Optimal. Leaf size=44 \[ -\frac{12 x^2+11}{2 \left (x^4+3 x^2+2\right )}-\frac{9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )+\log (x) \]
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Rubi [A] time = 0.0766938, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1663, 1646, 800} \[ -\frac{12 x^2+11}{2 \left (x^4+3 x^2+2\right )}-\frac{9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )+\log (x) \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1646
Rule 800
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x \left (2+3 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{4+x+3 x^2+5 x^3}{x \left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{11+12 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-2+7 x}{x \left (2+3 x+x^2\right )} \, dx,x,x^2\right )\\ &=-\frac{11+12 x^2}{2 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{x}+\frac{9}{1+x}-\frac{8}{2+x}\right ) \, dx,x,x^2\right )\\ &=-\frac{11+12 x^2}{2 \left (2+3 x^2+x^4\right )}+\log (x)-\frac{9}{2} \log \left (1+x^2\right )+4 \log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0215405, size = 44, normalized size = 1. \[ \frac{-12 x^2-11}{2 \left (x^4+3 x^2+2\right )}-\frac{9}{2} \log \left (x^2+1\right )+4 \log \left (x^2+2\right )+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 38, normalized size = 0.9 \begin{align*} 4\,\ln \left ({x}^{2}+2 \right ) -{\frac{13}{2\,{x}^{2}+4}}-{\frac{9\,\ln \left ({x}^{2}+1 \right ) }{2}}+{\frac{1}{2\,{x}^{2}+2}}+\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0193, size = 59, normalized size = 1.34 \begin{align*} -\frac{12 \, x^{2} + 11}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac{9}{2} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74922, size = 185, normalized size = 4.2 \begin{align*} -\frac{12 \, x^{2} - 8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 2\right ) + 9 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x^{2} + 1\right ) - 2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \log \left (x\right ) + 11}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.16426, size = 39, normalized size = 0.89 \begin{align*} - \frac{12 x^{2} + 11}{2 x^{4} + 6 x^{2} + 4} + \log{\left (x \right )} - \frac{9 \log{\left (x^{2} + 1 \right )}}{2} + 4 \log{\left (x^{2} + 2 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08408, size = 63, normalized size = 1.43 \begin{align*} \frac{x^{4} - 21 \, x^{2} - 20}{4 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} + 4 \, \log \left (x^{2} + 2\right ) - \frac{9}{2} \, \log \left (x^{2} + 1\right ) + \frac{1}{2} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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