Optimal. Leaf size=58 \[ \frac{25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{5}{4} \log \left (x^4+2 x^2+3\right )-\frac{23 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0669888, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {1663, 1660, 634, 618, 204, 628} \[ \frac{25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{5}{4} \log \left (x^4+2 x^2+3\right )-\frac{23 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1660
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{4+x+3 x^2+5 x^3}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{-6+40 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{5}{4} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )-\frac{23}{8} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{5}{4} \log \left (3+2 x^2+x^4\right )+\frac{23}{4} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac{25 \left (1+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{23 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{5}{4} \log \left (3+2 x^2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0223489, size = 58, normalized size = 1. \[ \frac{25 \left (x^2+1\right )}{8 \left (x^4+2 x^2+3\right )}+\frac{5}{4} \log \left (x^4+2 x^2+3\right )-\frac{23 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 54, normalized size = 0.9 \begin{align*}{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{25\,{x}^{2}}{4}}+{\frac{25}{4}} \right ) }+{\frac{5\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{4}}-{\frac{23\,\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 [A] time = 1.47022, size = 66, normalized size = 1.14 \begin{align*} -\frac{23}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{25 \,{\left (x^{2} + 1\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{5}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52721, size = 194, normalized size = 3.34 \begin{align*} -\frac{23 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 50 \, x^{2} - 20 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 50}{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 = 0.161755, size = 60, normalized size = 1.03 \begin{align*} \frac{25 x^{2} + 25}{8 x^{4} + 16 x^{2} + 24} + \frac{5 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - \frac{23 \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 = 1.10919, size = 66, normalized size = 1.14 \begin{align*} -\frac{23}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{25 \,{\left (x^{2} + 1\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{5}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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