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 = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {x^2}{2}+\frac {1}{2} \log \left (1-x^2\right )+x-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 260
Rule 1593
Rule 1802
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {x^2}{2}+\frac {1}{2} \log \left (1-x^2\right )+x-\log (x) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x-x^2+x^3+x^4}{-x+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.28, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{2} \, x^{2} + x + \frac {1}{2} \, \log \left (x^{2} - 1\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 26, normalized size = 1.04 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.96 \begin {gather*} \frac {x^{2}}{2}+x -\ln \relax (x )+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 23, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, x^{2} + x + \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 19, normalized size = 0.76 \begin {gather*} x+\frac {\ln \left (x^2-1\right )}{2}-\ln \relax (x)+\frac {x^2}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 17, normalized size = 0.68 \begin {gather*} \frac {x^{2}}{2} + x - \log {\relax (x )} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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