Optimal. Leaf size=68 \[ \frac {1}{12 (1-x)}+\frac {1}{36 (2-x)}-\frac {1}{36 (x+1)}+\frac {1}{18} \log (1-x)-\frac {35}{432} \log (2-x)+\frac {1}{54} \log (x+1)+\frac {1}{144} \log (x+2) \]
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Rubi [A] time = 0.06, antiderivative size = 68, 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, 2074} \begin {gather*} \frac {1}{12 (1-x)}+\frac {1}{36 (2-x)}-\frac {1}{36 (x+1)}+\frac {1}{18} \log (1-x)-\frac {35}{432} \log (2-x)+\frac {1}{54} \log (x+1)+\frac {1}{144} \log (x+2) \end {gather*}
Antiderivative was successfully verified.
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Rule 1586
Rule 2074
Rubi steps
\begin {align*} \int \frac {2+x}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac {1}{(2+x) \left (2-x-2 x^2+x^3\right )^2} \, dx\\ &=\int \left (\frac {1}{36 (-2+x)^2}-\frac {35}{432 (-2+x)}+\frac {1}{12 (-1+x)^2}+\frac {1}{18 (-1+x)}+\frac {1}{36 (1+x)^2}+\frac {1}{54 (1+x)}+\frac {1}{144 (2+x)}\right ) \, dx\\ &=\frac {1}{12 (1-x)}+\frac {1}{36 (2-x)}-\frac {1}{36 (1+x)}+\frac {1}{18} \log (1-x)-\frac {35}{432} \log (2-x)+\frac {1}{54} \log (1+x)+\frac {1}{144} \log (2+x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 60, normalized size = 0.88 \begin {gather*} \frac {1}{432} \left (\frac {12 \left (-5 x^2+6 x+5\right )}{x^3-2 x^2-x+2}+24 \log (1-x)-35 \log (2-x)+8 \log (x+1)+3 \log (x+2)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+x}{\left (4-5 x^2+x^4\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.03, size = 103, normalized size = 1.51 \begin {gather*} -\frac {60 \, x^{2} - 3 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x + 2\right ) - 8 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x + 1\right ) - 24 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x - 1\right ) + 35 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x - 2\right ) - 72 \, x - 60}{432 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 56, normalized size = 0.82 \begin {gather*} -\frac {5 \, x^{2} - 6 \, x - 5}{36 \, {\left (x + 1\right )} {\left (x - 1\right )} {\left (x - 2\right )}} + \frac {1}{144} \, \log \left ({\left | x + 2 \right |}\right ) + \frac {1}{54} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{18} \, \log \left ({\left | x - 1 \right |}\right ) - \frac {35}{432} \, \log \left ({\left | x - 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.69 \begin {gather*} \frac {\ln \left (x +2\right )}{144}-\frac {35 \ln \left (x -2\right )}{432}+\frac {\ln \left (x -1\right )}{18}+\frac {\ln \left (x +1\right )}{54}-\frac {1}{36 \left (x -2\right )}-\frac {1}{36 \left (x +1\right )}-\frac {1}{12 \left (x -1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 52, normalized size = 0.76 \begin {gather*} -\frac {5 \, x^{2} - 6 \, x - 5}{36 \, {\left (x^{3} - 2 \, x^{2} - x + 2\right )}} + \frac {1}{144} \, \log \left (x + 2\right ) + \frac {1}{54} \, \log \left (x + 1\right ) + \frac {1}{18} \, \log \left (x - 1\right ) - \frac {35}{432} \, \log \left (x - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 52, normalized size = 0.76 \begin {gather*} \frac {\ln \left (x-1\right )}{18}+\frac {\ln \left (x+1\right )}{54}-\frac {35\,\ln \left (x-2\right )}{432}+\frac {\ln \left (x+2\right )}{144}-\frac {-\frac {5\,x^2}{36}+\frac {x}{6}+\frac {5}{36}}{-x^3+2\,x^2+x-2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 53, normalized size = 0.78 \begin {gather*} \frac {- 5 x^{2} + 6 x + 5}{36 x^{3} - 72 x^{2} - 36 x + 72} - \frac {35 \log {\left (x - 2 \right )}}{432} + \frac {\log {\left (x - 1 \right )}}{18} + \frac {\log {\left (x + 1 \right )}}{54} + \frac {\log {\left (x + 2 \right )}}{144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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