Optimal. Leaf size=48 \[ \frac{1}{2 x^2}-\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0239909, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {1593, 325, 200, 31, 634, 618, 204, 628} \[ \frac{1}{2 x^2}-\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 325
Rule 200
Rule 31
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{-x^3+x^6} \, dx &=\int \frac{1}{x^3 \left (-1+x^3\right )} \, dx\\ &=\frac{1}{2 x^2}+\int \frac{1}{-1+x^3} \, dx\\ &=\frac{1}{2 x^2}+\frac{1}{3} \int \frac{1}{-1+x} \, dx+\frac{1}{3} \int \frac{-2-x}{1+x+x^2} \, dx\\ &=\frac{1}{2 x^2}+\frac{1}{3} \log (1-x)-\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 x^2}+\frac{1}{3} \log (1-x)-\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 )\\ &=\frac{1}{2 x^2}-\frac{\tan ^{-1}\left (\frac{1+2 x}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{1}{3} \log (1-x)-\frac{1}{6} \log \left (1+x+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0120447, size = 48, normalized size = 1. \[ \frac{1}{2 x^2}-\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 38, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( -1+x \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{1}{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43038, size = 50, normalized size = 1.04 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2 \, x^{2}} - \frac{1}{6} \, \log \left (x^{2} + 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 = 1.84714, size = 138, normalized size = 2.88 \begin{align*} -\frac{2 \, \sqrt{3} x^{2} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x^{2} \log \left (x^{2} + x + 1\right ) - 2 \, x^{2} \log \left (x - 1\right ) - 3}{6 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.129993, size = 48, normalized size = 1. \begin{align*} \frac{\log{\left (x - 1 \right )}}{3} - \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{1}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.05633, size = 51, normalized size = 1.06 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2 \, x^{2}} - \frac{1}{6} \, \log \left (x^{2} + x + 1\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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