Optimal. Leaf size=97 \[ \frac {x}{4 \left (x^4+1\right )}-\frac {3 \log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}+\frac {3 \log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} x+1\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {28, 199, 211, 1165, 628, 1162, 617, 204} \[ \frac {x}{4 \left (x^4+1\right )}-\frac {3 \log \left (x^2-\sqrt {2} x+1\right )}{16 \sqrt {2}}+\frac {3 \log \left (x^2+\sqrt {2} x+1\right )}{16 \sqrt {2}}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {3 \tan ^{-1}\left (\sqrt {2} x+1\right )}{8 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 28
Rule 199
Rule 204
Rule 211
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{1+2 x^4+x^8} \, dx &=\int \frac {1}{\left (1+x^4\right )^2} \, dx\\ &=\frac {x}{4 \left (1+x^4\right )}+\frac {3}{4} \int \frac {1}{1+x^4} \, dx\\ &=\frac {x}{4 \left (1+x^4\right )}+\frac {3}{8} \int \frac {1-x^2}{1+x^4} \, dx+\frac {3}{8} \int \frac {1+x^2}{1+x^4} \, dx\\ &=\frac {x}{4 \left (1+x^4\right )}+\frac {3}{16} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx+\frac {3}{16} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx-\frac {3 \int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}}-\frac {3 \int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{16 \sqrt {2}}\\ &=\frac {x}{4 \left (1+x^4\right )}-\frac {3 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{8 \sqrt {2}}\\ &=\frac {x}{4 \left (1+x^4\right )}-\frac {3 \tan ^{-1}\left (1-\sqrt {2} x\right )}{8 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\sqrt {2} x\right )}{8 \sqrt {2}}-\frac {3 \log \left (1-\sqrt {2} x+x^2\right )}{16 \sqrt {2}}+\frac {3 \log \left (1+\sqrt {2} x+x^2\right )}{16 \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 91, normalized size = 0.94 \[ \frac {1}{32} \left (\frac {8 x}{x^4+1}-3 \sqrt {2} \log \left (x^2-\sqrt {2} x+1\right )+3 \sqrt {2} \log \left (x^2+\sqrt {2} x+1\right )-6 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} x\right )+6 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 127, normalized size = 1.31 \[ -\frac {12 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + \sqrt {2} x + 1} - 1\right ) + 12 \, \sqrt {2} {\left (x^{4} + 1\right )} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - \sqrt {2} x + 1} + 1\right ) - 3 \, \sqrt {2} {\left (x^{4} + 1\right )} \log \left (x^{2} + \sqrt {2} x + 1\right ) + 3 \, \sqrt {2} {\left (x^{4} + 1\right )} \log \left (x^{2} - \sqrt {2} x + 1\right ) - 8 \, x}{32 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 82, normalized size = 0.85 \[ \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {3}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {3}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {x}{4 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 68, normalized size = 0.70 \[ \frac {x}{4 x^{4}+4}+\frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, x -1\right )}{16}+\frac {3 \sqrt {2}\, \arctan \left (\sqrt {2}\, x +1\right )}{16}+\frac {3 \sqrt {2}\, \ln \left (\frac {x^{2}+\sqrt {2}\, x +1}{x^{2}-\sqrt {2}\, x +1}\right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.00, size = 82, normalized size = 0.85 \[ \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {3}{16} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {3}{32} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {3}{32} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {x}{4 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 44, normalized size = 0.45 \[ \frac {x}{4\,\left (x^4+1\right )}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{16}+\frac {3}{16}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {3}{16}-\frac {3}{16}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.20, size = 88, normalized size = 0.91 \[ \frac {x}{4 x^{4} + 4} - \frac {3 \sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{32} + \frac {3 \sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{32} + \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{16} + \frac {3 \sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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