Optimal. Leaf size=30 \[ \frac{x^2}{2}+x+\frac{2}{1-x}+\log (1-x)-\log (x+1) \]
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Rubi [A] time = 0.0298122, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2074} \[ \frac{x^2}{2}+x+\frac{2}{1-x}+\log (1-x)-\log (x+1) \]
Antiderivative was successfully verified.
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Rule 2074
Rubi steps
\begin{align*} \int \frac{1+4 x-2 x^2+x^4}{1-x-x^2+x^3} \, dx &=\int \left (1+\frac{1}{-1-x}+\frac{2}{(-1+x)^2}+\frac{1}{-1+x}+x\right ) \, dx\\ &=\frac{2}{1-x}+x+\frac{x^2}{2}+\log (1-x)-\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0158656, size = 29, normalized size = 0.97 \[ \frac{1}{2} (x+1)^2-\frac{2}{x-1}+\log (1-x)-\log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 25, normalized size = 0.8 \begin{align*} x+{\frac{{x}^{2}}{2}}-\ln \left ( 1+x \right ) +\ln \left ( -1+x \right ) -2\, \left ( -1+x \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925685, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{2} \, x^{2} + x - \frac{2}{x - 1} - \log \left (x + 1\right ) + \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.07323, size = 109, normalized size = 3.63 \begin{align*} \frac{x^{3} + x^{2} - 2 \,{\left (x - 1\right )} \log \left (x + 1\right ) + 2 \,{\left (x - 1\right )} \log \left (x - 1\right ) - 2 \, x - 4}{2 \,{\left (x - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.090176, size = 20, normalized size = 0.67 \begin{align*} \frac{x^{2}}{2} + x + \log{\left (x - 1 \right )} - \log{\left (x + 1 \right )} - \frac{2}{x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05145, size = 35, normalized size = 1.17 \begin{align*} \frac{1}{2} \, x^{2} + x - \frac{2}{x - 1} - \log \left ({\left | x + 1 \right |}\right ) + \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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