Optimal. Leaf size=69 \[ \frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{6} \log \left (x^2+x+1\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.11063, antiderivative size = 69, normalized size of antiderivative = 1., 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 = {388, 210, 634, 618, 204, 628, 206} \[ \frac{1}{6} \log \left (x^2-x+1\right )-\frac{1}{6} \log \left (x^2+x+1\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 388
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{1+x^6}{-1+x^6} \, dx &=x+2 \int \frac{1}{-1+x^6} \, dx\\ &=x-\frac{2}{3} \int \frac{1}{1-x^2} \, dx-\frac{2}{3} \int \frac{1-\frac{x}{2}}{1-x+x^2} \, dx-\frac{2}{3} \int \frac{1+\frac{x}{2}}{1+x+x^2} \, dx\\ &=x-\frac{2}{3} \tanh ^{-1}(x)+\frac{1}{6} \int \frac{-1+2 x}{1-x+x^2} \, dx-\frac{1}{6} \int \frac{1+2 x}{1+x+x^2} \, dx-\frac{1}{2} \int \frac{1}{1-x+x^2} \, dx-\frac{1}{2} \int \frac{1}{1+x+x^2} \, dx\\ &=x-\frac{2}{3} \tanh ^{-1}(x)+\frac{1}{6} \log \left (1-x+x^2\right )-\frac{1}{6} \log \left (1+x+x^2\right )+\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )+\operatorname{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 )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}(x)+\frac{1}{6} \log \left (1-x+x^2\right )-\frac{1}{6} \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0135999, size = 78, normalized size = 1.13 \[ \frac{1}{6} \left (\log \left (x^2-x+1\right )-\log \left (x^2+x+1\right )+6 x+2 \log (1-x)-2 \log (x+1)-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 67, normalized size = 1. \begin{align*} x+{\frac{\ln \left ( x-1 \right ) }{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) }+{\frac{\ln \left ({x}^{2}-x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }-{\frac{\ln \left ( 1+x \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.51663, size = 89, normalized size = 1.29 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x - \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{3} \, \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02847, size = 232, normalized size = 3.36 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x - \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{3} \, \log \left (x + 1\right ) + \frac{1}{3} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.221452, size = 85, normalized size = 1.23 \begin{align*} x + \frac{\log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x + 1 \right )}}{3} + \frac{\log{\left (x^{2} - x + 1 \right )}}{6} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11525, size = 92, normalized size = 1.33 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + x - \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) - \frac{1}{3} \, \log \left ({\left | x + 1 \right |}\right ) + \frac{1}{3} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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