Optimal. Leaf size=99 \[ \frac{x^3}{4 \left (x^4+1\right )}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.050228, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {28, 290, 297, 1162, 617, 204, 1165, 628} \[ \frac{x^3}{4 \left (x^4+1\right )}+\frac{\log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2}{1+2 x^4+x^8} \, dx &=\int \frac{x^2}{\left (1+x^4\right )^2} \, dx\\ &=\frac{x^3}{4 \left (1+x^4\right )}+\frac{1}{4} \int \frac{x^2}{1+x^4} \, dx\\ &=\frac{x^3}{4 \left (1+x^4\right )}-\frac{1}{8} \int \frac{1-x^2}{1+x^4} \, dx+\frac{1}{8} \int \frac{1+x^2}{1+x^4} \, dx\\ &=\frac{x^3}{4 \left (1+x^4\right )}+\frac{1}{16} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx+\frac{1}{16} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx+\frac{\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}+\frac{\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}\\ &=\frac{x^3}{4 \left (1+x^4\right )}+\frac{\log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} x\right )}{8 \sqrt{2}}\\ &=\frac{x^3}{4 \left (1+x^4\right )}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (1+\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}-\frac{\log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0495853, size = 92, normalized size = 0.93 \[ \frac{1}{32} \left (\frac{8 x^3}{x^4+1}+\sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-\sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 70, normalized size = 0.7 \begin{align*}{\frac{{x}^{3}}{4\,{x}^{4}+4}}+{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{16}}+{\frac{\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{16}}+{\frac{\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}-x\sqrt{2}}{1+{x}^{2}+x\sqrt{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4797, size = 113, normalized size = 1.14 \begin{align*} \frac{x^{3}}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65132, size = 373, normalized size = 3.77 \begin{align*} \frac{8 \, x^{3} - 4 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} - 1\right ) - 4 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} + 1\right ) - \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} - \sqrt{2} x + 1\right )}{32 \,{\left (x^{4} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.178476, size = 83, normalized size = 0.84 \begin{align*} \frac{x^{3}}{4 x^{4} + 4} + \frac{\sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} - \frac{\sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10743, size = 113, normalized size = 1.14 \begin{align*} \frac{x^{3}}{4 \,{\left (x^{4} + 1\right )}} + \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{1}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{1}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{1}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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