Optimal. Leaf size=32 \[ \frac{1}{2} \log \left (x^2-x+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
[Out]
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Rubi [A] time = 0.046202, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{1}{2} \log \left (x^2-x+1\right )+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(-1 + x)/(1 - x + x^2),x]
[Out]
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Rubi in Sympy [A] time = 5.62089, size = 31, normalized size = 0.97 \[ \frac{\log{\left (x^{2} - x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+x)/(x**2-x+1),x)
[Out]
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Mathematica [A] time = 0.0158734, size = 33, normalized size = 1.03 \[ \frac{1}{2} \log \left (x^2-x+1\right )-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(-1 + x)/(1 - x + x^2),x]
[Out]
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Maple [A] time = 0.002, size = 29, normalized size = 0.9 \[{\frac{\ln \left ({x}^{2}-x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+x)/(x^2-x+1),x)
[Out]
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Maxima [A] time = 0.889137, size = 38, normalized size = 1.19 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \, \log \left (x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - 1)/(x^2 - x + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272673, size = 43, normalized size = 1.34 \[ \frac{1}{6} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} - x + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - 1)/(x^2 - x + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.208901, size = 34, normalized size = 1.06 \[ \frac{\log{\left (x^{2} - x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-1+x)/(x**2-x+1),x)
[Out]
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GIAC/XCAS [A] time = 0.259034, size = 38, normalized size = 1.19 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x - 1)/(x^2 - x + 1),x, algorithm="giac")
[Out]