Optimal. Leaf size=47 \[ \frac {1}{12} \left (4-5 \sqrt {6}\right ) \log \left (\sqrt {6}-3 x\right )+\frac {1}{12} \left (4+5 \sqrt {6}\right ) \log \left (\sqrt {6}+3 x\right ) \]
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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 = {647, 31}
\begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 647
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 = 0.02, size = 47, normalized size = 1.00 \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.00, size = 35, normalized size = 0.74 \begin {gather*} \frac {\left (4-5 \sqrt {6}\right ) \text {Log}\left [-\frac {\sqrt {6}}{3}+x\right ]}{12}+\frac {\left (4+5 \sqrt {6}\right ) \text {Log}\left [\frac {\sqrt {6}}{3}+x\right ]}{12} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, 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 = 0.35, size = 36, normalized size = 0.77 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 40, normalized size = 0.85 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 42, normalized size = 0.89 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 51, normalized size = 1.09 \begin {gather*} \frac {1}{12} \left (-5 \sqrt {6}+4\right ) \ln \left |x-\frac {\sqrt {6}}{3}\right |+\frac {1}{12} \left (5 \sqrt {6}+4\right ) \ln \left |x+\frac {\sqrt {6}}{3}\right | \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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