Optimal. Leaf size=113 \[ \frac{1}{4 x^7 \left (x^4+1\right )}+\frac{11}{12 x^3}-\frac{11}{28 x^7}-\frac{11 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{11 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{11 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{11 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0548459, 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, 211, 1165, 628, 1162, 617, 204} \[ \frac{1}{4 x^7 \left (x^4+1\right )}+\frac{11}{12 x^3}-\frac{11}{28 x^7}-\frac{11 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{11 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{11 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{11 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 290
Rule 325
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^8 \left (1+2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^8 \left (1+x^4\right )^2} \, dx\\ &=\frac{1}{4 x^7 \left (1+x^4\right )}+\frac{11}{4} \int \frac{1}{x^8 \left (1+x^4\right )} \, dx\\ &=-\frac{11}{28 x^7}+\frac{1}{4 x^7 \left (1+x^4\right )}-\frac{11}{4} \int \frac{1}{x^4 \left (1+x^4\right )} \, dx\\ &=-\frac{11}{28 x^7}+\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1+x^4\right )}+\frac{11}{4} \int \frac{1}{1+x^4} \, dx\\ &=-\frac{11}{28 x^7}+\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1+x^4\right )}+\frac{11}{8} \int \frac{1-x^2}{1+x^4} \, dx+\frac{11}{8} \int \frac{1+x^2}{1+x^4} \, dx\\ &=-\frac{11}{28 x^7}+\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1+x^4\right )}+\frac{11}{16} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx+\frac{11}{16} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx-\frac{11 \int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}-\frac{11 \int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}\\ &=-\frac{11}{28 x^7}+\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1+x^4\right )}-\frac{11 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{11 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{11 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} x\right )}{8 \sqrt{2}}\\ &=-\frac{11}{28 x^7}+\frac{11}{12 x^3}+\frac{1}{4 x^7 \left (1+x^4\right )}-\frac{11 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{11 \tan ^{-1}\left (1+\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{11 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}+\frac{11 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0813733, size = 101, normalized size = 0.89 \[ \frac{1}{672} \left (\frac{168 x}{x^4+1}+\frac{448}{x^3}-\frac{96}{x^7}-231 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )+231 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )-462 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )+462 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 78, normalized size = 0.7 \begin{align*} -{\frac{1}{7\,{x}^{7}}}+{\frac{2}{3\,{x}^{3}}}+{\frac{x}{4\,{x}^{4}+4}}+{\frac{11\,\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{16}}+{\frac{11\,\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{16}}+{\frac{11\,\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 = 1.49561, size = 128, normalized size = 1.13 \begin{align*} \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{11}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{11}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{77 \, x^{8} + 44 \, x^{4} - 12}{84 \,{\left (x^{11} + x^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64168, size = 433, normalized size = 3.83 \begin{align*} \frac{616 \, x^{8} + 352 \, x^{4} - 924 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} - 1\right ) - 924 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} + 1\right ) + 231 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) - 231 \, \sqrt{2}{\left (x^{11} + x^{7}\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) - 96}{672 \,{\left (x^{11} + x^{7}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.24912, size = 102, normalized size = 0.9 \begin{align*} - \frac{11 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} + \frac{11 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} + \frac{11 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x + 1 \right )}}{16} + \frac{77 x^{8} + 44 x^{4} - 12}{84 x^{11} + 84 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14518, size = 127, normalized size = 1.12 \begin{align*} \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) + \frac{11}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) + \frac{11}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) - \frac{11}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) + \frac{x}{4 \,{\left (x^{4} + 1\right )}} + \frac{14 \, x^{4} - 3}{21 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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