Optimal. Leaf size=66 \[ \frac{25 \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac{1}{9} \log \left (x^4+2 x^2+3\right )+\frac{89 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{72 \sqrt{2}}+\frac{4 \log (x)}{9} \]
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Rubi [A] time = 0.108079, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1663, 1646, 800, 634, 618, 204, 628} \[ \frac{25 \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}-\frac{1}{9} \log \left (x^4+2 x^2+3\right )+\frac{89 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{72 \sqrt{2}}+\frac{4 \log (x)}{9} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1646
Rule 800
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{x \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}{x \left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{\frac{32}{3}+\frac{70 x}{3}}{x \left (3+2 x+x^2\right )} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \left (\frac{32}{9 x}-\frac{2 (-73+16 x)}{9 \left (3+2 x+x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{4 \log (x)}{9}-\frac{1}{72} \operatorname{Subst}\left (\int \frac{-73+16 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{4 \log (x)}{9}-\frac{1}{9} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )+\frac{89}{72} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{25 \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{4 \log (x)}{9}-\frac{1}{9} \log \left (3+2 x^2+x^4\right )-\frac{89}{36} \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 )}{24 \left (3+2 x^2+x^4\right )}+\frac{89 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{72 \sqrt{2}}+\frac{4 \log (x)}{9}-\frac{1}{9} \log \left (3+2 x^2+x^4\right )\\ \end{align*}
Mathematica [C] time = 0.0602343, size = 93, normalized size = 1.41 \[ \frac{1}{288} \left (-\frac{300 \left (x^2-1\right )}{x^4+2 x^2+3}-\sqrt{2} \left (16 \sqrt{2}+89 i\right ) \log \left (x^2-i \sqrt{2}+1\right )+\sqrt{2} \left (-16 \sqrt{2}+89 i\right ) \log \left (x^2+i \sqrt{2}+1\right )+128 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 58, normalized size = 0.9 \begin{align*} -{\frac{1}{18\,{x}^{4}+36\,{x}^{2}+54} \left ({\frac{75\,{x}^{2}}{4}}-{\frac{75}{4}} \right ) }-{\frac{\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{9}}+{\frac{89\,\sqrt{2}}{144}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) }+{\frac{4\,\ln \left ( x \right ) }{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47864, size = 74, normalized size = 1.12 \begin{align*} \frac{89}{144} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (x^{2} - 1\right )}}{24 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{1}{9} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{2}{9} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48087, size = 238, normalized size = 3.61 \begin{align*} \frac{89 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - 150 \, x^{2} - 16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) + 64 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x\right ) + 150}{144 \,{\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 = 0.173623, size = 65, normalized size = 0.98 \begin{align*} - \frac{25 x^{2} - 25}{24 x^{4} + 48 x^{2} + 72} + \frac{4 \log{\left (x \right )}}{9} - \frac{\log{\left (x^{4} + 2 x^{2} + 3 \right )}}{9} + \frac{89 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{144} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07984, size = 84, normalized size = 1.27 \begin{align*} \frac{89}{144} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{8 \, x^{4} - 59 \, x^{2} + 99}{72 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{1}{9} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) + \frac{2}{9} \, \log \left (x^{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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