Optimal. Leaf size=47 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{1-x^2}\right )}{2 \sqrt {3}}-\frac {1}{6} \tanh ^{-1}\left (\frac {x}{x^2+1}\right )-\frac {1}{3} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.10, antiderivative size = 73, normalized size of antiderivative = 1.55, number of steps used = 10, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {210, 634, 618, 204, 628, 206} \[ \frac {1}{12} \log \left (x^2-x+1\right )-\frac {1}{12} \log \left (x^2+x+1\right )+\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{3} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 210
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{-1+x^6} \, dx &=-\left (\frac {1}{3} \int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx\right )-\frac {1}{3} \int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx-\frac {1}{3} \int \frac {1}{1-x^2} \, dx\\ &=-\frac {1}{3} \tanh ^{-1}(x)+\frac {1}{12} \int \frac {-1+2 x}{1-x+x^2} \, dx-\frac {1}{12} \int \frac {1+2 x}{1+x+x^2} \, dx-\frac {1}{4} \int \frac {1}{1-x+x^2} \, dx-\frac {1}{4} \int \frac {1}{1+x+x^2} \, dx\\ &=-\frac {1}{3} \tanh ^{-1}(x)+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {1}{12} \log \left (1+x+x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )\\ &=\frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{3} \tanh ^{-1}(x)+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {1}{12} \log \left (1+x+x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 75, normalized size = 1.60 \[ \frac {1}{12} \left (\log \left (x^2-x+1\right )-\log \left (x^2+x+1\right )+2 \log (1-x)-2 \log (x+1)-2 \sqrt {3} \tan ^{-1}\left (\frac {2 x-1}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 65, normalized size = 1.38 \[ -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) + \frac {1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.01, size = 67, normalized size = 1.43 \[ -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 66, normalized size = 1.40 \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{6}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x -1\right )}{6}-\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\ln \left (x^{2}+x +1\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 65, normalized size = 1.38 \[ -\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) - \frac {1}{12} \, \log \left (x^{2} + x + 1\right ) + \frac {1}{12} \, \log \left (x^{2} - x + 1\right ) - \frac {1}{6} \, \log \left (x + 1\right ) + \frac {1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 88, normalized size = 1.87 \[ -\frac {\mathrm {atanh}\relax (x)}{3}-\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{1+\sqrt {3}\,1{}\mathrm {i}}+\frac {\sqrt {3}\,x}{1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{6}+\frac {1}{6}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,1{}\mathrm {i}}{-1+\sqrt {3}\,1{}\mathrm {i}}-\frac {\sqrt {3}\,x}{-1+\sqrt {3}\,1{}\mathrm {i}}\right )\,\left (\frac {\sqrt {3}}{6}-\frac {1}{6}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.25, size = 83, normalized size = 1.77 \[ \frac {\log {\left (x - 1 \right )}}{6} - \frac {\log {\left (x + 1 \right )}}{6} + \frac {\log {\left (x^{2} - x + 1 \right )}}{12} - \frac {\log {\left (x^{2} + x + 1 \right )}}{12} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} - \frac {\sqrt {3}}{3} \right )}}{6} - \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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