3.209 \(\int \frac {-3+x}{(4+2 x+x^2)^2} \, dx\)

Optimal. Leaf size=39 \[ \frac {-4 x-7}{6 \left (x^2+2 x+4\right )}-\frac {2 \tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

[Out]

1/6*(-7-4*x)/(x^2+2*x+4)-2/9*arctan(1/3*(1+x)*3^(1/2))*3^(1/2)

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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} \[ -\frac {4 x+7}{6 \left (x^2+2 x+4\right )}-\frac {2 \tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[(-3 + x)/(4 + 2*x + x^2)^2,x]

[Out]

-(7 + 4*x)/(6*(4 + 2*x + x^2)) - (2*ArcTan[(1 + 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 {-3+x}{\left (4+2 x+x^2\right )^2} \, dx &=-\frac {7+4 x}{6 \left (4+2 x+x^2\right )}-\frac {2}{3} \int \frac {1}{4+2 x+x^2} \, dx\\ &=-\frac {7+4 x}{6 \left (4+2 x+x^2\right )}+\frac {4}{3} \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,2+2 x\right )\\ &=-\frac {7+4 x}{6 \left (4+2 x+x^2\right )}-\frac {2 \tan ^{-1}\left (\frac {1+x}{\sqrt {3}}\right )}{3 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 39, normalized size = 1.00 \[ \frac {-4 x-7}{6 \left (x^2+2 x+4\right )}-\frac {2 \tan ^{-1}\left (\frac {x+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Integrate[(-3 + x)/(4 + 2*x + x^2)^2,x]

[Out]

(-7 - 4*x)/(6*(4 + 2*x + x^2)) - (2*ArcTan[(1 + x)/Sqrt[3]])/(3*Sqrt[3])

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fricas [A]  time = 0.40, size = 39, normalized size = 1.00 \[ -\frac {4 \, \sqrt {3} {\left (x^{2} + 2 \, x + 4\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) + 12 \, x + 21}{18 \, {\left (x^{2} + 2 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)/(x^2+2*x+4)^2,x, algorithm="fricas")

[Out]

-1/18*(4*sqrt(3)*(x^2 + 2*x + 4)*arctan(1/3*sqrt(3)*(x + 1)) + 12*x + 21)/(x^2 + 2*x + 4)

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giac [A]  time = 0.93, size = 32, normalized size = 0.82 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {4 \, x + 7}{6 \, {\left (x^{2} + 2 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)/(x^2+2*x+4)^2,x, algorithm="giac")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(x + 1)) - 1/6*(4*x + 7)/(x^2 + 2*x + 4)

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maple [A]  time = 0.01, size = 35, normalized size = 0.90 \[ -\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 x +2\right ) \sqrt {3}}{6}\right )}{9}+\frac {-8 x -14}{12 x^{2}+24 x +48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-3)/(x^2+2*x+4)^2,x)

[Out]

1/12*(-8*x-14)/(x^2+2*x+4)-2/9*3^(1/2)*arctan(1/6*(2*x+2)*3^(1/2))

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maxima [A]  time = 1.10, size = 32, normalized size = 0.82 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x + 1\right )}\right ) - \frac {4 \, x + 7}{6 \, {\left (x^{2} + 2 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)/(x^2+2*x+4)^2,x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(x + 1)) - 1/6*(4*x + 7)/(x^2 + 2*x + 4)

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mupad [B]  time = 0.04, size = 36, normalized size = 0.92 \[ -\frac {\frac {2\,x}{3}+\frac {7}{6}}{x^2+2\,x+4}-\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x}{3}+\frac {\sqrt {3}}{3}\right )}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 3)/(2*x + x^2 + 4)^2,x)

[Out]

- ((2*x)/3 + 7/6)/(2*x + x^2 + 4) - (2*3^(1/2)*atan((3^(1/2)*x)/3 + 3^(1/2)/3))/9

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sympy [A]  time = 0.13, size = 41, normalized size = 1.05 \[ \frac {- 4 x - 7}{6 x^{2} + 12 x + 24} - \frac {2 \sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3+x)/(x**2+2*x+4)**2,x)

[Out]

(-4*x - 7)/(6*x**2 + 12*x + 24) - 2*sqrt(3)*atan(sqrt(3)*x/3 + sqrt(3)/3)/9

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