Optimal. Leaf size=18 \[ x-\log (1-x)+4 \log (2-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {717, 646, 31}
\begin {gather*} x-\log (1-x)+4 \log (2-x) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 646
Rule 717
Rubi steps
\begin {align*} \int \frac {x^2}{2-3 x+x^2} \, dx &=x+\int \frac {-2+3 x}{2-3 x+x^2} \, dx\\ &=x+4 \int \frac {1}{-2+x} \, dx-\int \frac {1}{-1+x} \, dx\\ &=x-\log (1-x)+4 \log (2-x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} x-\log (1-x)+4 \log (2-x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.63, size = 15, normalized size = 0.83
method | result | size |
default | \(x -\ln \left (x -1\right )+4 \ln \left (x -2\right )\) | \(15\) |
norman | \(x -\ln \left (x -1\right )+4 \ln \left (x -2\right )\) | \(15\) |
risch | \(x -\ln \left (x -1\right )+4 \ln \left (x -2\right )\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 14, normalized size = 0.78 \begin {gather*} x - \log \left (x - 1\right ) + 4 \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.20, size = 14, normalized size = 0.78 \begin {gather*} x - \log \left (x - 1\right ) + 4 \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 12, normalized size = 0.67 \begin {gather*} x + 4 \log {\left (x - 2 \right )} - \log {\left (x - 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.10, size = 16, normalized size = 0.89 \begin {gather*} x - \log \left ({\left | x - 1 \right |}\right ) + 4 \, \log \left ({\left | x - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 14, normalized size = 0.78 \begin {gather*} x-\ln \left (x-1\right )+4\,\ln \left (x-2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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