Optimal. Leaf size=73 \[ \frac {\tan ^{-1}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{3} \tanh ^{-1}(x)+\frac {1}{12} \log \left (1-x+x^2\right )-\frac {1}{12} \log \left (1+x+x^2\right ) \]
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Rubi [A]
time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, 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 = {216, 648, 632,
210, 642, 212} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-2 x}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{12} \log \left (x^2-x+1\right )-\frac {1}{12} \log \left (x^2+x+1\right )-\frac {1}{3} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 632
Rule 642
Rule 648
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} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x\right )+\frac {1}{2} \text {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 = 1.03 \begin {gather*} \frac {1}{12} \left (-2 \sqrt {3} \tan ^{-1}\left (\frac {-1+2 x}{\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )+2 \log (1-x)-2 \log (1+x)+\log \left (1-x+x^2\right )-\log \left (1+x+x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 66, normalized size = 0.90
method | result | size |
risch | \(-\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \sqrt {3}\, \left (x +\frac {1}{2}\right )}{3}\right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\) | \(62\) |
default | \(\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (x^{2}+x +1\right )}{12}-\frac {\arctan \left (\frac {\left (1+2 x \right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (1+x \right )}{6}+\frac {\ln \left (x^{2}-x +1\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(66\) |
meijerg | \(\frac {x \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}\right )-\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}\right )+\frac {\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{6 \left (x^{6}\right )^{\frac {1}{6}}}\) | \(116\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.62, size = 65, normalized size = 0.89 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.79, size = 65, normalized size = 0.89 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.12, size = 83, normalized size = 1.14 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.98, size = 67, normalized size = 0.92 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.09, size = 88, normalized size = 1.21 \begin {gather*} -\frac {\mathrm {atanh}\left (x\right )}{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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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