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

Optimal. Leaf size=18 \[ \frac{\sqrt{4 x^2-9}}{9 x} \]

[Out]

Sqrt[-9 + 4*x^2]/(9*x)

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Rubi [A]  time = 0.0029692, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {264} \[ \frac{\sqrt{4 x^2-9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[-9 + 4*x^2]),x]

[Out]

Sqrt[-9 + 4*x^2]/(9*x)

Rule 264

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

Rubi steps

\begin{align*} \int \frac{1}{x^2 \sqrt{-9+4 x^2}} \, dx &=\frac{\sqrt{-9+4 x^2}}{9 x}\\ \end{align*}

Mathematica [A]  time = 0.0023989, size = 18, normalized size = 1. \[ \frac{\sqrt{4 x^2-9}}{9 x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[-9 + 4*x^2]),x]

[Out]

Sqrt[-9 + 4*x^2]/(9*x)

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Maple [A]  time = 0.002, size = 25, normalized size = 1.4 \begin{align*}{\frac{ \left ( -3+2\,x \right ) \left ( 3+2\,x \right ) }{9\,x}{\frac{1}{\sqrt{4\,{x}^{2}-9}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 3.09762, size = 19, normalized size = 1.06 \begin{align*} \frac{\sqrt{4 \, x^{2} - 9}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*sqrt(4*x^2 - 9)/x

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Fricas [A]  time = 1.28741, size = 42, normalized size = 2.33 \begin{align*} \frac{2 \, x + \sqrt{4 \, x^{2} - 9}}{9 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*(2*x + sqrt(4*x^2 - 9))/x

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Sympy [A]  time = 0.745963, size = 37, normalized size = 2.06 \begin{align*} \begin{cases} \frac{2 i \sqrt{-1 + \frac{9}{4 x^{2}}}}{9} & \text{for}\: \frac{9}{4 \left |{x^{2}}\right |} > 1 \\\frac{2 \sqrt{1 - \frac{9}{4 x^{2}}}}{9} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*I*sqrt(-1 + 9/(4*x**2))/9, 9/(4*Abs(x**2)) > 1), (2*sqrt(1 - 9/(4*x**2))/9, True))

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Giac [A]  time = 1.83537, size = 31, normalized size = 1.72 \begin{align*} \frac{4}{{\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/((2*x - sqrt(4*x^2 - 9))^2 + 9)