Optimal. Leaf size=24 \[ \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1130, 203, 207} \[ \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 1130
Rubi steps
\begin {align*} \int \frac {x^2}{-2+x^2+x^4} \, dx &=\frac {1}{3} \int \frac {1}{-1+x^2} \, dx+\frac {2}{3} \int \frac {1}{2+x^2} \, dx\\ &=\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.33 \[ \frac {1}{6} \left (\log (1-x)-\log (x+1)+2 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 24, normalized size = 1.00 \[ \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 25, normalized size = 1.04 \[ \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{6} \, \log \left (x + 1\right ) + \frac {1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 27, normalized size = 1.12 \[ \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 26, normalized size = 1.08
method | result | size |
default | \(\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{3}+\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (1+x \right )}{6}\) | \(26\) |
risch | \(\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{3}+\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (1+x \right )}{6}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 25, normalized size = 1.04 \[ \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{6} \, \log \left (x + 1\right ) + \frac {1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 17, normalized size = 0.71 \[ \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{3}-\frac {\mathrm {atanh}\relax (x)}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 29, normalized size = 1.21 \[ \frac {\log {\left (x - 1 \right )}}{6} - \frac {\log {\left (x + 1 \right )}}{6} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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