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

Optimal. Leaf size=27 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0278542, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[(1 - x^2)^(-1)],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 0.693494, size = 20, normalized size = 0.74 \[ \sqrt{- x^{2} + 1} \sqrt{\frac{1}{- x^{2} + 1}} \operatorname{asin}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0115194, size = 27, normalized size = 1. \[ \sqrt{\frac{1}{1-x^2}} \sqrt{1-x^2} \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[(1 - x^2)^(-1)],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 30, normalized size = 1.1 \[ \sqrt{- \left ({x}^{2}-1 \right ) ^{-1}}\sqrt{{x}^{2}-1}\ln \left ( x+\sqrt{{x}^{2}-1} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1/(-x^2+1))^(1/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.7899, size = 3, normalized size = 0.11 \[ \arcsin \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsin(x)

_______________________________________________________________________________________

Fricas [A]  time = 0.268061, size = 42, normalized size = 1.56 \[ 2 \, \arctan \left (\frac{\sqrt{-\frac{1}{x^{2} - 1}} - 1}{x \sqrt{-\frac{1}{x^{2} - 1}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 2.9306, size = 7, normalized size = 0.26 \[ \begin{cases} \operatorname{asin}{\left (x \right )} & \text{for}\: x > -1 \wedge x < 1 \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((asin(x), (x > -1) & (x < 1)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270197, size = 14, normalized size = 0.52 \[ -\arcsin \left (x\right ){\rm sign}\left (x^{2} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-arcsin(x)*sign(x^2 - 1)