Optimal. Leaf size=50 \[ -x+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{1-x^2}\right )}{2 \sqrt {3}}+\frac {1}{3} \tanh ^{-1}(x)+\frac {1}{6} \tanh ^{-1}\left (\frac {x}{1+x^2}\right ) \]
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Rubi [A]
time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.52, number of steps
used = 11, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {327, 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 )-x+\frac {1}{3} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 216
Rule 327
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^6}{1-x^6} \, dx &=-x+\int \frac {1}{1-x^6} \, dx\\ &=-x+\frac {1}{3} \int \frac {1-\frac {x}{2}}{1-x+x^2} \, dx+\frac {1}{3} \int \frac {1+\frac {x}{2}}{1+x+x^2} \, dx+\frac {1}{3} \int \frac {1}{1-x^2} \, dx\\ &=-x+\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\\ &=-x+\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 )\\ &=-x-\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 = 78, normalized size = 1.56 \begin {gather*} \frac {1}{12} \left (-12 x+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 = 0.20, size = 69, normalized size = 1.38
method | result | size |
risch | \(-x -\frac {\ln \left (x -1\right )}{6}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x -\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}+\frac {\ln \left (x +1\right )}{6}+\frac {\ln \left (x^{2}+x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x +\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{6}\) | \(65\) |
default | \(-x +\frac {\ln \left (x^{2}+x +1\right )}{12}+\frac {\arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}-\frac {\ln \left (x -1\right )}{6}+\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x^{2}-x +1\right )}{12}+\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{6}\) | \(69\) |
meijerg | \(\frac {\left (-1\right )^{\frac {5}{6}} \left (6 x \left (-1\right )^{\frac {1}{6}}+\frac {x \left (-1\right )^{\frac {1}{6}} \left (\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}\right )-\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}\right )+\frac {\ln \left (1-\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2-\left (x^{6}\right )^{\frac {1}{6}}}\right )-\frac {\ln \left (1+\left (x^{6}\right )^{\frac {1}{6}}+\left (x^{6}\right )^{\frac {1}{3}}\right )}{2}-\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{6}\right )^{\frac {1}{6}}}{2+\left (x^{6}\right )^{\frac {1}{6}}}\right )\right )}{\left (x^{6}\right )^{\frac {1}{6}}}\right )}{6}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 68, normalized size = 1.36 \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 ) - x + \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 = 0.38, size = 68, normalized size = 1.36 \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 ) - x + \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 85 vs.
\(2 (36) = 72\).
time = 0.12, size = 85, normalized size = 1.70 \begin {gather*} - x - \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 = 1.56, size = 70, normalized size = 1.40 \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 ) - x + \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 = 1.06, size = 93, normalized size = 1.86 \begin {gather*} -x-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{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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