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

Optimal. Leaf size=34 \[ -\frac{\sqrt{-4 x^2-9}}{x}-2 \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0076738, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {277, 217, 203} \[ -\frac{\sqrt{-4 x^2-9}}{x}-2 \tan ^{-1}\left (\frac{2 x}{\sqrt{-4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 - 4*x^2]/x^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0069708, size = 49, normalized size = 1.44 \[ \frac{\sqrt{-4 x^2-9} \left (2 x \sinh ^{-1}\left (\frac{2 x}{3}\right )-\sqrt{4 x^2+9}\right )}{x \sqrt{4 x^2+9}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 - 4*x^2]/x^2,x]

[Out]

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

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Maple [A]  time = 0.003, size = 43, normalized size = 1.3 \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\,\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((-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*arctan(2*x/(-4*x^2-9)^(1/2))

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Maxima [C]  time = 3.47615, size = 28, normalized size = 0.82 \begin{align*} -\frac{\sqrt{-4 \, x^{2} - 9}}{x} + 2 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^2,x, algorithm="maxima")

[Out]

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

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Fricas [C]  time = 1.34476, size = 147, normalized size = 4.32 \begin{align*} \frac{-i \, x \log \left (-\frac{8 \, x + 4 i \, \sqrt{-4 \, x^{2} - 9}}{x}\right ) + i \, x \log \left (-\frac{8 \, x - 4 i \, \sqrt{-4 \, x^{2} - 9}}{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]

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

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Sympy [A]  time = 0.420157, size = 32, normalized size = 0.94 \begin{align*} - 2 \operatorname{atan}{\left (\frac{2 x}{\sqrt{- 4 x^{2} - 9}} \right )} - \frac{\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)

[Out]

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

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Giac [C]  time = 1.33912, size = 58, normalized size = 1.71 \begin{align*} -\frac{i \, \sqrt{4 \, x^{2} + 9} + 3 i}{2 \, x} - \frac{8 \, x}{-4 i \, \sqrt{4 \, x^{2} + 9} - 12 i} + 2 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^2,x, algorithm="giac")

[Out]

-1/2*(I*sqrt(4*x^2 + 9) + 3*I)/x - 8*x/(-4*I*sqrt(4*x^2 + 9) - 12*I) + 2*I*arcsin(2/3*I*x)