Optimal. Leaf size=12 \[ -\frac {1}{2} \log \left (1-x^2\right ) \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1598, 266}
\begin {gather*} -\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^2}{x-x^3} \, dx &=\int \frac {x}{1-x^2} \, dx\\ &=-\frac {1}{2} \log \left (1-x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 14, normalized size = 1.17
method | result | size |
risch | \(-\frac {\ln \left (x^{2}-1\right )}{2}\) | \(9\) |
meijerg | \(-\frac {\ln \left (-x^{2}+1\right )}{2}\) | \(11\) |
default | \(-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}\) | \(14\) |
norman | \(-\frac {\ln \left (x -1\right )}{2}-\frac {\ln \left (x +1\right )}{2}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 1.08 \begin {gather*} -\frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.71, size = 8, normalized size = 0.67 \begin {gather*} -\frac {1}{2} \, \log \left (x^{2} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 8, normalized size = 0.67 \begin {gather*} - \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.26, size = 15, normalized size = 1.25 \begin {gather*} -\frac {1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 8, normalized size = 0.67 \begin {gather*} -\frac {\ln \left (x^2-1\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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