Optimal. Leaf size=97 \[ \frac{x}{4 \left (x^4+1\right )}-\frac{3 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0469694, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {28, 199, 211, 1165, 628, 1162, 617, 204} \[ \frac{x}{4 \left (x^4+1\right )}-\frac{3 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 199
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{1+2 x^4+x^8} \, dx &=\int \frac{1}{\left (1+x^4\right )^2} \, dx\\ &=\frac{x}{4 \left (1+x^4\right )}+\frac{3}{4} \int \frac{1}{1+x^4} \, dx\\ &=\frac{x}{4 \left (1+x^4\right )}+\frac{3}{8} \int \frac{1-x^2}{1+x^4} \, dx+\frac{3}{8} \int \frac{1+x^2}{1+x^4} \, dx\\ &=\frac{x}{4 \left (1+x^4\right )}+\frac{3}{16} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx+\frac{3}{16} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx-\frac{3 \int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}-\frac{3 \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{3 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{3 \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{3 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1+\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{3 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{3 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0495273, size = 91, normalized size = 0.94 \[ \frac{1}{32} \left (\frac{8 x}{x^4+1}-3 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+3 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
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{3\,\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{16}}+{\frac{3\,\sqrt{2}}{32}\ln \left ({\frac{1+{x}^{2}+x\sqrt{2}}{1+{x}^{2}-x\sqrt{2}}} \right ) }+{\frac{3\,\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 = 1.50572, size = 111, normalized size = 1.14 \begin{align*} \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{3}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{3}{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 = 1.5888, size = 379, normalized size = 3.91 \begin{align*} -\frac{12 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} - 1\right ) + 12 \, \sqrt{2}{\left (x^{4} + 1\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} + 1\right ) - 3 \, \sqrt{2}{\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) + 3 \, \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 = 0.180556, size = 88, normalized size = 0.91 \begin{align*} \frac{x}{4 x^{4} + 4} - \frac{3 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{3 \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 = 1.10491, size = 111, normalized size = 1.14 \begin{align*} \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{3}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{3}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{3}{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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