### 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]

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

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Rubi [A]  time = 0.0141822, 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} $-\frac{4 x+7}{6 \left (x^2+2 x+4\right )}-\frac{2 \tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{3 \sqrt{3}}$

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 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{-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*}

Mathematica [A]  time = 0.0224098, size = 39, normalized size = 1. $\frac{-4 x-7}{6 \left (x^2+2 x+4\right )}-\frac{2 \tan ^{-1}\left (\frac{x+1}{\sqrt{3}}\right )}{3 \sqrt{3}}$

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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Maple [A]  time = 0.004, size = 35, normalized size = 0.9 \begin{align*}{\frac{-8\,x-14}{12\,{x}^{2}+24\,x+48}}-{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( 2\,x+2 \right ) \sqrt{3}}{6}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-3+x)/(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 = 1.42095, size = 43, normalized size = 1.1 \begin{align*} -\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 )}} \end{align*}

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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Fricas [A]  time = 2.01193, size = 123, normalized size = 3.15 \begin{align*} -\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 )}} \end{align*}

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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Sympy [A]  time = 0.115488, size = 41, normalized size = 1.05 \begin{align*} - \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} \end{align*}

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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Giac [A]  time = 1.0604, size = 43, normalized size = 1.1 \begin{align*} -\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 )}} \end{align*}

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)