3.148 \(\int \sqrt{3-x^2} \, dx\)

Optimal. Leaf size=29 \[ \frac{1}{2} \sqrt{3-x^2} x+\frac{3}{2} \sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0114883, antiderivative size = 29, 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 \[ \frac{1}{2} \sqrt{3-x^2} x+\frac{3}{2} \sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 - x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 0.597389, size = 24, normalized size = 0.83 \[ \frac{x \sqrt{- x^{2} + 3}}{2} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.013566, size = 29, normalized size = 1. \[ \frac{1}{2} \sqrt{3-x^2} x+\frac{3}{2} \sin ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 - x^2],x]

[Out]

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

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Maple [A]  time = 0.005, size = 23, normalized size = 0.8 \[{\frac{3}{2}\arcsin \left ({\frac{x\sqrt{3}}{3}} \right ) }+{\frac{x}{2}\sqrt{-{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+3)^(1/2),x)

[Out]

3/2*arcsin(1/3*x*3^(1/2))+1/2*x*(-x^2+3)^(1/2)

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Maxima [A]  time = 1.51561, size = 30, normalized size = 1.03 \[ \frac{1}{2} \, \sqrt{-x^{2} + 3} x + \frac{3}{2} \, \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.211993, size = 39, normalized size = 1.34 \[ \frac{1}{2} \, \sqrt{-x^{2} + 3} x - \frac{3}{2} \, \arctan \left (\frac{\sqrt{-x^{2} + 3}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.227111, size = 24, normalized size = 0.83 \[ \frac{x \sqrt{- x^{2} + 3}}{2} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{3} x}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+3)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216836, size = 30, normalized size = 1.03 \[ \frac{1}{2} \, \sqrt{-x^{2} + 3} x + \frac{3}{2} \, \arcsin \left (\frac{1}{3} \, \sqrt{3} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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