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

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

[Out]

ArcSin[x]

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Rubi [A]  time = 0.0010183, antiderivative size = 2, 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, Rules used = {216} \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - x^2],x]

[Out]

ArcSin[x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{1-x^2}} \, dx &=\sin ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0036128, size = 2, normalized size = 1. \[ \sin ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - x^2],x]

[Out]

ArcSin[x]

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Maple [A]  time = 0.003, size = 3, normalized size = 1.5 \begin{align*} \arcsin \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-x^2+1)^(1/2),x)

[Out]

arcsin(x)

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Maxima [A]  time = 1.85818, size = 3, normalized size = 1.5 \begin{align*} \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

arcsin(x)

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Fricas [B]  time = 1.40812, size = 47, normalized size = 23.5 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/2),x, algorithm="fricas")

[Out]

-2*arctan((sqrt(-x^2 + 1) - 1)/x)

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Sympy [A]  time = 0.124319, size = 2, normalized size = 1. \begin{align*} \operatorname{asin}{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x**2+1)**(1/2),x)

[Out]

asin(x)

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Giac [A]  time = 1.12543, size = 3, normalized size = 1.5 \begin{align*} \arcsin \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-x^2+1)^(1/2),x, algorithm="giac")

[Out]

arcsin(x)