3.2394 \(\int \frac{1}{(-2+x) \sqrt{3-4 x+x^2}} \, dx\)

Optimal. Leaf size=13 \[ \tan ^{-1}\left (\sqrt{x^2-4 x+3}\right ) \]

[Out]

ArcTan[Sqrt[3 - 4*x + x^2]]

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Rubi [A]  time = 0.0266645, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \tan ^{-1}\left (\sqrt{x^2-4 x+3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[Sqrt[3 - 4*x + x^2]]

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Rubi in Sympy [A]  time = 5.51505, size = 12, normalized size = 0.92 \[ \operatorname{atan}{\left (\sqrt{x^{2} - 4 x + 3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(x**2 - 4*x + 3))

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Mathematica [A]  time = 0.0121926, size = 14, normalized size = 1.08 \[ -\tan ^{-1}\left (\frac{1}{\sqrt{(x-2)^2-1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[1/Sqrt[-1 + (-2 + x)^2]]

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Maple [A]  time = 0.008, size = 13, normalized size = 1. \[ -\arctan \left ({\frac{1}{\sqrt{ \left ( x-2 \right ) ^{2}-1}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.746995, size = 12, normalized size = 0.92 \[ -\arcsin \left (\frac{1}{{\left | x - 2 \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(1/abs(x - 2))

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Fricas [A]  time = 0.219727, size = 24, normalized size = 1.85 \[ 2 \, \arctan \left (-x + \sqrt{x^{2} - 4 \, x + 3} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(-x + sqrt(x^2 - 4*x + 3) + 2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (x - 3\right ) \left (x - 1\right )} \left (x - 2\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt((x - 3)*(x - 1))*(x - 2)), x)

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GIAC/XCAS [A]  time = 0.207598, size = 24, normalized size = 1.85 \[ 2 \, \arctan \left (-x + \sqrt{x^{2} - 4 \, x + 3} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*arctan(-x + sqrt(x^2 - 4*x + 3) + 2)