Optimal. Leaf size=17 \[ \frac{1}{2} \log \left (1-x^2\right )-\log (x) \]
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Rubi [A] time = 0.007101, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {1593, 266, 36, 31, 29} \[ \frac{1}{2} \log \left (1-x^2\right )-\log (x) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 266
Rule 36
Rule 31
Rule 29
Rubi steps
\begin{align*} \int \frac{1}{-x+x^3} \, dx &=\int \frac{1}{x \left (-1+x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(-1+x) x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,x^2\right )-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )\\ &=-\log (x)+\frac{1}{2} \log \left (1-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0020843, size = 17, normalized size = 1. \[ \frac{1}{2} \log \left (1-x^2\right )-\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 18, normalized size = 1.1 \begin{align*} -\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 = 0.929392, size = 23, normalized size = 1.35 \begin{align*} \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 = 0.877403, size = 36, normalized size = 2.12 \begin{align*} \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 = 0.083429, size = 10, normalized size = 0.59 \begin{align*} - \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 = 1.07797, size = 22, normalized size = 1.29 \begin{align*} -\frac{1}{2} \, \log \left (x^{2}\right ) + \frac{1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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