Optimal. Leaf size=25 \[ \frac{x^2}{2}+\frac{1}{2} \log \left (1-x^2\right )+x-\log (x) \]
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Rubi [A] time = 0.0501107, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1593, 1802, 260} \[ \frac{x^2}{2}+\frac{1}{2} \log \left (1-x^2\right )+x-\log (x) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1802
Rule 260
Rubi steps
\begin{align*} \int \frac{1-x-x^2+x^3+x^4}{-x+x^3} \, dx &=\int \frac{1-x-x^2+x^3+x^4}{x \left (-1+x^2\right )} \, dx\\ &=\int \left (1-\frac{1}{x}+x+\frac{x}{-1+x^2}\right ) \, dx\\ &=x+\frac{x^2}{2}-\log (x)+\int \frac{x}{-1+x^2} \, dx\\ &=x+\frac{x^2}{2}-\log (x)+\frac{1}{2} \log \left (1-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0052752, size = 25, normalized size = 1. \[ \frac{x^2}{2}+\frac{1}{2} \log \left (1-x^2\right )+x-\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 24, normalized size = 1. \begin{align*} x+{\frac{{x}^{2}}{2}}-\ln \left ( x \right ) +{\frac{\ln \left ( 1+x \right ) }{2}}+{\frac{\ln \left ( -1+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.925505, size = 31, normalized size = 1.24 \begin{align*} \frac{1}{2} \, x^{2} + x + \frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (x - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.9284, size = 55, normalized size = 2.2 \begin{align*} \frac{1}{2} \, x^{2} + x + \frac{1}{2} \, \log \left (x^{2} - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086741, size = 17, normalized size = 0.68 \begin{align*} \frac{x^{2}}{2} + x - \log{\left (x \right )} + \frac{\log{\left (x^{2} - 1 \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05335, size = 35, normalized size = 1.4 \begin{align*} \frac{1}{2} \, x^{2} + x + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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