3.2247 \(\int \frac{x}{\left (1+x+x^2\right )^3} \, dx\)

Optimal. Leaf size=54 \[ -\frac{x+2}{6 \left (x^2+x+1\right )^2}-\frac{2 x+1}{6 \left (x^2+x+1\right )}-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

-(2 + x)/(6*(1 + x + x^2)^2) - (1 + 2*x)/(6*(1 + x + x^2)) - (2*ArcTan[(1 + 2*x)
/Sqrt[3]])/(3*Sqrt[3])

_______________________________________________________________________________________

Rubi [A]  time = 0.0533882, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{x+2}{6 \left (x^2+x+1\right )^2}-\frac{2 x+1}{6 \left (x^2+x+1\right )}-\frac{2 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x + x^2)^3,x]

[Out]

-(2 + x)/(6*(1 + x + x^2)^2) - (1 + 2*x)/(6*(1 + x + x^2)) - (2*ArcTan[(1 + 2*x)
/Sqrt[3]])/(3*Sqrt[3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.06445, size = 53, normalized size = 0.98 \[ - \frac{x + 2}{6 \left (x^{2} + x + 1\right )^{2}} - \frac{2 x + 1}{6 \left (x^{2} + x + 1\right )} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3} + \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x + 2)/(6*(x**2 + x + 1)**2) - (2*x + 1)/(6*(x**2 + x + 1)) - 2*sqrt(3)*atan(s
qrt(3)*(2*x/3 + 1/3))/9

_______________________________________________________________________________________

Mathematica [A]  time = 0.0522228, size = 49, normalized size = 0.91 \[ \frac{1}{18} \left (-\frac{3 \left (2 x^3+3 x^2+4 x+3\right )}{\left (x^2+x+1\right )^2}-4 \sqrt{3} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + x + x^2)^3,x]

[Out]

((-3*(3 + 4*x + 3*x^2 + 2*x^3))/(1 + x + x^2)^2 - 4*Sqrt[3]*ArcTan[(1 + 2*x)/Sqr
t[3]])/18

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 48, normalized size = 0.9 \[{\frac{-2-x}{6\, \left ({x}^{2}+x+1 \right ) ^{2}}}-{\frac{1+2\,x}{6\,{x}^{2}+6\,x+6}}-{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^2+x+1)^3,x)

[Out]

1/6*(-2-x)/(x^2+x+1)^2-1/6*(1+2*x)/(x^2+x+1)-2/9*arctan(1/3*(1+2*x)*3^(1/2))*3^(
1/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.749513, size = 73, normalized size = 1.35 \[ -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{2 \, x^{3} + 3 \, x^{2} + 4 \, x + 3}{6 \,{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*(2*x^3 + 3*x^2 + 4*x + 3)/(x^4
+ 2*x^3 + 3*x^2 + 2*x + 1)

_______________________________________________________________________________________

Fricas [A]  time = 0.200228, size = 103, normalized size = 1.91 \[ -\frac{\sqrt{3}{\left (4 \,{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \sqrt{3}{\left (2 \, x^{3} + 3 \, x^{2} + 4 \, x + 3\right )}\right )}}{18 \,{\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(4*(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)*arctan(1/3*sqrt(3)*(2*x + 1)) +
sqrt(3)*(2*x^3 + 3*x^2 + 4*x + 3))/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)

_______________________________________________________________________________________

Sympy [A]  time = 0.358888, size = 63, normalized size = 1.17 \[ - \frac{2 x^{3} + 3 x^{2} + 4 x + 3}{6 x^{4} + 12 x^{3} + 18 x^{2} + 12 x + 6} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} + \frac{\sqrt{3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**2+x+1)**3,x)

[Out]

-(2*x**3 + 3*x**2 + 4*x + 3)/(6*x**4 + 12*x**3 + 18*x**2 + 12*x + 6) - 2*sqrt(3)
*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.203502, size = 57, normalized size = 1.06 \[ -\frac{2}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{2 \, x^{3} + 3 \, x^{2} + 4 \, x + 3}{6 \,{\left (x^{2} + x + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/6*(2*x^3 + 3*x^2 + 4*x + 3)/(x^2
+ x + 1)^2