Optimal. Leaf size=56 \[ -\frac{3 x+5}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} \log (1-x)+\frac{1}{144} \log (2-x)-\frac{7}{36} \log (x+1)+\frac{31}{144} \log (x+2) \]
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Rubi [A] time = 0.0565911, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {1586, 974, 1072, 632, 31} \[ -\frac{3 x+5}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} \log (1-x)+\frac{1}{144} \log (2-x)-\frac{7}{36} \log (x+1)+\frac{31}{144} \log (x+2) \]
Antiderivative was successfully verified.
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Rule 1586
Rule 974
Rule 1072
Rule 632
Rule 31
Rubi steps
\begin{align*} \int \frac{2-3 x+x^2}{\left (4-5 x^2+x^4\right )^2} \, dx &=\int \frac{1}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )^2} \, dx\\ &=-\frac{5+3 x}{12 \left (2+3 x+x^2\right )}+\frac{1}{72} \int \frac{-18+48 x-18 x^2}{\left (2-3 x+x^2\right ) \left (2+3 x+x^2\right )} \, dx\\ &=-\frac{5+3 x}{12 \left (2+3 x+x^2\right )}+\frac{\int \frac{252-108 x}{2-3 x+x^2} \, dx}{5184}+\frac{\int \frac{-900+108 x}{2+3 x+x^2} \, dx}{5184}\\ &=-\frac{5+3 x}{12 \left (2+3 x+x^2\right )}+\frac{1}{144} \int \frac{1}{-2+x} \, dx-\frac{1}{36} \int \frac{1}{-1+x} \, dx-\frac{7}{36} \int \frac{1}{1+x} \, dx+\frac{31}{144} \int \frac{1}{2+x} \, dx\\ &=-\frac{5+3 x}{12 \left (2+3 x+x^2\right )}-\frac{1}{36} \log (1-x)+\frac{1}{144} \log (2-x)-\frac{7}{36} \log (1+x)+\frac{31}{144} \log (2+x)\\ \end{align*}
Mathematica [A] time = 0.0248618, size = 48, normalized size = 0.86 \[ \frac{1}{144} \left (-\frac{12 (3 x+5)}{x^2+3 x+2}-4 \log (1-x)+\log (2-x)-28 \log (x+1)+31 \log (x+2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 40, normalized size = 0.7 \begin{align*} -{\frac{1}{24+12\,x}}+{\frac{31\,\ln \left ( 2+x \right ) }{144}}-{\frac{1}{6+6\,x}}-{\frac{7\,\ln \left ( 1+x \right ) }{36}}+{\frac{\ln \left ( x-2 \right ) }{144}}-{\frac{\ln \left ( x-1 \right ) }{36}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.950417, size = 57, normalized size = 1.02 \begin{align*} -\frac{3 \, x + 5}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} + \frac{31}{144} \, \log \left (x + 2\right ) - \frac{7}{36} \, \log \left (x + 1\right ) - \frac{1}{36} \, \log \left (x - 1\right ) + \frac{1}{144} \, \log \left (x - 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82589, size = 215, normalized size = 3.84 \begin{align*} \frac{31 \,{\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 2\right ) - 28 \,{\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 1\right ) - 4 \,{\left (x^{2} + 3 \, x + 2\right )} \log \left (x - 1\right ) +{\left (x^{2} + 3 \, x + 2\right )} \log \left (x - 2\right ) - 36 \, x - 60}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.266986, size = 44, normalized size = 0.79 \begin{align*} - \frac{3 x + 5}{12 x^{2} + 36 x + 24} + \frac{\log{\left (x - 2 \right )}}{144} - \frac{\log{\left (x - 1 \right )}}{36} - \frac{7 \log{\left (x + 1 \right )}}{36} + \frac{31 \log{\left (x + 2 \right )}}{144} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08075, size = 62, normalized size = 1.11 \begin{align*} -\frac{3 \, x + 5}{12 \,{\left (x + 2\right )}{\left (x + 1\right )}} + \frac{31}{144} \, \log \left ({\left | x + 2 \right |}\right ) - \frac{7}{36} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \, \log \left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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