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 = 0.0540768, antiderivative size = 23, normalized size of antiderivative = 1., 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 1594
Rule 1628
Rule 628
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*}
Mathematica [A] time = 0.0074805, size = 23, normalized size = 1. \[ \frac{x^2}{2}-\frac{1}{2} \log \left (x^2-2 x+3\right )+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2}}+\ln \left ( x \right ) -{\frac{\ln \left ({x}^{2}-2\,x+3 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993679, size = 26, normalized size = 1.13 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.4448, size = 58, normalized size = 2.52 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098815, size = 19, normalized size = 0.83 \begin{align*} \frac{x^{2}}{2} + \log{\left (x \right )} - \frac{\log{\left (x^{2} - 2 x + 3 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.38314, size = 27, normalized size = 1.17 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, \log \left (x^{2} - 2 \, x + 3\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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