### 3.163 $$\int \frac{1+x+x^2}{(1-x+x^2)^2} \, dx$$

Optimal. Leaf size=41 $-\frac{2 (2-x)}{3 \left (x^2-x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}$

[Out]

(-2*(2 - x))/(3*(1 - x + x^2)) - (10*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0329209, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.235, Rules used = {1660, 12, 618, 204} $-\frac{2 (2-x)}{3 \left (x^2-x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(1 + x + x^2)/(1 - x + x^2)^2,x]

[Out]

(-2*(2 - x))/(3*(1 - x + x^2)) - (10*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[3])

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1+x+x^2}{\left (1-x+x^2\right )^2} \, dx &=-\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{1}{3} \int \frac{5}{1-x+x^2} \, dx\\ &=-\frac{2 (2-x)}{3 \left (1-x+x^2\right )}+\frac{5}{3} \int \frac{1}{1-x+x^2} \, dx\\ &=-\frac{2 (2-x)}{3 \left (1-x+x^2\right )}-\frac{10}{3} \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,-1+2 x\right )\\ &=-\frac{2 (2-x)}{3 \left (1-x+x^2\right )}-\frac{10 \tan ^{-1}\left (\frac{1-2 x}{\sqrt{3}}\right )}{3 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.021879, size = 39, normalized size = 0.95 $\frac{2 (x-2)}{3 \left (x^2-x+1\right )}+\frac{10 \tan ^{-1}\left (\frac{2 x-1}{\sqrt{3}}\right )}{3 \sqrt{3}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*(-2 + x))/(3*(1 - x + x^2)) + (10*ArcTan[(-1 + 2*x)/Sqrt[3]])/(3*Sqrt[3])

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 34, normalized size = 0.8 \begin{align*}{\frac{1}{{x}^{2}-x+1} \left ({\frac{2\,x}{3}}-{\frac{4}{3}} \right ) }+{\frac{10\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x-1 \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+x+1)/(x^2-x+1)^2,x)

[Out]

(2/3*x-4/3)/(x^2-x+1)+10/9*3^(1/2)*arctan(1/3*(2*x-1)*3^(1/2))

________________________________________________________________________________________

Maxima [A]  time = 1.51796, size = 43, normalized size = 1.05 \begin{align*} \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{2 \,{\left (x - 2\right )}}{3 \,{\left (x^{2} - x + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)/(x^2-x+1)^2,x, algorithm="maxima")

[Out]

10/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 2/3*(x - 2)/(x^2 - x + 1)

________________________________________________________________________________________

Fricas [A]  time = 1.90323, size = 115, normalized size = 2.8 \begin{align*} \frac{2 \,{\left (5 \, \sqrt{3}{\left (x^{2} - x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + 3 \, x - 6\right )}}{9 \,{\left (x^{2} - x + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)/(x^2-x+1)^2,x, algorithm="fricas")

[Out]

2/9*(5*sqrt(3)*(x^2 - x + 1)*arctan(1/3*sqrt(3)*(2*x - 1)) + 3*x - 6)/(x^2 - x + 1)

________________________________________________________________________________________

Sympy [A]  time = 0.125305, size = 41, normalized size = 1. \begin{align*} \frac{2 x - 4}{3 x^{2} - 3 x + 3} + \frac{10 \sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{9} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+x+1)/(x**2-x+1)**2,x)

[Out]

(2*x - 4)/(3*x**2 - 3*x + 3) + 10*sqrt(3)*atan(2*sqrt(3)*x/3 - sqrt(3)/3)/9

________________________________________________________________________________________

Giac [A]  time = 1.28502, size = 43, normalized size = 1.05 \begin{align*} \frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{2 \,{\left (x - 2\right )}}{3 \,{\left (x^{2} - x + 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+x+1)/(x^2-x+1)^2,x, algorithm="giac")

[Out]

10/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 2/3*(x - 2)/(x^2 - x + 1)