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]]

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Rubi [A]  time = 0.0277707, 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 \[ 2 \tanh ^{-1}\left (\frac{2 x}{\sqrt{4 x^2-9}}\right )-\frac{\sqrt{4 x^2-9}}{x} \]

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]]

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Rubi in Sympy [A]  time = 3.50471, size = 27, normalized size = 0.79 \[ 2 \operatorname{atanh}{\left (\frac{2 x}{\sqrt{4 x^{2} - 9}} \right )} - \frac{\sqrt{4 x^{2} - 9}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0135292, size = 35, normalized size = 1.03 \[ 2 \log \left (\sqrt{4 x^2-9}+2 x\right )-\frac{\sqrt{4 x^2-9}}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-9 + 4*x^2]/x^2,x]

[Out]

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

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Maple [A]  time = 0.004, size = 48, normalized size = 1.4 \[{\frac{1}{9\,x} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{4\,x}{9}\sqrt{4\,{x}^{2}-9}}+\ln \left ( x\sqrt{4}+\sqrt{4\,{x}^{2}-9} \right ) \sqrt{4} \]

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
)

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Maxima [A]  time = 1.48656, size = 45, normalized size = 1.32 \[ -\frac{\sqrt{4 \, x^{2} - 9}}{x} + 2 \, \log \left (8 \, x + 4 \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.228035, size = 78, normalized size = 2.29 \[ -\frac{2 \,{\left (2 \, x^{2} - \sqrt{4 \, x^{2} - 9} x\right )} \log \left (-2 \, x + \sqrt{4 \, x^{2} - 9}\right ) + 9}{2 \, x^{2} - \sqrt{4 \, x^{2} - 9} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.681614, size = 19, normalized size = 0.56 \[ 2 \operatorname{acosh}{\left (\frac{2 x}{3} \right )} - \frac{\sqrt{4 x^{2} - 9}}{x} \]

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

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GIAC/XCAS [A]  time = 0.207545, size = 59, normalized size = 1.74 \[ -\frac{36}{{\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2} + 9} -{\rm ln}\left ({\left (2 \, x - \sqrt{4 \, x^{2} - 9}\right )}^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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