Optimal. Leaf size=37 \[ \frac {1}{6} \log \left (3 x^2-4 x+3\right )+\frac {\tan ^{-1}\left (\frac {2-3 x}{\sqrt {5}}\right )}{3 \sqrt {5}} \]
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Rubi [A] time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {634, 618, 204, 628} \[ \frac {1}{6} \log \left (3 x^2-4 x+3\right )+\frac {\tan ^{-1}\left (\frac {2-3 x}{\sqrt {5}}\right )}{3 \sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {-1+x}{3-4 x+3 x^2} \, dx &=\frac {1}{6} \int \frac {-4+6 x}{3-4 x+3 x^2} \, dx-\frac {1}{3} \int \frac {1}{3-4 x+3 x^2} \, dx\\ &=\frac {1}{6} \log \left (3-4 x+3 x^2\right )+\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-20-x^2} \, dx,x,-4+6 x\right )\\ &=\frac {\tan ^{-1}\left (\frac {2-3 x}{\sqrt {5}}\right )}{3 \sqrt {5}}+\frac {1}{6} \log \left (3-4 x+3 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ \frac {1}{6} \log \left (3 x^2-4 x+3\right )-\frac {\tan ^{-1}\left (\frac {3 x-2}{\sqrt {5}}\right )}{3 \sqrt {5}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 30, normalized size = 0.81 \[ -\frac {1}{15} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (3 \, x - 2\right )}\right ) + \frac {1}{6} \, \log \left (3 \, x^{2} - 4 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 30, normalized size = 0.81 \[ -\frac {1}{15} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (3 \, x - 2\right )}\right ) + \frac {1}{6} \, \log \left (3 \, x^{2} - 4 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.84 \[ -\frac {\sqrt {5}\, \arctan \left (\frac {\left (6 x -4\right ) \sqrt {5}}{10}\right )}{15}+\frac {\ln \left (3 x^{2}-4 x +3\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 30, normalized size = 0.81 \[ -\frac {1}{15} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} {\left (3 \, x - 2\right )}\right ) + \frac {1}{6} \, \log \left (3 \, x^{2} - 4 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.22, size = 30, normalized size = 0.81 \[ \frac {\ln \left (x^2-\frac {4\,x}{3}+1\right )}{6}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {3\,\sqrt {5}\,x}{5}-\frac {2\,\sqrt {5}}{5}\right )}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 39, normalized size = 1.05 \[ \frac {\log {\left (x^{2} - \frac {4 x}{3} + 1 \right )}}{6} - \frac {\sqrt {5} \operatorname {atan}{\left (\frac {3 \sqrt {5} x}{5} - \frac {2 \sqrt {5}}{5} \right )}}{15} \]
Verification of antiderivative is not currently implemented for this CAS.
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