Optimal. Leaf size=17 \[ \frac {1}{2} \log \left (1-x^2\right )-\log (x) \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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 29
Rule 31
Rule 36
Rule 266
Rule 1593
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*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \[ \frac {1}{2} \log \left (1-x^2\right )-\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.38, size = 13, normalized size = 0.76 \[ \frac {1}{2} \, \log \left (x^{2} - 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.01, size = 16, normalized size = 0.94 \[ -\frac {1}{2} \, \log \left (x^{2}\right ) + \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 18, normalized size = 1.06 \[ -\ln \relax (x )+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 17, normalized size = 1.00 \[ \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 13, normalized size = 0.76 \[ \frac {\ln \left (x^2-1\right )}{2}-\ln \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 10, normalized size = 0.59 \[ - \log {\relax (x )} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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