3.1.79 \(\int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx\) [79]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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 = {632, 210} \begin {gather*} -\text {ArcTan}\left (\sqrt {3}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 - 2*Sqrt[3]*x + x^2)^(-1),x]

[Out]

-ArcTan[Sqrt[3] - x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

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]

Rubi steps

\begin {align*} \int \frac {1}{4-2 \sqrt {3} x+x^2} \, dx &=-\left (2 \text {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*}

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Mathematica [A]
time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\sqrt {3}-x\right ) \end {gather*}

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 = 1.05, size = 9, normalized size = 0.75

method	result	size
default	\(\arctan \left (x -\sqrt {3}\right )\)	\(9\)
risch	\(\arctan \left (x -\sqrt {3}\right )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 8, normalized size = 0.67 \begin {gather*} \arctan \left (x - \sqrt {3}\right ) \end {gather*}

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 = 1.33, size = 10, normalized size = 0.83 \begin {gather*} -\arctan \left (-x + \sqrt {3}\right ) \end {gather*}

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 = 0.07, size = 7, normalized size = 0.58 \begin {gather*} \operatorname {atan}{\left (x - \sqrt {3} \right )} \end {gather*}

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 = 1.02, size = 8, normalized size = 0.67 \begin {gather*} \arctan \left (x - \sqrt {3}\right ) \end {gather*}

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))

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Mupad [B]
time = 0.27, size = 8, normalized size = 0.67 \begin {gather*} \mathrm {atan}\left (x-\sqrt {3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2 - 2*3^(1/2)*x + 4),x)

[Out]

atan(x - 3^(1/2))

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