3.562 \(\int \frac{1}{x^3 \sqrt{-9+4 x^2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]

[Out]

Sqrt[-9 + 4*x^2]/(18*x^2) + (2*ArcTan[Sqrt[-9 + 4*x^2]/3])/27

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Rubi [A]  time = 0.0507426, 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 \[ \frac{\sqrt{4 x^2-9}}{18 x^2}+\frac{2}{27} \tan ^{-1}\left (\frac{1}{3} \sqrt{4 x^2-9}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[-9 + 4*x^2]),x]

[Out]

Sqrt[-9 + 4*x^2]/(18*x^2) + (2*ArcTan[Sqrt[-9 + 4*x^2]/3])/27

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Rubi in Sympy [A]  time = 5.46903, size = 31, normalized size = 0.79 \[ \frac{2 \operatorname{atan}{\left (\frac{\sqrt{4 x^{2} - 9}}{3} \right )}}{27} + \frac{\sqrt{4 x^{2} - 9}}{18 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atan(sqrt(4*x**2 - 9)/3)/27 + sqrt(4*x**2 - 9)/(18*x**2)

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Mathematica [A]  time = 0.0227073, size = 37, normalized size = 0.95 \[ \frac{\sqrt{4 x^2-9}}{18 x^2}-\frac{2}{27} \tan ^{-1}\left (\frac{3}{\sqrt{4 x^2-9}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[-9 + 4*x^2]),x]

[Out]

Sqrt[-9 + 4*x^2]/(18*x^2) - (2*ArcTan[3/Sqrt[-9 + 4*x^2]])/27

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Maple [A]  time = 0.005, size = 30, normalized size = 0.8 \[{\frac{1}{18\,{x}^{2}}\sqrt{4\,{x}^{2}-9}}-{\frac{2}{27}\arctan \left ( 3\,{\frac{1}{\sqrt{4\,{x}^{2}-9}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(4*x^2-9)^(1/2),x)

[Out]

1/18*(4*x^2-9)^(1/2)/x^2-2/27*arctan(3/(4*x^2-9)^(1/2))

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Maxima [A]  time = 1.59341, size = 32, normalized size = 0.82 \[ \frac{\sqrt{4 \, x^{2} - 9}}{18 \, x^{2}} - \frac{2}{27} \, \arcsin \left (\frac{3}{2 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 - 9)*x^3),x, algorithm="maxima")

[Out]

1/18*sqrt(4*x^2 - 9)/x^2 - 2/27*arcsin(3/2/abs(x))

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Fricas [A]  time = 0.219368, size = 134, normalized size = 3.44 \[ -\frac{48 \, x^{3} - 8 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} - 9} x^{3} - 9 \, x^{2}\right )} \arctan \left (-\frac{2}{3} \, x + \frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) - 3 \,{\left (8 \, x^{2} - 9\right )} \sqrt{4 \, x^{2} - 9} - 108 \, x}{54 \,{\left (8 \, x^{4} - 4 \, \sqrt{4 \, x^{2} - 9} x^{3} - 9 \, x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 - 9)*x^3),x, algorithm="fricas")

[Out]

-1/54*(48*x^3 - 8*(8*x^4 - 4*sqrt(4*x^2 - 9)*x^3 - 9*x^2)*arctan(-2/3*x + 1/3*sq
rt(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 = 8.16821, size = 100, normalized size = 2.56 \[ \begin{cases} \frac{2 i \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{27} - \frac{i}{9 x \sqrt{-1 + \frac{9}{4 x^{2}}}} + \frac{i}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9 \left |{\frac{1}{x^{2}}}\right |}{4} > 1 \\- \frac{2 \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{27} + \frac{1}{9 x \sqrt{1 - \frac{9}{4 x^{2}}}} - \frac{1}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*I*acosh(3/(2*x))/27 - I/(9*x*sqrt(-1 + 9/(4*x**2))) + I/(4*x**3*sqr
t(-1 + 9/(4*x**2))), 9*Abs(x**(-2))/4 > 1), (-2*asin(3/(2*x))/27 + 1/(9*x*sqrt(1
- 9/(4*x**2))) - 1/(4*x**3*sqrt(1 - 9/(4*x**2))), True))

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GIAC/XCAS [A]  time = 0.216261, size = 39, normalized size = 1. \[ \frac{\sqrt{4 \, x^{2} - 9}}{18 \, x^{2}} + \frac{2}{27} \, \arctan \left (\frac{1}{3} \, \sqrt{4 \, x^{2} - 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(4*x^2 - 9)*x^3),x, algorithm="giac")

[Out]

1/18*sqrt(4*x^2 - 9)/x^2 + 2/27*arctan(1/3*sqrt(4*x^2 - 9))