3.479 \(\int \sqrt{-9-4 x^2} \, dx\)

Optimal. Leaf size=36 \[ \frac{1}{2} x \sqrt{-4 x^2-9}-\frac{9}{4} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

[Out]

(x*Sqrt[-9 - 4*x^2])/2 - (9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/4

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Rubi [A]  time = 0.0057094, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {195, 217, 203} \[ \frac{1}{2} x \sqrt{-4 x^2-9}-\frac{9}{4} \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 - 4*x^2],x]

[Out]

(x*Sqrt[-9 - 4*x^2])/2 - (9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/4

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [A]  time = 0.0075753, size = 36, normalized size = 1. \[ \frac{1}{4} \left (2 x \sqrt{-4 x^2-9}-9 \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 - 4*x^2],x]

[Out]

(2*x*Sqrt[-9 - 4*x^2] - 9*ArcTan[(2*x)/Sqrt[-9 - 4*x^2]])/4

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Maple [A]  time = 0.002, size = 29, normalized size = 0.8 \begin{align*} -{\frac{9}{4}\arctan \left ( 2\,{\frac{x}{\sqrt{-4\,{x}^{2}-9}}} \right ) }+{\frac{x}{2}\sqrt{-4\,{x}^{2}-9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C]  time = 4.25974, size = 26, normalized size = 0.72 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} - 9} x + \frac{9}{4} i \, \operatorname{arsinh}\left (\frac{2}{3} \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C]  time = 1.26692, size = 154, normalized size = 4.28 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} - 9} x - \frac{9}{8} i \, \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) + \frac{9}{8} 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((-4*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.379129, size = 34, normalized size = 0.94 \begin{align*} \frac{x \sqrt{- 4 x^{2} - 9}}{2} - \frac{9 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \right )}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*sqrt(-4*x**2 - 9)/2 - 9*atan(2*x/sqrt(-4*x**2 - 9))/4

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Giac [C]  time = 2.36686, size = 26, normalized size = 0.72 \begin{align*} \frac{1}{2} \, \sqrt{-4 \, x^{2} - 9} x + \frac{9}{4} i \, \arcsin \left (\frac{2}{3} i \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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