Optimal. Leaf size=25 \[ \frac{x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]
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Rubi [A] time = 0.0372468, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {1593, 1802} \[ \frac{x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1802
Rubi steps
\begin{align*} \int \frac{1+2 x^2+x^5}{-x+x^3} \, dx &=\int \frac{1+2 x^2+x^5}{x \left (-1+x^2\right )} \, dx\\ &=\int \left (1+\frac{2}{-1+x}-\frac{1}{x}+x^2+\frac{1}{1+x}\right ) \, dx\\ &=x+\frac{x^3}{3}+2 \log (1-x)-\log (x)+\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0053016, size = 25, normalized size = 1. \[ \frac{x^3}{3}+x+2 \log (1-x)-\log (x)+\log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 22, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3}}+x+2\,\ln \left ( x-1 \right ) -\ln \left ( x \right ) +\ln \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988588, size = 28, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 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 = 1.39819, size = 68, normalized size = 2.72 \begin{align*} \frac{1}{3} \, x^{3} + x + \log \left (x + 1\right ) + 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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Sympy [A] time = 0.115942, size = 20, normalized size = 0.8 \begin{align*} \frac{x^{3}}{3} + x - \log{\left (x \right )} + 2 \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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Giac [A] time = 1.14471, size = 32, normalized size = 1.28 \begin{align*} \frac{1}{3} \, x^{3} + x + \log \left ({\left | x + 1 \right |}\right ) + 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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