Optimal. Leaf size=29 \[ -\frac {1}{2} \log (1-x)+\frac {1}{3} \log (2-x)+\frac {1}{6} \log (x+1) \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1586, 2058} \[ -\frac {1}{2} \log (1-x)+\frac {1}{3} \log (2-x)+\frac {1}{6} \log (x+1) \]
Antiderivative was successfully verified.
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Rule 1586
Rule 2058
Rubi steps
\begin {align*} \int \frac {2+x}{4-5 x^2+x^4} \, dx &=\int \frac {1}{2-x-2 x^2+x^3} \, dx\\ &=\int \left (\frac {1}{3 (-2+x)}-\frac {1}{2 (-1+x)}+\frac {1}{6 (1+x)}\right ) \, dx\\ &=-\frac {1}{2} \log (1-x)+\frac {1}{3} \log (2-x)+\frac {1}{6} \log (1+x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ -\frac {1}{2} \log (1-x)+\frac {1}{3} \log (2-x)+\frac {1}{6} \log (x+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 19, normalized size = 0.66 \[ \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) + \frac {1}{3} \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 22, normalized size = 0.76 \[ \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 0.69 \[ \frac {\ln \left (x -2\right )}{3}-\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 19, normalized size = 0.66 \[ \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (x - 1\right ) + \frac {1}{3} \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 19, normalized size = 0.66 \[ \frac {\ln \left (x+1\right )}{6}-\frac {\ln \left (x-1\right )}{2}+\frac {\ln \left (x-2\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 19, normalized size = 0.66 \[ \frac {\log {\left (x - 2 \right )}}{3} - \frac {\log {\left (x - 1 \right )}}{2} + \frac {\log {\left (x + 1 \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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