Optimal. Leaf size=99 \[ \frac {\log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {\log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}}+\frac {x^3}{4 \left (x^4+1\right )}-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} x+1\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.05, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {28, 290, 297, 1162, 617, 204, 1165, 628} \[ \frac {x^3}{4 \left (x^4+1\right )}+\frac {\log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {\log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} x+1\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 204
Rule 290
Rule 297
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^2}{1+2 x^4+x^8} \, dx &=\int \frac {x^2}{\left (1+x^4\right )^2} \, dx\\ &=\frac {x^3}{4 \left (1+x^4\right )}+\frac {1}{4} \int \frac {x^2}{1+x^4} \, dx\\ &=\frac {x^3}{4 \left (1+x^4\right )}-\frac {1}{8} \int \frac {1-x^2}{1+x^4} \, dx+\frac {1}{8} \int \frac {1+x^2}{1+x^4} \, dx\\ &=\frac {x^3}{4 \left (1+x^4\right )}+\frac {1}{16} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx+\frac {1}{16} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx+\frac {\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}}\\ &=\frac {x^3}{4 \left (1+x^4\right )}+\frac {\log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{8 \sqrt {2}}\\ &=\frac {x^3}{4 \left (1+x^4\right )}-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {\log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 92, normalized size = 0.93 \[ \frac {1}{32} \left (\sqrt {2} \log \left (x^2-\sqrt {2} x+1\right )-\sqrt {2} \log \left (x^2+\sqrt {2} x+1\right )+\frac {8 x^3}{x^4+1}-2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} x\right )+2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 128, normalized size = 1.29 \[ \frac {8 \, x^{3} - 4 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + \sqrt {2} x + 1} - 1\right ) - 4 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - \sqrt {2} x + 1} + 1\right ) - \sqrt {2} {\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \sqrt {2} {\left (x^{4} + 1\right )} \log \left (x^{2} - \sqrt {2} x + 1\right )}{32 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 84, normalized size = 0.85 \[ \frac {x^{3}}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 70, normalized size = 0.71 \[ \frac {x^{3}}{4 x^{4}+4}+\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x -1\right )}{16}+\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x +1\right )}{16}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\sqrt {2}\, x +1}{x^{2}+\sqrt {2}\, x +1}\right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.06, size = 84, normalized size = 0.85 \[ \frac {x^{3}}{4 \, {\left (x^{4} + 1\right )}} + \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) - \frac {1}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) + \frac {1}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 46, normalized size = 0.46 \[ \frac {x^3}{4\,\left (x^4+1\right )}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{16}-\frac {1}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{16}+\frac {1}{16}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 83, normalized size = 0.84 \[ \frac {x^{3}}{4 x^{4} + 4} + \frac {\sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{32} - \frac {\sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{32} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{16} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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