3.6.48 \(\int \frac {1}{\sqrt {9-4 x^2}} \, dx\) [548]

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 = {222} \begin {gather*} \frac {1}{2} \text {ArcSin}\left (\frac {2 x}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[9 - 4*x^2],x]

[Out]

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

Rule 222

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 [C] Result contains complex when optimal does not.
time = 0.02, size = 24, normalized size = 2.40 \begin {gather*} \frac {1}{2} i \log \left (-2 i x+\sqrt {9-4 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[9 - 4*x^2],x]

[Out]

(I/2)*Log[(-2*I)*x + Sqrt[9 - 4*x^2]]

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Maple [A]
time = 0.11, size = 7, normalized size = 0.70

method	result	size
default	\(\frac {\arcsin \left (\frac {2 x}{3}\right )}{2}\)	\(7\)
meijerg	\(\frac {\arcsin \left (\frac {2 x}{3}\right )}{2}\)	\(7\)
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-4 x^{2}+9}+2 x \right )}{2}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 6, normalized size = 0.60 \begin {gather*} \frac {1}{2} \, \arcsin \left (\frac {2}{3} \, x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (6) = 12\).
time = 1.31, size = 19, normalized size = 1.90 \begin {gather*} -\arctan \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{2 \, x}\right ) \end {gather*}

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 = 0.05, size = 7, normalized size = 0.70 \begin {gather*} \frac {\operatorname {asin}{\left (\frac {2 x}{3} \right )}}{2} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (6) = 12\).
time = 0.98, size = 19, normalized size = 1.90 \begin {gather*} \frac {1}{2} \, \sqrt {-4 \, x^{2} + 9} x + \frac {9}{4} \, \arcsin \left (\frac {2}{3} \, x\right ) \end {gather*}

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

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

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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