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

Optimal. Leaf size=25 \[ -\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

[Out]

-(Sqrt[9 - 4*x^2]/x) - 2*ArcSin[(2*x)/3]

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Rubi [A]  time = 0.0050271, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {277, 216} \[ -\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[9 - 4*x^2]/x^2,x]

[Out]

-(Sqrt[9 - 4*x^2]/x) - 2*ArcSin[(2*x)/3]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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

Mathematica [A]  time = 0.0064594, size = 25, normalized size = 1. \[ -\frac{\sqrt{9-4 x^2}}{x}-2 \sin ^{-1}\left (\frac{2 x}{3}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[9 - 4*x^2]/x^2,x]

[Out]

-(Sqrt[9 - 4*x^2]/x) - 2*ArcSin[(2*x)/3]

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Maple [A]  time = 0.002, size = 34, normalized size = 1.4 \begin{align*} -{\frac{1}{9\,x} \left ( -4\,{x}^{2}+9 \right ) ^{{\frac{3}{2}}}}-{\frac{4\,x}{9}\sqrt{-4\,{x}^{2}+9}}-2\,\arcsin \left ( 2/3\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 3.75001, size = 28, normalized size = 1.12 \begin{align*} -\frac{\sqrt{-4 \, x^{2} + 9}}{x} - 2 \, \arcsin \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^2,x, algorithm="maxima")

[Out]

-sqrt(-4*x^2 + 9)/x - 2*arcsin(2/3*x)

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Fricas [A]  time = 1.5143, size = 88, normalized size = 3.52 \begin{align*} \frac{4 \, x \arctan \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x}\right ) - \sqrt{-4 \, x^{2} + 9}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.231266, size = 20, normalized size = 0.8 \begin{align*} - 2 \operatorname{asin}{\left (\frac{2 x}{3} \right )} - \frac{\sqrt{9 - 4 x^{2}}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*asin(2*x/3) - sqrt(9 - 4*x**2)/x

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Giac [A]  time = 2.33949, size = 53, normalized size = 2.12 \begin{align*} \frac{2 \, x}{\sqrt{-4 \, x^{2} + 9} - 3} - \frac{\sqrt{-4 \, x^{2} + 9} - 3}{2 \, x} - 2 \, \arcsin \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^2,x, algorithm="giac")

[Out]

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