Optimal. Leaf size=41 \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.0395499, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.857 \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(-1 + x^3)^(-1),x]
[Out]
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Rubi in Sympy [A] time = 2.50005, size = 37, normalized size = 0.9 \[ \frac{\log{\left (- x + 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x**3-1),x)
[Out]
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Mathematica [A] time = 0.00903536, size = 41, normalized size = 1. \[ -\frac{1}{6} \log \left (x^2+x+1\right )+\frac{1}{3} \log (1-x)-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + x^3)^(-1),x]
[Out]
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Maple [A] time = 0.003, size = 33, normalized size = 0.8 \[{\frac{\ln \left ( -1+x \right ) }{3}}-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{6}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x^3-1),x)
[Out]
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Maxima [A] time = 1.51389, size = 43, normalized size = 1.05 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{3} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^3 - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203758, size = 53, normalized size = 1.29 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} + x + 1\right ) - 2 \, \sqrt{3} \log \left (x - 1\right ) + 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^3 - 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.171504, size = 41, normalized size = 1. \[ \frac{\log{\left (x - 1 \right )}}{3} - \frac{\log{\left (x^{2} + x + 1 \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x**3-1),x)
[Out]
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GIAC/XCAS [A] time = 0.205249, size = 45, normalized size = 1.1 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{1}{6} \,{\rm ln}\left (x^{2} + x + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x^3 - 1),x, algorithm="giac")
[Out]