3.548 \(\int \frac{1}{\sqrt{9-4 x^2}} \, dx\)

Optimal. Leaf size=10 \[ \frac{1}{2} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

ArcSin[(2*x)/3]/2

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Rubi [A]  time = 0.0014207, antiderivative size = 10, 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} \[ \frac{1}{2} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 - 4*x^2],x]

[Out]

ArcSin[(2*x)/3]/2

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{9-4 x^2}} \, dx &=\frac{1}{2} \sin ^{-1}\left (\frac{2 x}{3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0036364, size = 10, normalized size = 1. \[ \frac{1}{2} \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 - 4*x^2],x]

[Out]

ArcSin[(2*x)/3]/2

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Maple [A]  time = 0.003, size = 7, normalized size = 0.7 \begin{align*}{\frac{1}{2}\arcsin \left ({\frac{2\,x}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-4*x^2+9)^(1/2),x)

[Out]

1/2*arcsin(2/3*x)

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Maxima [A]  time = 3.71767, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{2} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+9)^(1/2),x, algorithm="maxima")

[Out]

1/2*arcsin(2/3*x)

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Fricas [B]  time = 1.26488, size = 53, normalized size = 5.3 \begin{align*} -\arctan \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+9)^(1/2),x, algorithm="fricas")

[Out]

-arctan(1/2*(sqrt(-4*x^2 + 9) - 3)/x)

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Sympy [A]  time = 0.134488, size = 7, normalized size = 0.7 \begin{align*} \frac{\operatorname{asin}{\left (\frac{2 x}{3} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x**2+9)**(1/2),x)

[Out]

asin(2*x/3)/2

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Giac [A]  time = 2.09543, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{2} \, \arcsin \left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-4*x^2+9)^(1/2),x, algorithm="giac")

[Out]

1/2*arcsin(2/3*x)