3.305 \(\int \sqrt{x-\sqrt{-4+x^2}} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{3} \left (x-\sqrt{x^2-4}\right )^{3/2}+\frac{4}{\sqrt{x-\sqrt{x^2-4}}} \]

[Out]

4/Sqrt[x - Sqrt[-4 + x^2]] + (x - Sqrt[-4 + x^2])^(3/2)/3

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Rubi [A]  time = 0.0319785, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{1}{3} \left (x-\sqrt{x^2-4}\right )^{3/2}+\frac{4}{\sqrt{x-\sqrt{x^2-4}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x - Sqrt[-4 + x^2]],x]

[Out]

4/Sqrt[x - Sqrt[-4 + x^2]] + (x - Sqrt[-4 + x^2])^(3/2)/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x - \sqrt{x^{2} - 4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x - sqrt(x**2 - 4)), x)

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Mathematica [A]  time = 0.0244141, size = 34, normalized size = 0.83 \[ \frac{2}{3} \sqrt{x-\sqrt{x^2-4}} \left (\sqrt{x^2-4}+2 x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x - Sqrt[-4 + x^2]],x]

[Out]

(2*Sqrt[x - Sqrt[-4 + x^2]]*(2*x + Sqrt[-4 + x^2]))/3

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int \sqrt{x-\sqrt{{x}^{2}-4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x-(x^2-4)^(1/2))^(1/2),x)

[Out]

int((x-(x^2-4)^(1/2))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x - \sqrt{x^{2} - 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x - sqrt(x^2 - 4)),x, algorithm="maxima")

[Out]

integrate(sqrt(x - sqrt(x^2 - 4)), x)

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Fricas [A]  time = 0.280239, size = 41, normalized size = 1. \[ \frac{2 \,{\left (x^{2} - \sqrt{x^{2} - 4} x + 4\right )}}{3 \, \sqrt{x - \sqrt{x^{2} - 4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x - sqrt(x^2 - 4)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.902583, size = 42, normalized size = 1.02 \[ \frac{4 x \sqrt{x - \sqrt{x^{2} - 4}}}{3} + \frac{2 \sqrt{x - \sqrt{x^{2} - 4}} \sqrt{x^{2} - 4}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*x*sqrt(x - sqrt(x**2 - 4))/3 + 2*sqrt(x - sqrt(x**2 - 4))*sqrt(x**2 - 4)/3

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x - \sqrt{x^{2} - 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x - sqrt(x^2 - 4)),x, algorithm="giac")

[Out]

integrate(sqrt(x - sqrt(x^2 - 4)), x)