Optimal. Leaf size=25 \[ \frac {1}{2} \log (2-x)+\frac {\log (x)}{6}+\frac {1}{3} \log (x+3) \]
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Rubi [A] time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1594, 1628} \[ \frac {1}{2} \log (2-x)+\frac {\log (x)}{6}+\frac {1}{3} \log (x+3) \]
Antiderivative was successfully verified.
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Rule 1594
Rule 1628
Rubi steps
\begin {align*} \int \frac {-1+x+x^2}{-6 x+x^2+x^3} \, dx &=\int \frac {-1+x+x^2}{x \left (-6+x+x^2\right )} \, dx\\ &=\int \left (\frac {1}{2 (-2+x)}+\frac {1}{6 x}+\frac {1}{3 (3+x)}\right ) \, dx\\ &=\frac {1}{2} \log (2-x)+\frac {\log (x)}{6}+\frac {1}{3} \log (3+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ \frac {1}{2} \log (2-x)+\frac {\log (x)}{6}+\frac {1}{3} \log (x+3) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 23, normalized size = 0.92 \[ \frac {1}{2} \log (x-2)+\frac {\log (x)}{6}+\frac {1}{3} \log (x+3) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 17, normalized size = 0.68 \[ \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{2} \, \log \left (x - 2\right ) + \frac {1}{6} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.88, size = 20, normalized size = 0.80 \[ \frac {1}{3} \, \log \left ({\left | x + 3 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x - 2 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 18, normalized size = 0.72
method | result | size |
default | \(\frac {\ln \left (3+x \right )}{3}+\frac {\ln \relax (x )}{6}+\frac {\ln \left (-2+x \right )}{2}\) | \(18\) |
norman | \(\frac {\ln \left (3+x \right )}{3}+\frac {\ln \relax (x )}{6}+\frac {\ln \left (-2+x \right )}{2}\) | \(18\) |
risch | \(\frac {\ln \left (3+x \right )}{3}+\frac {\ln \relax (x )}{6}+\frac {\ln \left (-2+x \right )}{2}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 17, normalized size = 0.68 \[ \frac {1}{3} \, \log \left (x + 3\right ) + \frac {1}{2} \, \log \left (x - 2\right ) + \frac {1}{6} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 17, normalized size = 0.68 \[ \frac {\ln \left (x-2\right )}{2}+\frac {\ln \left (x+3\right )}{3}+\frac {\ln \relax (x)}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 17, normalized size = 0.68 \[ \frac {\log {\relax (x )}}{6} + \frac {\log {\left (x - 2 \right )}}{2} + \frac {\log {\left (x + 3 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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