3.144 \(\int \sqrt{2 x-x^2} \, dx\)

Optimal. Leaf size=33 \[ -\frac{1}{2} \sqrt{2 x-x^2} (1-x)-\frac{1}{2} \sin ^{-1}(1-x) \]

[Out]

-((1 - x)*Sqrt[2*x - x^2])/2 - ArcSin[1 - x]/2

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Rubi [A]  time = 0.017401, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{2} \sqrt{2 x-x^2} (1-x)-\frac{1}{2} \sin ^{-1}(1-x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[2*x - x^2],x]

[Out]

-((1 - x)*Sqrt[2*x - x^2])/2 - ArcSin[1 - x]/2

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Rubi in Sympy [A]  time = 0.798937, size = 22, normalized size = 0.67 \[ - \frac{\left (- 2 x + 2\right ) \sqrt{- x^{2} + 2 x}}{4} + \frac{\operatorname{asin}{\left (x - 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*x + 2)*sqrt(-x**2 + 2*x)/4 + asin(x - 1)/2

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Mathematica [A]  time = 0.0408164, size = 45, normalized size = 1.36 \[ \frac{1}{2} \sqrt{-(x-2) x} \left (x-\frac{2 \log \left (\sqrt{x-2}+\sqrt{x}\right )}{\sqrt{x-2} \sqrt{x}}-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[2*x - x^2],x]

[Out]

(Sqrt[-((-2 + x)*x)]*(-1 + x - (2*Log[Sqrt[-2 + x] + Sqrt[x]])/(Sqrt[-2 + x]*Sqr
t[x])))/2

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Maple [A]  time = 0.006, size = 26, normalized size = 0.8 \[ -{\frac{-2\,x+2}{4}\sqrt{-{x}^{2}+2\,x}}+{\frac{\arcsin \left ( -1+x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+2*x)^(1/2),x)

[Out]

-1/4*(-2*x+2)*(-x^2+2*x)^(1/2)+1/2*arcsin(-1+x)

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Maxima [A]  time = 1.53843, size = 49, normalized size = 1.48 \[ \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \arcsin \left (-x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^2 + 2*x)*x - 1/2*sqrt(-x^2 + 2*x) - 1/2*arcsin(-x + 1)

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Fricas [A]  time = 0.203513, size = 47, normalized size = 1.42 \[ \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x - 1\right )} - \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^2 + 2*x)*(x - 1) - arctan(sqrt(-x^2 + 2*x)/x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+2*x)**(1/2),x)

[Out]

Integral(sqrt(-x**2 + 2*x), x)

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GIAC/XCAS [A]  time = 0.213038, size = 31, normalized size = 0.94 \[ \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x - 1\right )} + \frac{1}{2} \, \arcsin \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^2 + 2*x)*(x - 1) + 1/2*arcsin(x - 1)