3.97 \(\int \sqrt{1-x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]

[Out]

(x*Sqrt[1 - x^2])/2 + ArcSin[x]/2

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Rubi [A]  time = 0.0086933, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - x^2],x]

[Out]

(x*Sqrt[1 - x^2])/2 + ArcSin[x]/2

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Rubi in Sympy [A]  time = 0.564831, size = 15, normalized size = 0.65 \[ \frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(-x**2 + 1)/2 + asin(x)/2

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Mathematica [A]  time = 0.00871474, size = 20, normalized size = 0.87 \[ \frac{1}{2} \left (\sqrt{1-x^2} x+\sin ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - x^2],x]

[Out]

(x*Sqrt[1 - x^2] + ArcSin[x])/2

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Maple [A]  time = 0.004, size = 18, normalized size = 0.8 \[{\frac{\arcsin \left ( x \right ) }{2}}+{\frac{x}{2}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.53721, size = 23, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^2 + 1)*x + 1/2*arcsin(x)

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Fricas [A]  time = 0.204438, size = 109, normalized size = 4.74 \[ -\frac{2 \, x^{3} + 2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (x^{3} - 2 \, x\right )} \sqrt{-x^{2} + 1} - 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*x^3 + 2*(x^2 + 2*sqrt(-x^2 + 1) - 2)*arctan((sqrt(-x^2 + 1) - 1)/x) - (x
^3 - 2*x)*sqrt(-x^2 + 1) - 2*x)/(x^2 + 2*sqrt(-x^2 + 1) - 2)

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Sympy [A]  time = 0.220365, size = 15, normalized size = 0.65 \[ \frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)**(1/2),x)

[Out]

x*sqrt(-x**2 + 1)/2 + asin(x)/2

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GIAC/XCAS [A]  time = 0.217321, size = 23, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(-x^2 + 1)*x + 1/2*arcsin(x)