3.471 \(\int \frac{\sqrt{-9+4 x^2}}{x^3} \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right )-\frac{\sqrt{4 x^2-9}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0502188, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2}{3} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right )-\frac{\sqrt{4 x^2-9}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-9 + 4*x^2]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.8086, size = 31, normalized size = 0.79 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{4 x^{2} - 9}}{3} \right )}}{3} - \frac{\sqrt{4 x^{2} - 9}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**2-9)**(1/2)/x**3,x)

[Out]

2*atan(sqrt(4*x**2 - 9)/3)/3 - sqrt(4*x**2 - 9)/(2*x**2)

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Mathematica [A]  time = 0.0159016, size = 37, normalized size = 0.95 \[ -\frac{\sqrt{4 x^2-9}}{2 x^2}-\frac{2}{3} \tan ^{-1}\left (\frac{3}{\sqrt{4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-9 + 4*x^2]/x^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 41, normalized size = 1.1 \[{\frac{1}{18\,{x}^{2}} \left ( 4\,{x}^{2}-9 \right ) ^{{\frac{3}{2}}}}-{\frac{2}{9}\sqrt{4\,{x}^{2}-9}}-{\frac{2}{3}\arctan \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}-9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^2-9)^(1/2)/x^3,x)

[Out]

1/18/x^2*(4*x^2-9)^(3/2)-2/9*(4*x^2-9)^(1/2)-2/3*arctan(3/(4*x^2-9)^(1/2))

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Maxima [A]  time = 1.48651, size = 47, normalized size = 1.21 \[ -\frac{2}{9} \, \sqrt{4 \, x^{2} - 9} + \frac{{\left (4 \, x^{2} - 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} - \frac{2}{3} \, \arcsin \left (\frac{3}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9)/x^3,x, algorithm="maxima")

[Out]

-2/9*sqrt(4*x^2 - 9) + 1/18*(4*x^2 - 9)^(3/2)/x^2 - 2/3*arcsin(3/2/abs(x))

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Fricas [A]  time = 0.230625, size = 134, normalized size = 3.44 \[ \frac{48 \, x^{3} + 8 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} - 9} x^{3} - 9 \, x^{2}\right )} \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) - 3 \,{\left (8 \, x^{2} - 9\right )} \sqrt{4 \, x^{2} - 9} - 108 \, x}{6 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} - 9} x^{3} - 9 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9)/x^3,x, algorithm="fricas")

[Out]

1/6*(48*x^3 + 8*(8*x^4 - 4*sqrt(4*x^2 - 9)*x^3 - 9*x^2)*arctan(-2/3*x + 1/3*sqrt
(4*x^2 - 9)) - 3*(8*x^2 - 9)*sqrt(4*x^2 - 9) - 108*x)/(8*x^4 - 4*sqrt(4*x^2 - 9)
*x^3 - 9*x^2)

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Sympy [A]  time = 6.07295, size = 99, normalized size = 2.54 \[ \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{3} + \frac{i}{x \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9 i}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9 \left |{\frac{1}{x^{2}}}\right |}{4} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{3} - \frac{1}{x \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**2-9)**(1/2)/x**3,x)

[Out]

Piecewise((2*I*acosh(3/(2*x))/3 + I/(x*sqrt(-1 + 9/(4*x**2))) - 9*I/(4*x**3*sqrt
(-1 + 9/(4*x**2))), 9*Abs(x**(-2))/4 > 1), (-2*asin(3/(2*x))/3 - 1/(x*sqrt(1 - 9
/(4*x**2))) + 9/(4*x**3*sqrt(1 - 9/(4*x**2))), True))

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GIAC/XCAS [A]  time = 0.204703, size = 39, normalized size = 1. \[ -\frac{\sqrt{4 \, x^{2} - 9}}{2 \, x^{2}} + \frac{2}{3} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(4*x^2 - 9)/x^3,x, algorithm="giac")

[Out]

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