Optimal. Leaf size=97 \[ -\frac {\log \left (x^2-\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {\log \left (x^2+\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {1}{4} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} x+1\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {214, 212, 206, 203, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\log \left (x^2-\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {\log \left (x^2+\sqrt {2} x+1\right )}{8 \sqrt {2}}+\frac {1}{4} \tan ^{-1}(x)-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} x+1\right )}{4 \sqrt {2}}+\frac {1}{4} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 204
Rule 206
Rule 211
Rule 212
Rule 214
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {1}{1-x^8} \, dx &=\frac {1}{2} \int \frac {1}{1-x^4} \, dx+\frac {1}{2} \int \frac {1}{1+x^4} \, dx\\ &=\frac {1}{4} \int \frac {1}{1-x^2} \, dx+\frac {1}{4} \int \frac {1}{1+x^2} \, dx+\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 {1}{4} \tan ^{-1}(x)+\frac {1}{4} \tanh ^{-1}(x)+\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 {1}{4} \tan ^{-1}(x)+\frac {1}{4} \tanh ^{-1}(x)-\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 {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{4 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{4 \sqrt {2}}\\ &=\frac {1}{4} \tan ^{-1}(x)-\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 {1}{4} \tanh ^{-1}(x)-\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 = 0.03, size = 98, normalized size = 1.01 \[ \frac {1}{16} \left (-\sqrt {2} \log \left (x^2-\sqrt {2} x+1\right )+\sqrt {2} \log \left (x^2+\sqrt {2} x+1\right )-2 \log (1-x)+2 \log (x+1)+4 \tan ^{-1}(x)-2 \sqrt {2} \tan ^{-1}\left (1-\sqrt {2} x\right )+2 \sqrt {2} \tan ^{-1}\left (\sqrt {2} x+1\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 111, normalized size = 1.14 \[ -\frac {1}{4} \, \sqrt {2} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} + \sqrt {2} x + 1} - 1\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - \sqrt {2} x + 1} + 1\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {1}{4} \, \arctan \relax (x) + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 90, normalized size = 0.93 \[ \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {1}{4} \, \arctan \relax (x) + \frac {1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 66, normalized size = 0.68 \[ \frac {\arctanh \relax (x )}{4}+\frac {\arctan \relax (x )}{4}+\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x -1\right )}{8}+\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x +1\right )}{8}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}+\sqrt {2}\, x +1}{x^{2}-\sqrt {2}\, x +1}\right )}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.27, size = 88, normalized size = 0.91 \[ \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{8} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (x^{2} + \sqrt {2} x + 1\right ) - \frac {1}{16} \, \sqrt {2} \log \left (x^{2} - \sqrt {2} x + 1\right ) + \frac {1}{4} \, \arctan \relax (x) + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 45, normalized size = 0.46 \[ \frac {\mathrm {atan}\relax (x)}{4}-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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