Optimal. Leaf size=80 \[ -\frac{1}{5 x^5}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.110031, antiderivative size = 80, 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 = {325, 210, 634, 618, 204, 628, 206} \[ -\frac{1}{5 x^5}-\frac{1}{12} \log \left (x^2-x+1\right )+\frac{1}{12} \log \left (x^2+x+1\right )-\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{2 \sqrt{3}}+\frac{1}{3} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 325
Rule 210
Rule 634
Rule 618
Rule 204
Rule 628
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^6 \left (1-x^6\right )} \, dx &=-\frac{1}{5 x^5}+\int \frac{1}{1-x^6} \, dx\\ &=-\frac{1}{5 x^5}+\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\\ &=-\frac{1}{5 x^5}+\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}{5 x^5}+\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{1}{5 x^5}-\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*}
Mathematica [A] time = 0.0234388, size = 82, normalized size = 1.02 \[ \frac{1}{60} \left (-\frac{12}{x^5}-5 \log \left (x^2-x+1\right )+5 \log \left (x^2+x+1\right )-10 \log (1-x)+10 \log (x+1)+10 \sqrt{3} \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )+10 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 71, normalized size = 0.9 \begin{align*} -{\frac{1}{5\,{x}^{5}}}-{\frac{\ln \left ({x}^{2}-x+1 \right ) }{12}}+{\frac{\sqrt{3}}{6}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) }+{\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}}{6}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4865, size = 95, normalized size = 1.19 \begin{align*} \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}{5 \, x^{5}} + \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8117, size = 269, normalized size = 3.36 \begin{align*} \frac{10 \, \sqrt{3} x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 10 \, \sqrt{3} x^{5} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 5 \, x^{5} \log \left (x^{2} + x + 1\right ) - 5 \, x^{5} \log \left (x^{2} - x + 1\right ) + 10 \, x^{5} \log \left (x + 1\right ) - 10 \, x^{5} \log \left (x - 1\right ) - 12}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.257557, size = 90, normalized size = 1.12 \begin{align*} - \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} - \frac{1}{5 x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12217, size = 97, normalized size = 1.21 \begin{align*} \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}{5 \, x^{5}} + \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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