3.128 \(\int \sqrt{1-4 x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{1}{2} \sqrt{1-4 x^2} x+\frac{1}{4} \sin ^{-1}(2 x) \]

[Out]

(x*Sqrt[1 - 4*x^2])/2 + ArcSin[2*x]/4

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Rubi [A]  time = 0.00996907, antiderivative size = 25, 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 \[ \frac{1}{2} \sqrt{1-4 x^2} x+\frac{1}{4} \sin ^{-1}(2 x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 - 4*x^2],x]

[Out]

(x*Sqrt[1 - 4*x^2])/2 + ArcSin[2*x]/4

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Rubi in Sympy [A]  time = 0.618543, size = 19, normalized size = 0.76 \[ \frac{x \sqrt{- 4 x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (2 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(-4*x**2 + 1)/2 + asin(2*x)/4

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Mathematica [A]  time = 0.0108913, size = 25, normalized size = 1. \[ \frac{1}{2} \sqrt{1-4 x^2} x+\frac{1}{4} \sin ^{-1}(2 x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 - 4*x^2],x]

[Out]

(x*Sqrt[1 - 4*x^2])/2 + ArcSin[2*x]/4

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Maple [A]  time = 0.004, size = 20, normalized size = 0.8 \[{\frac{\arcsin \left ( 2\,x \right ) }{4}}+{\frac{x}{2}\sqrt{-4\,{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4*x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.49218, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, \sqrt{-4 \, x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.208477, size = 112, normalized size = 4.48 \[ -\frac{4 \, x^{3} +{\left (2 \, x^{2} + \sqrt{-4 \, x^{2} + 1} - 1\right )} \arctan \left (\frac{\sqrt{-4 \, x^{2} + 1} - 1}{2 \, x}\right ) -{\left (2 \, x^{3} - x\right )} \sqrt{-4 \, x^{2} + 1} - x}{2 \,{\left (2 \, x^{2} + \sqrt{-4 \, x^{2} + 1} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(4*x^3 + (2*x^2 + sqrt(-4*x^2 + 1) - 1)*arctan(1/2*(sqrt(-4*x^2 + 1) - 1)/x
) - (2*x^3 - x)*sqrt(-4*x^2 + 1) - x)/(2*x^2 + sqrt(-4*x^2 + 1) - 1)

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Sympy [A]  time = 0.243862, size = 19, normalized size = 0.76 \[ \frac{x \sqrt{- 4 x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (2 x \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-4*x**2+1)**(1/2),x)

[Out]

x*sqrt(-4*x**2 + 1)/2 + asin(2*x)/4

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GIAC/XCAS [A]  time = 0.206941, size = 26, normalized size = 1.04 \[ \frac{1}{2} \, \sqrt{-4 \, x^{2} + 1} x + \frac{1}{4} \, \arcsin \left (2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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