Optimal. Leaf size=106 \[ \frac{1}{4 x^3 \left (x^4+1\right )}-\frac{7}{12 x^3}+\frac{7 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{7 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.0521337, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 12, 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} \[ \frac{1}{4 x^3 \left (x^4+1\right )}-\frac{7}{12 x^3}+\frac{7 \log \left (x^2-\sqrt{2} x+1\right )}{16 \sqrt{2}}-\frac{7 \log \left (x^2+\sqrt{2} x+1\right )}{16 \sqrt{2}}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{7 \tan ^{-1}\left (\sqrt{2} x+1\right )}{8 \sqrt{2}} \]
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^4 \left (1+2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^4 \left (1+x^4\right )^2} \, dx\\ &=\frac{1}{4 x^3 \left (1+x^4\right )}+\frac{7}{4} \int \frac{1}{x^4 \left (1+x^4\right )} \, dx\\ &=-\frac{7}{12 x^3}+\frac{1}{4 x^3 \left (1+x^4\right )}-\frac{7}{4} \int \frac{1}{1+x^4} \, dx\\ &=-\frac{7}{12 x^3}+\frac{1}{4 x^3 \left (1+x^4\right )}-\frac{7}{8} \int \frac{1-x^2}{1+x^4} \, dx-\frac{7}{8} \int \frac{1+x^2}{1+x^4} \, dx\\ &=-\frac{7}{12 x^3}+\frac{1}{4 x^3 \left (1+x^4\right )}-\frac{7}{16} \int \frac{1}{1-\sqrt{2} x+x^2} \, dx-\frac{7}{16} \int \frac{1}{1+\sqrt{2} x+x^2} \, dx+\frac{7 \int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}+\frac{7 \int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx}{16 \sqrt{2}}\\ &=-\frac{7}{12 x^3}+\frac{1}{4 x^3 \left (1+x^4\right )}+\frac{7 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}-\frac{7 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} x\right )}{8 \sqrt{2}}\\ &=-\frac{7}{12 x^3}+\frac{1}{4 x^3 \left (1+x^4\right )}+\frac{7 \tan ^{-1}\left (1-\sqrt{2} x\right )}{8 \sqrt{2}}-\frac{7 \tan ^{-1}\left (1+\sqrt{2} x\right )}{8 \sqrt{2}}+\frac{7 \log \left (1-\sqrt{2} x+x^2\right )}{16 \sqrt{2}}-\frac{7 \log \left (1+\sqrt{2} x+x^2\right )}{16 \sqrt{2}}\\ \end{align*}
Mathematica [A] time = 0.0738577, size = 96, normalized size = 0.91 \[ \frac{1}{96} \left (-\frac{24 x}{x^4+1}-\frac{32}{x^3}+21 \sqrt{2} \log \left (x^2-\sqrt{2} x+1\right )-21 \sqrt{2} \log \left (x^2+\sqrt{2} x+1\right )+42 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} x\right )-42 \sqrt{2} \tan ^{-1}\left (\sqrt{2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 73, normalized size = 0.7 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-{\frac{x}{4\,{x}^{4}+4}}-{\frac{7\,\arctan \left ( 1+x\sqrt{2} \right ) \sqrt{2}}{16}}-{\frac{7\,\arctan \left ( -1+x\sqrt{2} \right ) \sqrt{2}}{16}}-{\frac{7\,\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 = 1.49873, size = 122, normalized size = 1.15 \begin{align*} -\frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{7}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{7}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{7 \, x^{4} + 4}{12 \,{\left (x^{7} + x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52469, size = 406, normalized size = 3.83 \begin{align*} -\frac{56 \, x^{4} - 84 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} + \sqrt{2} x + 1} - 1\right ) - 84 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \arctan \left (-\sqrt{2} x + \sqrt{2} \sqrt{x^{2} - \sqrt{2} x + 1} + 1\right ) + 21 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \log \left (x^{2} + \sqrt{2} x + 1\right ) - 21 \, \sqrt{2}{\left (x^{7} + x^{3}\right )} \log \left (x^{2} - \sqrt{2} x + 1\right ) + 32}{96 \,{\left (x^{7} + x^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.219176, size = 97, normalized size = 0.92 \begin{align*} - \frac{7 x^{4} + 4}{12 x^{7} + 12 x^{3}} + \frac{7 \sqrt{2} \log{\left (x^{2} - \sqrt{2} x + 1 \right )}}{32} - \frac{7 \sqrt{2} \log{\left (x^{2} + \sqrt{2} x + 1 \right )}}{32} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - 1 \right )}}{16} - \frac{7 \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 = 1.12046, size = 117, normalized size = 1.1 \begin{align*} -\frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x + \sqrt{2}\right )}\right ) - \frac{7}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - \sqrt{2}\right )}\right ) - \frac{7}{32} \, \sqrt{2} \log \left (x^{2} + \sqrt{2} x + 1\right ) + \frac{7}{32} \, \sqrt{2} \log \left (x^{2} - \sqrt{2} x + 1\right ) - \frac{x}{4 \,{\left (x^{4} + 1\right )}} - \frac{1}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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