Optimal. Leaf size=113 \[ -\frac {9}{20 x^5}+\frac {9 \log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {9 \log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}}+\frac {1}{4 x^5 \left (x^4+1\right )}+\frac {9}{4 x}-\frac {9 \tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {9 \tan ^{-1}\left (\sqrt {2} x+1\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, 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 204
Rule 290
Rule 297
Rule 325
Rule 617
Rule 628
Rule 1162
Rule 1165
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*}
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Mathematica [A] time = 0.08, size = 103, normalized size = 0.91 \[ \frac {1}{160} \left (-\frac {32}{x^5}+45 \sqrt {2} \log \left (x^2-\sqrt {2} x+1\right )-45 \sqrt {2} \log \left (x^2+\sqrt {2} x+1\right )+\frac {40 x^3}{x^4+1}+\frac {320}{x}-90 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} x\right )+90 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.94, size = 145, normalized size = 1.28 \[ \frac {360 \, x^{8} + 288 \, x^{4} - 180 \, \sqrt {2} {\left (x^{9} + x^{5}\right )} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + \sqrt {2} x + 1} - 1\right ) - 180 \, \sqrt {2} {\left (x^{9} + x^{5}\right )} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - \sqrt {2} x + 1} + 1\right ) - 45 \, \sqrt {2} {\left (x^{9} + x^{5}\right )} \log \left (x^{2} + \sqrt {2} x + 1\right ) + 45 \, \sqrt {2} {\left (x^{9} + x^{5}\right )} \log \left (x^{2} - \sqrt {2} x + 1\right ) - 32}{160 \, {\left (x^{9} + x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 96, normalized size = 0.85 \[ \frac {x^{3}}{4 \, {\left (x^{4} + 1\right )}} + \frac {9}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {9}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {9}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {9}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {10 \, x^{4} - 1}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 80, normalized size = 0.71 \[ \frac {x^{3}}{4 x^{4}+4}+\frac {9 \sqrt {2}\, \arctan \left (\sqrt {2}\, x -1\right )}{16}+\frac {9 \sqrt {2}\, \arctan \left (\sqrt {2}\, x +1\right )}{16}+\frac {9 \sqrt {2}\, \ln \left (\frac {x^{2}-\sqrt {2}\, x +1}{x^{2}+\sqrt {2}\, x +1}\right )}{32}+\frac {2}{x}-\frac {1}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.01, size = 95, normalized size = 0.84 \[ \frac {9}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {9}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {9}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {9}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {45 \, x^{8} + 36 \, x^{4} - 4}{20 \, {\left (x^{9} + x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 55, normalized size = 0.49 \[ \frac {\frac {9\,x^8}{4}+\frac {9\,x^4}{5}-\frac {1}{5}}{x^9+x^5}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {9}{16}-\frac {9}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {9}{16}+\frac {9}{16}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.23, size = 102, normalized size = 0.90 \[ \frac {9 \sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{32} - \frac {9 \sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{32} + \frac {9 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{16} + \frac {9 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{16} + \frac {45 x^{8} + 36 x^{4} - 4}{20 x^{9} + 20 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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