3.907 \(\int \frac{1+x}{\left (1-x+x^2\right )^3} \, dx\)

Optimal. Leaf size=58 \[ -\frac{1-2 x}{2 \left (x^2-x+1\right )}-\frac{1-x}{2 \left (x^2-x+1\right )^2}-\frac{2 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0480157, antiderivative size = 58, 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 \[ -\frac{1-2 x}{2 \left (x^2-x+1\right )}-\frac{1-x}{2 \left (x^2-x+1\right )^2}-\frac{2 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/(1 - x + x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 4.95726, size = 53, normalized size = 0.91 \[ - \frac{- 3 x + 3}{6 \left (x^{2} - x + 1\right )^{2}} - \frac{- 2 x + 1}{2 \left (x^{2} - x + 1\right )} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(x**2-x+1)**3,x)

[Out]

-(-3*x + 3)/(6*(x**2 - x + 1)**2) - (-2*x + 1)/(2*(x**2 - x + 1)) + 2*sqrt(3)*at
an(sqrt(3)*(2*x/3 - 1/3))/3

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Mathematica [A]  time = 0.0528474, size = 49, normalized size = 0.84 \[ \frac{2 x^3-3 x^2+4 x-2}{2 \left (x^2-x+1\right )^2}+\frac{2 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)/(1 - x + x^2)^3,x]

[Out]

(-2 + 4*x - 3*x^2 + 2*x^3)/(2*(1 - x + x^2)^2) + (2*ArcTan[(-1 + 2*x)/Sqrt[3]])/
Sqrt[3]

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Maple [A]  time = 0.006, size = 52, normalized size = 0.9 \[{\frac{3\,x-3}{6\, \left ({x}^{2}-x+1 \right ) ^{2}}}+{\frac{2\,x-1}{2\,{x}^{2}-2\,x+2}}+{\frac{2\,\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^2-x+1)^3,x)

[Out]

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

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Maxima [A]  time = 0.764578, size = 73, normalized size = 1.26 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{2 \, x^{3} - 3 \, x^{2} + 4 \, x - 2}{2 \,{\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^2 - x + 1)^3,x, algorithm="maxima")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*(2*x^3 - 3*x^2 + 4*x - 2)/(x^4 -
2*x^3 + 3*x^2 - 2*x + 1)

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Fricas [A]  time = 0.3029, size = 103, normalized size = 1.78 \[ \frac{\sqrt{3}{\left (4 \,{\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \sqrt{3}{\left (2 \, x^{3} - 3 \, x^{2} + 4 \, x - 2\right )}\right )}}{6 \,{\left (x^{4} - 2 \, x^{3} + 3 \, x^{2} - 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^2 - x + 1)^3,x, algorithm="fricas")

[Out]

1/6*sqrt(3)*(4*(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)*arctan(1/3*sqrt(3)*(2*x - 1)) + s
qrt(3)*(2*x^3 - 3*x^2 + 4*x - 2))/(x^4 - 2*x^3 + 3*x^2 - 2*x + 1)

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Sympy [A]  time = 0.358881, size = 61, normalized size = 1.05 \[ \frac{2 x^{3} - 3 x^{2} + 4 x - 2}{2 x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 2} + \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(x**2-x+1)**3,x)

[Out]

(2*x**3 - 3*x**2 + 4*x - 2)/(2*x**4 - 4*x**3 + 6*x**2 - 4*x + 2) + 2*sqrt(3)*ata
n(2*sqrt(3)*x/3 - sqrt(3)/3)/3

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GIAC/XCAS [A]  time = 0.268956, size = 59, normalized size = 1.02 \[ \frac{2}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{2 \, x^{3} - 3 \, x^{2} + 4 \, x - 2}{2 \,{\left (x^{2} - x + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(x^2 - x + 1)^3,x, algorithm="giac")

[Out]

2/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 1/2*(2*x^3 - 3*x^2 + 4*x - 2)/(x^2 -
x + 1)^2