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

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

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Rubi [A]  time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.02, size = 39, normalized size = 1.00 \begin {gather*} \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}} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+3 x}{\left (1+x+x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 37, normalized size = 0.95 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.15, size = 32, normalized size = 0.82 \begin {gather*} -\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 {gather*}

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)

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maple [A]  time = 0.04, size = 33, normalized size = 0.85 \begin {gather*} -\frac {10 \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{9}+\frac {-5 x -7}{3 x^{2}+3 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.30, size = 32, normalized size = 0.82 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.04, size = 34, normalized size = 0.87 \begin {gather*} -\frac {10\,\sqrt {3}\,\mathrm {atan}\left (\frac {2\,\sqrt {3}\,x}{3}+\frac {\sqrt {3}}{3}\right )}{9}-\frac {\frac {5\,x}{3}+\frac {7}{3}}{x^2+x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x - 1)/(x + x^2 + 1)^2,x)

[Out]

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

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sympy [A]  time = 0.14, size = 42, normalized size = 1.08 \begin {gather*} \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 {gather*}

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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