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 = 0.0168321, 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 \[ -\tan ^{-1}\left (\sqrt{3}-x\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[Sqrt[3] - x]

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Rubi in Sympy [A]  time = 1.35754, size = 7, normalized size = 0.58 \[ \operatorname{atan}{\left (x - \sqrt{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(4+x**2-2*x*3**(1/2)),x)

[Out]

atan(x - sqrt(3))

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Mathematica [A]  time = 0.0230932, 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 = 0.018, size = 9, normalized size = 0.8 \[ \arctan \left ( x-\sqrt{3} \right ) \]

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 = 0.792211, size = 11, normalized size = 0.92 \[ \arctan \left (x - \sqrt{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x - sqrt(3))

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.392541, size = 7, normalized size = 0.58 \[ \operatorname{atan}{\left (x - \sqrt{3} \right )} \]

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/XCAS [A]  time = 0.20952, size = 11, normalized size = 0.92 \[ \arctan \left (x - \sqrt{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x - sqrt(3))