Optimal. Leaf size=23 \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]
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Rubi [A] time = 0.05, 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 = {1594, 1628, 628} \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 628
Rule 1594
Rule 1628
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 \[ \frac {x^2}{2}-\frac {1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 19, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 20, normalized size = 0.87 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.87 \[ \frac {x^{2}}{2}+\ln \relax (x )-\frac {\ln \left (x^{2}-2 x +3\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.12, size = 19, normalized size = 0.83 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \relax (x) \]
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 \[ \ln \relax (x)-\frac {\ln \left (x^2-2\,x+3\right )}{2}+\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.83 \[ \frac {x^{2}}{2} + \log {\relax (x )} - \frac {\log {\left (x^{2} - 2 x + 3 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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