Optimal. Leaf size=55 \[ \frac{11 x^2+9}{4 \left (x^4+3 x^2+2\right )}-\frac{1}{2 x^2}+5 \log \left (x^2+1\right )-\frac{29}{8} \log \left (x^2+2\right )-\frac{11 \log (x)}{4} \]
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Rubi [A] time = 0.104323, antiderivative size = 55, 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, 1628} \[ \frac{11 x^2+9}{4 \left (x^4+3 x^2+2\right )}-\frac{1}{2 x^2}+5 \log \left (x^2+1\right )-\frac{29}{8} \log \left (x^2+2\right )-\frac{11 \log (x)}{4} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1646
Rule 1628
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x^3 \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^2 \left (2+3 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{9+11 x^2}{4 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-2+\frac{5 x}{2}-\frac{11 x^2}{2}}{x^2 \left (2+3 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac{9+11 x^2}{4 \left (2+3 x^2+x^4\right )}-\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{1}{x^2}+\frac{11}{4 x}-\frac{10}{1+x}+\frac{29}{4 (2+x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 x^2}+\frac{9+11 x^2}{4 \left (2+3 x^2+x^4\right )}-\frac{11 \log (x)}{4}+5 \log \left (1+x^2\right )-\frac{29}{8} \log \left (2+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0251518, size = 50, normalized size = 0.91 \[ \frac{1}{8} \left (\frac{22 x^2+18}{x^4+3 x^2+2}-\frac{4}{x^2}+40 \log \left (x^2+1\right )-29 \log \left (x^2+2\right )-22 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 45, normalized size = 0.8 \begin{align*} -{\frac{29\,\ln \left ({x}^{2}+2 \right ) }{8}}+{\frac{13}{4\,{x}^{2}+8}}+5\,\ln \left ({x}^{2}+1 \right ) -{\frac{1}{2\,{x}^{2}+2}}-{\frac{1}{2\,{x}^{2}}}-{\frac{11\,\ln \left ( x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00124, size = 72, normalized size = 1.31 \begin{align*} \frac{9 \, x^{4} + 3 \, x^{2} - 4}{4 \,{\left (x^{6} + 3 \, x^{4} + 2 \, x^{2}\right )}} - \frac{29}{8} \, \log \left (x^{2} + 2\right ) + 5 \, \log \left (x^{2} + 1\right ) - \frac{11}{8} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.85876, size = 219, normalized size = 3.98 \begin{align*} \frac{18 \, x^{4} + 6 \, x^{2} - 29 \,{\left (x^{6} + 3 \, x^{4} + 2 \, x^{2}\right )} \log \left (x^{2} + 2\right ) + 40 \,{\left (x^{6} + 3 \, x^{4} + 2 \, x^{2}\right )} \log \left (x^{2} + 1\right ) - 22 \,{\left (x^{6} + 3 \, x^{4} + 2 \, x^{2}\right )} \log \left (x\right ) - 8}{8 \,{\left (x^{6} + 3 \, x^{4} + 2 \, x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.190424, size = 51, normalized size = 0.93 \begin{align*} \frac{9 x^{4} + 3 x^{2} - 4}{4 x^{6} + 12 x^{4} + 8 x^{2}} - \frac{11 \log{\left (x \right )}}{4} + 5 \log{\left (x^{2} + 1 \right )} - \frac{29 \log{\left (x^{2} + 2 \right )}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1098, size = 72, normalized size = 1.31 \begin{align*} \frac{9 \, x^{4} + 3 \, x^{2} - 4}{4 \,{\left (x^{6} + 3 \, x^{4} + 2 \, x^{2}\right )}} - \frac{29}{8} \, \log \left (x^{2} + 2\right ) + 5 \, \log \left (x^{2} + 1\right ) - \frac{11}{8} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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