Optimal. Leaf size=113 \[ \frac{1}{4 x^5 \left (x^4+1\right )}-\frac{9}{20 x^5}+\frac{9 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{9 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{9}{4 x}-\frac{9 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{9 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0543694, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.562, Rules used = {28, 290, 325, 297, 1162, 617, 204, 1165, 628} \[ \frac{1}{4 x^5 \left (x^4+1\right )}-\frac{9}{20 x^5}+\frac{9 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{9 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{9}{4 x}-\frac{9 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{9 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 325
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (1+2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^6 \left (1+x^4\right )^2} \, dx\\ &=\frac{1}{4 x^5 \left (1+x^4\right )}+\frac{9}{4} \int \frac{1}{x^6 \left (1+x^4\right )} \, dx\\ &=-\frac{9}{20 x^5}+\frac{1}{4 x^5 \left (1+x^4\right )}-\frac{9}{4} \int \frac{1}{x^2 \left (1+x^4\right )} \, dx\\ &=-\frac{9}{20 x^5}+\frac{9}{4 x}+\frac{1}{4 x^5 \left (1+x^4\right )}+\frac{9}{4} \int \frac{x^2}{1+x^4} \, dx\\ &=-\frac{9}{20 x^5}+\frac{9}{4 x}+\frac{1}{4 x^5 \left (1+x^4\right )}-\frac{9}{8} \int \frac{1-x^2}{1+x^4} \, dx+\frac{9}{8} \int \frac{1+x^2}{1+x^4} \, dx\\ &=-\frac{9}{20 x^5}+\frac{9}{4 x}+\frac{1}{4 x^5 \left (1+x^4\right )}+\frac{9}{16} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx+\frac{9}{16} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx+\frac{9 \int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}+\frac{9 \int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}\\ &=-\frac{9}{20 x^5}+\frac{9}{4 x}+\frac{1}{4 x^5 \left (1+x^4\right )}+\frac{9 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}-\frac{9 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{9 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} x\right )}{8 \sqrt{2}}\\ &=-\frac{9}{20 x^5}+\frac{9}{4 x}+\frac{1}{4 x^5 \left (1+x^4\right )}-\frac{9 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{9 \tan ^{-1}\left (1+\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{9 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}-\frac{9 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0776377, size = 103, normalized size = 0.91 \[ \frac{1}{160} \left (\frac{40 x^3}{x^4+1}-\frac{32}{x^5}+45 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-45 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )+\frac{320}{x}-90 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+90 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 80, normalized size = 0.7 \begin{align*} -{\frac{1}{5\,{x}^{5}}}+2\,{x}^{-1}+{\frac{{x}^{3}}{4\,{x}^{4}+4}}+{\frac{9\,\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{16}}+{\frac{9\,\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{16}}+{\frac{9\,\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 = 1.47472, size = 128, normalized size = 1.13 \begin{align*} \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{9}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{9}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{45 \, x^{8} + 36 \, x^{4} - 4}{20 \,{\left (x^{9} + x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57116, size = 424, normalized size = 3.75 \begin{align*} \frac{360 \, x^{8} + 288 \, x^{4} - 180 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} - 1\right ) - 180 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} + 1\right ) - 45 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) + 45 \, \sqrt{2}{\left (x^{9} + x^{5}\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) - 32}{160 \,{\left (x^{9} + x^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.245177, size = 102, normalized size = 0.9 \begin{align*} \frac{9 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} - \frac{9 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} + \frac{45 x^{8} + 36 x^{4} - 4}{20 x^{9} + 20 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12316, size = 130, normalized size = 1.15 \begin{align*} \frac{x^{3}}{4 \,{\left (x^{4} + 1\right )}} + \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{9}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{9}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{10 \, x^{4} - 1}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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