meijerg | \(\frac {5 \sqrt {6}\, \arctanh \left (\frac {x \sqrt {2}\, \sqrt {3}}{2}\right )}{6}+\frac {\ln \left (1-\frac {3 x^{2}}{2}\right )}{3}\) | \(27\) |
risch | \(\frac {\ln \left (3 x +\sqrt {6}\right )}{3}+\frac {5 \ln \left (3 x +\sqrt {6}\right ) \sqrt {6}}{12}+\frac {\ln \left (3 x -\sqrt {6}\right )}{3}-\frac {5 \ln \left (3 x -\sqrt {6}\right ) \sqrt {6}}{12}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 36, normalized size = 0.77 \[ -\frac {5}{12} \, \sqrt {6} \log \left (\frac {3 \, x - \sqrt {6}}{3 \, x + \sqrt {6}}\right ) + \frac {1}{3} \, \log \left (3 \, x^{2} - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 47, normalized size = 1.00 \[ \frac {\ln \left (x-\frac {\sqrt {6}}{3}\right )}{3}+\frac {\ln \left (x+\frac {\sqrt {6}}{3}\right )}{3}-\frac {5\,\sqrt {6}\,\ln \left (x-\frac {\sqrt {6}}{3}\right )}{12}+\frac {5\,\sqrt {6}\,\ln \left (x+\frac {\sqrt {6}}{3}\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 42, normalized size = 0.89 \[ \left (\frac {1}{3} - \frac {5 \sqrt {6}}{12}\right ) \log {\left (x - \frac {\sqrt {6}}{3} \right )} + \left (\frac {1}{3} + \frac {5 \sqrt {6}}{12}\right ) \log {\left (x + \frac {\sqrt {6}}{3} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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