Optimal. Leaf size=45 \[ \frac{9 x^2+5}{8 \left (x^4+2 x^2+3\right )}+\frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0485355, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1593, 1247, 638, 618, 204} \[ \frac{9 x^2+5}{8 \left (x^4+2 x^2+3\right )}+\frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1247
Rule 638
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{11 x+2 x^3}{\left (3+2 x^2+x^4\right )^2} \, dx &=\int \frac{x \left (11+2 x^2\right )}{\left (3+2 x^2+x^4\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{11+2 x}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{5+9 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac{9}{8} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{5+9 x^2}{8 \left (3+2 x^2+x^4\right )}-\frac{9}{4} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac{5+9 x^2}{8 \left (3+2 x^2+x^4\right )}+\frac{9 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{8 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0246261, size = 45, normalized size = 1. \[ \frac{9 x^2+5}{8 \left (x^4+2 x^2+3\right )}+\frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 41, normalized size = 0.9 \begin{align*}{\frac{18\,{x}^{2}+10}{16\,{x}^{4}+32\,{x}^{2}+48}}+{\frac{9\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{9 \, x^{2} + 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{9}{4} \, \int \frac{x}{x^{4} + 2 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47967, size = 132, normalized size = 2.93 \begin{align*} \frac{9 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 18 \, x^{2} + 10}{16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.152507, size = 44, normalized size = 0.98 \begin{align*} \frac{9 x^{2} + 5}{8 x^{4} + 16 x^{2} + 24} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12985, size = 51, normalized size = 1.13 \begin{align*} \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{9 \, x^{2} + 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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