3.312 \(\int \frac{1}{\sqrt{16-x^2}} \, dx\)

Optimal. Leaf size=6 \[ \sin ^{-1}\left (\frac{x}{4}\right ) \]

[Out]

ArcSin[x/4]

_______________________________________________________________________________________

Rubi [A]  time = 0.00402379, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \sin ^{-1}\left (\frac{x}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[16 - x^2],x]

[Out]

ArcSin[x/4]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.528762, size = 3, normalized size = 0.5 \[ \operatorname{asin}{\left (\frac{x}{4} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-x**2+16)**(1/2),x)

[Out]

asin(x/4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00652445, size = 6, normalized size = 1. \[ \sin ^{-1}\left (\frac{x}{4}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[16 - x^2],x]

[Out]

ArcSin[x/4]

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 5, normalized size = 0.8 \[ \arcsin \left ({\frac{x}{4}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^2+16)^(1/2),x)

[Out]

arcsin(1/4*x)

_______________________________________________________________________________________

Maxima [A]  time = 1.49938, size = 5, normalized size = 0.83 \[ \arcsin \left (\frac{1}{4} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + 16),x, algorithm="maxima")

[Out]

arcsin(1/4*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.22106, size = 24, normalized size = 4. \[ -2 \, \arctan \left (\frac{\sqrt{-x^{2} + 16} - 4}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + 16),x, algorithm="fricas")

[Out]

-2*arctan((sqrt(-x^2 + 16) - 4)/x)

_______________________________________________________________________________________

Sympy [A]  time = 0.154797, size = 3, normalized size = 0.5 \[ \operatorname{asin}{\left (\frac{x}{4} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**2+16)**(1/2),x)

[Out]

asin(x/4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219416, size = 5, normalized size = 0.83 \[ \arcsin \left (\frac{1}{4} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(-x^2 + 16),x, algorithm="giac")

[Out]

arcsin(1/4*x)