Optimal. Leaf size=97 \[ -\frac{x}{4 \left (x^4+1\right )}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0511466, antiderivative size = 97, 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, 288, 211, 1165, 628, 1162, 617, 204} \[ -\frac{x}{4 \left (x^4+1\right )}-\frac{\log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{\log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{\tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{\tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 288
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{x^4}{1+2 x^4+x^8} \, dx &=\int \frac{x^4}{\left (1+x^4\right )^2} \, dx\\ &=-\frac{x}{4 \left (1+x^4\right )}+\frac{1}{4} \int \frac{1}{1+x^4} \, dx\\ &=-\frac{x}{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}{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}{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}{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 = 0.0719407, size = 90, normalized size = 0.93 \[ \frac{1}{32} \left (-\frac{8 x}{x^4+1}-\sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+\sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-2 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+2 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 68, normalized size = 0.7 \begin{align*} -{\frac{x}{4\,{x}^{4}+4}}+{\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 ) }+{\frac{\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46151, size = 111, normalized size = 1.14 \begin{align*} \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 ) - \frac{x}{4 \,{\left (x^{4} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.80504, size = 371, normalized size = 3.82 \begin{align*} -\frac{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 ) + 8 \, x}{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 = 0.173247, size = 82, normalized size = 0.85 \begin{align*} - \frac{x}{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 = 1.0967, size = 111, normalized size = 1.14 \begin{align*} \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 ) - \frac{x}{4 \,{\left (x^{4} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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