Optimal. Leaf size=97 \[ \frac {x}{2 \left (1+x^4\right )}-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} x\right )}{4 \sqrt {2}}-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+x^2\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {28, 393, 217,
1179, 642, 1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\text {ArcTan}\left (\sqrt {2} x+1\right )}{4 \sqrt {2}}+\frac {x}{2 \left (x^4+1\right )}-\frac {\log \left (x^2-\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {\log \left (x^2+\sqrt {2} x+1\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 210
Rule 217
Rule 393
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1-x^4}{1+2 x^4+x^8} \, dx &=\int \frac {1-x^4}{\left (1+x^4\right )^2} \, dx\\ &=\frac {x}{2 \left (1+x^4\right )}+\frac {1}{2} \int \frac {1}{1+x^4} \, dx\\ &=\frac {x}{2 \left (1+x^4\right )}+\frac {1}{4} \int \frac {1-x^2}{1+x^4} \, dx+\frac {1}{4} \int \frac {1+x^2}{1+x^4} \, dx\\ &=\frac {x}{2 \left (1+x^4\right )}+\frac {1}{8} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx+\frac {1}{8} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx-\frac {\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx}{8 \sqrt {2}}-\frac {\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx}{8 \sqrt {2}}\\ &=\frac {x}{2 \left (1+x^4\right )}-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+x^2\right )}{8 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{4 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{4 \sqrt {2}}\\ &=\frac {x}{2 \left (1+x^4\right )}-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} x\right )}{4 \sqrt {2}}-\frac {\log \left (1-\sqrt {2} x+x^2\right )}{8 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} x+x^2\right )}{8 \sqrt {2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 90, normalized size = 0.93 \begin {gather*} \frac {1}{16} \left (\frac {8 x}{1+x^4}-2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} x\right )+2 \sqrt {2} \tan ^{-1}\left (1+\sqrt {2} x\right )-\sqrt {2} \log \left (1-\sqrt {2} x+x^2\right )+\sqrt {2} \log \left (1+\sqrt {2} x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 63, normalized size = 0.65
method | result | size |
risch | \(\frac {x}{2 x^{4}+2}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{8}\) | \(33\) |
default | \(\frac {x}{2 x^{4}+2}+\frac {\sqrt {2}\, \left (\ln \left (\frac {1+x^{2}+\sqrt {2}\, x}{1+x^{2}-\sqrt {2}\, x}\right )+2 \arctan \left (\sqrt {2}\, x +1\right )+2 \arctan \left (\sqrt {2}\, x -1\right )\right )}{16}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 82, normalized size = 0.85 \begin {gather*} \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {x}{2 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 131, normalized size = 1.35 \begin {gather*} -\frac {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 (4 \, x^{2} + 4 \, \sqrt {2} x + 4\right ) + \sqrt {2} {\left (x^{4} + 1\right )} \log \left (4 \, x^{2} - 4 \, \sqrt {2} x + 4\right ) - 8 \, x}{16 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 82, normalized size = 0.85 \begin {gather*} \frac {x}{2 x^{4} + 2} - \frac {\sqrt {2} \log {\left (x^{2} - \sqrt {2} x + 1 \right )}}{16} + \frac {\sqrt {2} \log {\left (x^{2} + \sqrt {2} x + 1 \right )}}{16} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x - 1 \right )}}{8} + \frac {\sqrt {2} \operatorname {atan}{\left (\sqrt {2} x + 1 \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.00, size = 82, normalized size = 0.85 \begin {gather*} \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {x}{2 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.62, size = 44, normalized size = 0.45 \begin {gather*} \frac {x}{2\,\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}{8}+\frac {1}{8}{}\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}{8}-\frac {1}{8}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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