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 = 0.0160932, 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 \[ \frac{\sqrt{4 x^2-9}}{9 x} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[-9 + 4*x^2]),x]

[Out]

Sqrt[-9 + 4*x^2]/(9*x)

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Rubi in Sympy [A]  time = 3.04114, size = 12, normalized size = 0.67 \[ \frac{\sqrt{4 x^{2} - 9}}{9 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(4*x**2-9)**(1/2),x)

[Out]

sqrt(4*x**2 - 9)/(9*x)

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Mathematica [A]  time = 0.00913327, 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 = 0.005, size = 25, normalized size = 1.4 \[{\frac{ \left ( 2\,x-3 \right ) \left ( 2\,x+3 \right ) }{9\,x}{\frac{1}{\sqrt{4\,{x}^{2}-9}}}} \]

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

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Maxima [A]  time = 1.4843, size = 19, normalized size = 1.06 \[ \frac{\sqrt{4 \, x^{2} - 9}}{9 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*sqrt(4*x^2 - 9)/x

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Fricas [A]  time = 0.218556, size = 27, normalized size = 1.5 \[ \frac{1}{2 \, x^{2} - \sqrt{4 \, x^{2} - 9} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(2*x^2 - sqrt(4*x^2 - 9)*x)

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Sympy [A]  time = 2.59728, size = 39, normalized size = 2.17 \[ \begin{cases} \frac{2 i \sqrt{-1 + \frac{9}{4 x^{2}}}}{9} & \text{for}\: \frac{9 \left |{\frac{1}{x^{2}}}\right |}{4} > 1 \\\frac{2 \sqrt{1 - \frac{9}{4 x^{2}}}}{9} & \text{otherwise} \end{cases} \]

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

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GIAC/XCAS [A]  time = 0.219679, size = 31, normalized size = 1.72 \[ \frac{4}{{\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/((2*x - sqrt(4*x^2 - 9))^2 + 9)