3.2248 \(\int \frac{x^2}{1+x+x^2} \, dx\)

Optimal. Leaf size=32 \[ -\frac{1}{2} \log \left (x^2+x+1\right )+x-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0523575, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{1}{2} \log \left (x^2+x+1\right )+x-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(1 + x + x^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.54613, size = 32, normalized size = 1. \[ x - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**2+x+1),x)

[Out]

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

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Mathematica [A]  time = 0.0127356, size = 32, normalized size = 1. \[ -\frac{1}{2} \log \left (x^2+x+1\right )+x-\frac{\tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(1 + x + x^2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \[ x-{\frac{\ln \left ({x}^{2}+x+1 \right ) }{2}}-{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^2+x+1),x)

[Out]

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

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Maxima [A]  time = 0.750324, size = 36, normalized size = 1.12 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x - \frac{1}{2} \, \log \left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^2 + x + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.204595, size = 50, normalized size = 1.56 \[ \frac{1}{6} \, \sqrt{3}{\left (2 \, \sqrt{3} x - \sqrt{3} \log \left (x^{2} + x + 1\right ) - 2 \, \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^2 + x + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.212872, size = 36, normalized size = 1.12 \[ x - \frac{\log{\left (x^{2} + x + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**2+x+1),x)

[Out]

x - 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 = 0.204056, size = 36, normalized size = 1.12 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + x - \frac{1}{2} \,{\rm ln}\left (x^{2} + x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^2 + x + 1),x, algorithm="giac")

[Out]

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