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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0079967, 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, 206} \[ 2 \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right )-\frac{\sqrt{4 x^2-9}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[-9 + 4*x^2]/x^2,x]

[Out]

-(Sqrt[-9 + 4*x^2]/x) + 2*ArcTanh[(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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[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 \tanh ^{-1}\left (\frac{2 x}{\sqrt{-9+4 x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.009073, size = 48, normalized size = 1.41 \[ -\frac{\sqrt{4 x^2-9}+\frac{2 x \sqrt{4 x^2-9} \sin ^{-1}\left (\frac{2 x}{3}\right )}{\sqrt{9-4 x^2}}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-9 + 4*x^2]/x^2,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 48, 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}}+\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) \sqrt{4} \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)+ln(x*4^(1/2)+(4*x^2-9)^(1/2))*4^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 2.81638, size = 45, normalized size = 1.32 \begin{align*} -\frac{\sqrt{4 \, x^{2} - 9}}{x} + 2 \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\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*log(8*x + 4*sqrt(4*x^2 - 9))

________________________________________________________________________________________

Fricas [A]  time = 1.49452, size = 84, normalized size = 2.47 \begin{align*} -\frac{2 \, x \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) + 2 \, x + \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]

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

________________________________________________________________________________________

Sympy [A]  time = 0.232093, size = 19, normalized size = 0.56 \begin{align*} 2 \operatorname{acosh}{\left (\frac{2 x}{3} \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*acosh(2*x/3) - sqrt(4*x**2 - 9)/x

________________________________________________________________________________________

Giac [A]  time = 2.20901, size = 59, normalized size = 1.74 \begin{align*} -\frac{36}{{\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9} - \log \left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2}\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]

-36/((2*x - sqrt(4*x^2 - 9))^2 + 9) - log((2*x - sqrt(4*x^2 - 9))^2)