Optimal. Leaf size=23 \[ \frac {x^2}{2}+\log (x)-\frac {1}{2} \log \left (3-2 x+x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1608, 1642,
642} \begin {gather*} \frac {x^2}{2}-\frac {1}{2} \log \left (x^2-2 x+3\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 642
Rule 1608
Rule 1642
Rubi steps
\begin {align*} \int \frac {3-x+3 x^2-2 x^3+x^4}{3 x-2 x^2+x^3} \, dx &=\int \frac {3-x+3 x^2-2 x^3+x^4}{x \left (3-2 x+x^2\right )} \, dx\\ &=\int \left (\frac {1}{x}+x+\frac {1-x}{3-2 x+x^2}\right ) \, dx\\ &=\frac {x^2}{2}+\log (x)+\int \frac {1-x}{3-2 x+x^2} \, dx\\ &=\frac {x^2}{2}+\log (x)-\frac {1}{2} \log \left (3-2 x+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 23, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}+\log (x)-\frac {1}{2} \log \left (3-2 x+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 20, normalized size = 0.87
method | result | size |
default | \(\frac {x^{2}}{2}+\ln \left (x \right )-\frac {\ln \left (x^{2}-2 x +3\right )}{2}\) | \(20\) |
norman | \(\frac {x^{2}}{2}+\ln \left (x \right )-\frac {\ln \left (x^{2}-2 x +3\right )}{2}\) | \(20\) |
risch | \(\frac {x^{2}}{2}+\ln \left (x \right )-\frac {\ln \left (x^{2}-2 x +3\right )}{2}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 19, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 19, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 19, normalized size = 0.83 \begin {gather*} \frac {x^{2}}{2} + \log {\left (x \right )} - \frac {\log {\left (x^{2} - 2 x + 3 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.54, size = 20, normalized size = 0.87 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 19, normalized size = 0.83 \begin {gather*} \ln \left (x\right )-\frac {\ln \left (x^2-2\,x+3\right )}{2}+\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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