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

Optimal. Leaf size=19 \[ \frac{1}{2} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

[Out]

ArcTan[(2*x)/Sqrt[-9 - 4*x^2]]/2

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Rubi [A]  time = 0.0032778, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {217, 203} \[ \frac{1}{2} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-9 - 4*x^2],x]

[Out]

ArcTan[(2*x)/Sqrt[-9 - 4*x^2]]/2

Rule 217

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-9-4 x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+4 x^2} \, dx,x,\frac{x}{\sqrt{-9-4 x^2}}\right )\\ &=\frac{1}{2} \tan ^{-1}\left (\frac{2 x}{\sqrt{-9-4 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0031859, size = 19, normalized size = 1. \[ \frac{1}{2} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[-9 - 4*x^2],x]

[Out]

ArcTan[(2*x)/Sqrt[-9 - 4*x^2]]/2

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*}{\frac{1}{2}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \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*arctan(2*x/(-4*x^2-9)^(1/2))

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Maxima [C]  time = 2.55582, size = 8, normalized size = 0.42 \begin{align*} -\frac{1}{2} i \, \operatorname{arsinh}\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*I*arcsinh(2/3*x)

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Fricas [C]  time = 1.26577, size = 120, normalized size = 6.32 \begin{align*} \frac{1}{4} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) - \frac{1}{4} i \, \log \left (-\frac{8 \, x - 4 i \, \sqrt{-4 \, x^{2} - 9}}{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]

1/4*I*log(-(8*x + 4*I*sqrt(-4*x^2 - 9))/x) - 1/4*I*log(-(8*x - 4*I*sqrt(-4*x^2 - 9))/x)

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Sympy [A]  time = 0.318525, size = 17, normalized size = 0.89 \begin{align*} \frac{\operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \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]

atan(2*x/sqrt(-4*x**2 - 9))/2

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Giac [C]  time = 2.44089, size = 8, normalized size = 0.42 \begin{align*} -\frac{1}{2} i \, \arcsin \left (\frac{2}{3} i \, 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*I*arcsin(2/3*I*x)