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]

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

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Rubi [A]  time = 0.00, antiderivative size = 10, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 1.00 \[ \frac {1}{2} \sin ^{-1}\left (\frac {2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 - 4*x^2],x]

[Out]

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

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fricas [B]  time = 0.94, size = 19, normalized size = 1.90 \[ -\arctan \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \]

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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giac [A]  time = 1.12, size = 6, normalized size = 0.60 \[ \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, x\right ) \]

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)

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maple [A]  time = 0.00, size = 7, normalized size = 0.70 \[ \frac {\arcsin \left (\frac {2 x}{3}\right )}{2} \]

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 = 2.98, size = 6, normalized size = 0.60 \[ \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, x\right ) \]

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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mupad [B]  time = 0.01, size = 6, normalized size = 0.60 \[ \frac {\mathrm {asin}\left (\frac {2\,x}{3}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(9 - 4*x^2)^(1/2),x)

[Out]

asin((2*x)/3)/2

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sympy [A]  time = 0.15, size = 7, normalized size = 0.70 \[ \frac {\operatorname {asin}{\left (\frac {2 x}{3} \right )}}{2} \]

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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