3.415 \(\int \frac{-1+x}{1-x+x^2} \, dx\)

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]

ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] + Log[1 - x + x^2]/2

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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]

ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] + Log[1 - x + x^2]/2

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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]

log(x**2 - x + 1)/2 - sqrt(3)*atan(sqrt(3)*(2*x/3 - 1/3))/3

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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]

-(ArcTan[(-1 + 2*x)/Sqrt[3]]/Sqrt[3]) + Log[1 - x + x^2]/2

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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]

1/2*ln(x^2-x+1)-1/3*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

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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]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*log(x^2 - x + 1)

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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]

1/6*sqrt(3)*(sqrt(3)*log(x^2 - x + 1) - 2*arctan(1/3*sqrt(3)*(2*x - 1)))

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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]

log(x**2 - x + 1)/2 - sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/3

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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]

-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*ln(x^2 - x + 1)