Optimal. Leaf size=224 \[ \frac{25 x \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}+\frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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Rubi [A] time = 0.215101, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1678, 1169, 634, 618, 204, 628} \[ \frac{25 x \left (1-x^2\right )}{24 \left (x^4+2 x^2+3\right )}+\frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{96} \sqrt{\frac{1}{6} \left (11567+12897 \sqrt{3}\right )} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{48} \sqrt{\frac{1}{6} \left (12897 \sqrt{3}-11567\right )} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
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Rule 1678
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{4+x^2+3 x^4+5 x^6}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac{25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{1}{48} \int \frac{14+190 x^2}{3+2 x^2+x^4} \, dx\\ &=\frac{25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{\int \frac{14 \sqrt{2 \left (-1+\sqrt{3}\right )}-\left (14-190 \sqrt{3}\right ) x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{96 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{\int \frac{14 \sqrt{2 \left (-1+\sqrt{3}\right )}+\left (14-190 \sqrt{3}\right ) x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{96 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=\frac{25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{\left (7-95 \sqrt{3}\right ) \int \frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{96 \sqrt{6 \left (-1+\sqrt{3}\right )}}+\frac{1}{288} \left (285+7 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{1}{288} \left (285+7 \sqrt{3}\right ) \int \frac{1}{\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx+\frac{\left (-7+95 \sqrt{3}\right ) \int \frac{-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2} \, dx}{96 \sqrt{6 \left (-1+\sqrt{3}\right )}}\\ &=\frac{25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}+\frac{1}{96} \sqrt{\frac{11567}{6}+\frac{4299 \sqrt{3}}{2}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{96} \sqrt{\frac{11567}{6}+\frac{4299 \sqrt{3}}{2}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{144} \left (285+7 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,-\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )-\frac{1}{144} \left (285+7 \sqrt{3}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 \left (1+\sqrt{3}\right )-x^2} \, dx,x,\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x\right )\\ &=\frac{25 x \left (1-x^2\right )}{24 \left (3+2 x^2+x^4\right )}-\frac{1}{48} \sqrt{\frac{1}{6} \left (-11567+12897 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{48} \sqrt{\frac{1}{6} \left (-11567+12897 \sqrt{3}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (-1+\sqrt{3}\right )}+2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{1}{96} \sqrt{\frac{11567}{6}+\frac{4299 \sqrt{3}}{2}} \log \left (\sqrt{3}-\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )-\frac{1}{96} \sqrt{\frac{11567}{6}+\frac{4299 \sqrt{3}}{2}} \log \left (\sqrt{3}+\sqrt{2 \left (-1+\sqrt{3}\right )} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.273742, size = 115, normalized size = 0.51 \[ \frac{1}{48} \left (-\frac{50 x \left (x^2-1\right )}{x^4+2 x^2+3}+\frac{\left (95+44 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{\sqrt{1-i \sqrt{2}}}+\frac{\left (95-44 i \sqrt{2}\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{\sqrt{1+i \sqrt{2}}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.02, size = 408, normalized size = 1.8 \begin{align*}{\frac{1}{{x}^{4}+2\,{x}^{2}+3} \left ( -{\frac{25\,{x}^{3}}{24}}+{\frac{25\,x}{24}} \right ) }+{\frac{139\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{576}}+{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{48}}+{\frac{ \left ( -278+278\,\sqrt{3} \right ) \sqrt{3}}{288\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-22+22\,\sqrt{3}}{24\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{139\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{576}}-{\frac{11\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{48}}+{\frac{ \left ( -278+278\,\sqrt{3} \right ) \sqrt{3}}{288\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{-22+22\,\sqrt{3}}{24\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{7\,\sqrt{3}}{72\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \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{25 \,{\left (x^{3} - x\right )}}{24 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} + \frac{1}{24} \, \int \frac{95 \, x^{2} + 7}{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 [B] time = 1.72396, size = 1993, normalized size = 8.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.522246, size = 48, normalized size = 0.21 \begin{align*} - \frac{25 x^{3} - 25 x}{24 x^{4} + 48 x^{2} + 72} + \operatorname{RootSum}{\left (28311552 t^{4} - 23689216 t^{2} + 18481401, \left ( t \mapsto t \log{\left (\frac{40992768 t^{3}}{19364129} - \frac{48423104 t}{58092387} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{5 \, x^{6} + 3 \, x^{4} + x^{2} + 4}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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