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

Optimal. Leaf size=39 \[ -\frac{5 x+7}{3 \left (x^2+x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

[Out]

-(7 + 5*x)/(3*(1 + x + x^2)) - (10*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3])

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Rubi [A]  time = 0.0131921, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {638, 618, 204} \[ -\frac{5 x+7}{3 \left (x^2+x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 3*x)/(1 + x + x^2)^2,x]

[Out]

-(7 + 5*x)/(3*(1 + x + x^2)) - (10*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3])

Rule 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

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+3 x}{\left (1+x+x^2\right )^2} \, dx &=-\frac{7+5 x}{3 \left (1+x+x^2\right )}-\frac{5}{3} \int \frac{1}{1+x+x^2} \, dx\\ &=-\frac{7+5 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{7+5 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 = 0.0225667, size = 39, normalized size = 1. \[ \frac{-5 x-7}{3 \left (x^2+x+1\right )}-\frac{10 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7 - 5*x)/(3*(1 + x + x^2)) - (10*ArcTan[(1 + 2*x)/Sqrt[3]])/(3*Sqrt[3])

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Maple [A]  time = 0.004, size = 33, normalized size = 0.9 \begin{align*}{\frac{-7-5\,x}{3\,{x}^{2}+3\,x+3}}-{\frac{10\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{3}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+3*x)/(x^2+x+1)^2,x)

[Out]

1/3*(-7-5*x)/(x^2+x+1)-10/9*arctan(1/3*(1+2*x)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 1.58553, size = 43, normalized size = 1.1 \begin{align*} -\frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5 \, x + 7}{3 \,{\left (x^{2} + x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/3*(5*x + 7)/(x^2 + x + 1)

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Fricas [A]  time = 1.56255, size = 120, normalized size = 3.08 \begin{align*} -\frac{10 \, \sqrt{3}{\left (x^{2} + x + 1\right )} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + 15 \, x + 21}{9 \,{\left (x^{2} + x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*(10*sqrt(3)*(x^2 + x + 1)*arctan(1/3*sqrt(3)*(2*x + 1)) + 15*x + 21)/(x^2 + x + 1)

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Sympy [A]  time = 0.152542, size = 42, normalized size = 1.08 \begin{align*} - \frac{5 x + 7}{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((-1+3*x)/(x**2+x+1)**2,x)

[Out]

-(5*x + 7)/(3*x**2 + 3*x + 3) - 10*sqrt(3)*atan(2*sqrt(3)*x/3 + sqrt(3)/3)/9

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Giac [A]  time = 1.28852, size = 43, normalized size = 1.1 \begin{align*} -\frac{10}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) - \frac{5 \, x + 7}{3 \,{\left (x^{2} + x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*x + 1)) - 1/3*(5*x + 7)/(x^2 + x + 1)