Optimal. Leaf size=17 \[ \frac {1}{3} \tanh ^{-1}(x)-\frac {1}{6} \tanh ^{-1}\left (\frac {x}{2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1093, 207} \[ \frac {1}{3} \tanh ^{-1}(x)-\frac {1}{6} \tanh ^{-1}\left (\frac {x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 1093
Rubi steps
\begin {align*} \int \frac {1}{4-5 x^2+x^4} \, dx &=\frac {1}{3} \int \frac {1}{-4+x^2} \, dx-\frac {1}{3} \int \frac {1}{-1+x^2} \, dx\\ &=-\frac {1}{6} \tanh ^{-1}\left (\frac {x}{2}\right )+\frac {1}{3} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [B] time = 0.01, size = 37, normalized size = 2.18 \[ -\frac {1}{6} \log (1-x)+\frac {1}{12} \log (2-x)+\frac {1}{6} \log (x+1)-\frac {1}{12} \log (x+2) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 25, normalized size = 1.47 \[ -\frac {1}{12} \, \log \left (x + 2\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 29, normalized size = 1.71 \[ -\frac {1}{12} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{12} \, \log \left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 26, normalized size = 1.53 \[ \frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x +2\right )}{12}+\frac {\ln \left (x -2\right )}{12}-\frac {\ln \left (x -1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.37, size = 25, normalized size = 1.47 \[ -\frac {1}{12} \, \log \left (x + 2\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) + \frac {1}{12} \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 11, normalized size = 0.65 \[ \frac {\mathrm {atanh}\relax (x)}{3}-\frac {\mathrm {atanh}\left (\frac {x}{2}\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.18, size = 26, normalized size = 1.53 \[ \frac {\log {\left (x - 2 \right )}}{12} - \frac {\log {\left (x - 1 \right )}}{6} + \frac {\log {\left (x + 1 \right )}}{6} - \frac {\log {\left (x + 2 \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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