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

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

[Out]

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

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Rubi [A]  time = 0.0014107, 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} \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 - 9*x^2],x]

[Out]

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

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

Mathematica [A]  time = 0.004271, size = 10, normalized size = 1. \[ \frac{1}{3} \sin ^{-1}\left (\frac{3 x}{2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 - 9*x^2],x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.65214, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.49721, size = 58, normalized size = 5.8 \begin{align*} -\frac{2}{3} \, \arctan \left (\frac{\sqrt{-9 \, x^{2} + 4} - 2}{3 \, x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*arctan(1/3*(sqrt(-9*x^2 + 4) - 2)/x)

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Sympy [A]  time = 0.133744, size = 7, normalized size = 0.7 \begin{align*} \frac{\operatorname{asin}{\left (\frac{3 x}{2} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

asin(3*x/2)/3

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Giac [A]  time = 1.12351, size = 8, normalized size = 0.8 \begin{align*} \frac{1}{3} \, \arcsin \left (\frac{3}{2} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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