Optimal. Leaf size=19 \[ \frac {\log (x)}{2}+\frac {1}{4} \log \left (2-x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1607, 457, 78}
\begin {gather*} \frac {1}{4} \log \left (2-x^2\right )+\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1607
Rubi steps
\begin {align*} \int \frac {-1+x^2}{-2 x+x^3} \, dx &=\int \frac {-1+x^2}{x \left (-2+x^2\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {-1+x}{(-2+x) x} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{2 (-2+x)}+\frac {1}{2 x}\right ) \, dx,x,x^2\right )\\ &=\frac {\log (x)}{2}+\frac {1}{4} \log \left (2-x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 19, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{2}+\frac {1}{4} \log \left (2-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.21, size = 14, normalized size = 0.74
method | result | size |
default | \(\frac {\ln \left (x \right )}{2}+\frac {\ln \left (x^{2}-2\right )}{4}\) | \(14\) |
norman | \(\frac {\ln \left (x \right )}{2}+\frac {\ln \left (x^{2}-2\right )}{4}\) | \(14\) |
risch | \(\frac {\ln \left (x \right )}{2}+\frac {\ln \left (x^{2}-2\right )}{4}\) | \(14\) |
meijerg | \(\frac {\ln \left (1-\frac {x^{2}}{2}\right )}{4}+\frac {\ln \left (x \right )}{2}-\frac {\ln \left (2\right )}{4}+\frac {i \pi }{4}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{4} \, \log \left (x^{2} - 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 13, normalized size = 0.68 \begin {gather*} \frac {1}{4} \, \log \left (x^{2} - 2\right ) + \frac {1}{2} \, \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 12, normalized size = 0.63 \begin {gather*} \frac {\log {\left (x \right )}}{2} + \frac {\log {\left (x^{2} - 2 \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.63, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{4} \, \log \left (x^{2}\right ) + \frac {1}{4} \, \log \left ({\left | x^{2} - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.34, size = 13, normalized size = 0.68 \begin {gather*} \frac {\ln \left (x^2-2\right )}{4}+\frac {\ln \left (x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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