Optimal. Leaf size=30 \[ \frac {1}{3 (1-x)}+\frac {2}{9} \log (1-x)-\frac {2}{9} \log (x+2) \]
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Rubi [A] time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2074} \[ \frac {1}{3 (1-x)}+\frac {2}{9} \log (1-x)-\frac {2}{9} \log (x+2) \]
Antiderivative was successfully verified.
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Rule 2074
Rubi steps
\begin {align*} \int \frac {x}{2-3 x+x^3} \, dx &=\int \left (\frac {1}{3 (-1+x)^2}+\frac {2}{9 (-1+x)}-\frac {2}{9 (2+x)}\right ) \, dx\\ &=\frac {1}{3 (1-x)}+\frac {2}{9} \log (1-x)-\frac {2}{9} \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.93 \[ -\frac {1}{3 (x-1)}+\frac {2}{9} \log (1-x)-\frac {2}{9} \log (x+2) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 27, normalized size = 0.90 \[ -\frac {2 \, {\left (x - 1\right )} \log \left (x + 2\right ) - 2 \, {\left (x - 1\right )} \log \left (x - 1\right ) + 3}{9 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 22, normalized size = 0.73 \[ -\frac {1}{3 \, {\left (x - 1\right )}} - \frac {2}{9} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {2}{9} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 21, normalized size = 0.70 \[ \frac {2 \ln \left (x -1\right )}{9}-\frac {2 \ln \left (x +2\right )}{9}-\frac {1}{3 \left (x -1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 20, normalized size = 0.67 \[ -\frac {1}{3 \, {\left (x - 1\right )}} - \frac {2}{9} \, \log \left (x + 2\right ) + \frac {2}{9} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 18, normalized size = 0.60 \[ -\frac {4\,\mathrm {atanh}\left (\frac {2\,x}{3}+\frac {1}{3}\right )}{9}-\frac {1}{3\,\left (x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 22, normalized size = 0.73 \[ \frac {2 \log {\left (x - 1 \right )}}{9} - \frac {2 \log {\left (x + 2 \right )}}{9} - \frac {1}{3 x - 3} \]
Verification of antiderivative is not currently implemented for this CAS.
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