3.79 \(\int \frac{1}{4-2 \sqrt{3} x+x^2} \, dx\)

Optimal. Leaf size=12 \[ -\tan ^{-1}\left (\sqrt{3}-x\right ) \]

[Out]

-ArcTan[Sqrt[3] - x]

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Rubi [A]  time = 0.0086108, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {618, 204} \[ -\tan ^{-1}\left (\sqrt{3}-x\right ) \]

Antiderivative was successfully verified.

[In]

Int[(4 - 2*Sqrt[3]*x + x^2)^(-1),x]

[Out]

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

Mathematica [A]  time = 0.0142963, size = 12, normalized size = 1. \[ -\tan ^{-1}\left (\sqrt{3}-x\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(4 - 2*Sqrt[3]*x + x^2)^(-1),x]

[Out]

-ArcTan[Sqrt[3] - x]

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Maple [A]  time = 0.052, size = 9, normalized size = 0.8 \begin{align*} \arctan \left ( x-\sqrt{3} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(4+x^2-2*x*3^(1/2)),x)

[Out]

arctan(x-3^(1/2))

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Maxima [A]  time = 1.70056, size = 11, normalized size = 0.92 \begin{align*} \arctan \left (x - \sqrt{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x^2-2*x*3^(1/2)),x, algorithm="maxima")

[Out]

arctan(x - sqrt(3))

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Fricas [A]  time = 2.27582, size = 28, normalized size = 2.33 \begin{align*} \arctan \left (x - \sqrt{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x^2-2*x*3^(1/2)),x, algorithm="fricas")

[Out]

arctan(x - sqrt(3))

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Sympy [A]  time = 0.247866, size = 7, normalized size = 0.58 \begin{align*} \operatorname{atan}{\left (x - \sqrt{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x**2-2*x*3**(1/2)),x)

[Out]

atan(x - sqrt(3))

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Giac [A]  time = 1.30593, size = 11, normalized size = 0.92 \begin{align*} \arctan \left (x - \sqrt{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(4+x^2-2*x*3^(1/2)),x, algorithm="giac")

[Out]

arctan(x - sqrt(3))