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

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

[Out]

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

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Rubi [A]  time = 0.0518891, 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 \[ \frac{2}{3} \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right )-\frac{\sqrt{9-4 x^2}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[9 - 4*x^2]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.75176, size = 31, normalized size = 0.79 \[ \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{- 4 x^{2} + 9}}{3} \right )}}{3} - \frac{\sqrt{- 4 x^{2} + 9}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*atanh(sqrt(-4*x**2 + 9)/3)/3 - sqrt(-4*x**2 + 9)/(2*x**2)

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Mathematica [A]  time = 0.0220078, size = 43, normalized size = 1.1 \[ -\frac{\sqrt{9-4 x^2}}{2 x^2}+\frac{2}{3} \log \left (\sqrt{9-4 x^2}+3\right )-\frac{2 \log (x)}{3} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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

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Maxima [A]  time = 1.49023, size = 69, normalized size = 1.77 \[ -\frac{2}{9} \, \sqrt{-4 \, x^{2} + 9} - \frac{{\left (-4 \, x^{2} + 9\right )}^{\frac{3}{2}}}{18 \, x^{2}} + \frac{2}{3} \, \log \left (\frac{6 \, \sqrt{-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \]

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*log(6*sqrt(-4*x^2 + 9)
/abs(x) + 18/abs(x))

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Fricas [A]  time = 0.232708, size = 131, normalized size = 3.36 \[ \frac{36 \, x^{2} - 4 \,{\left (2 \, x^{4} + 3 \, \sqrt{-4 \, x^{2} + 9} x^{2} - 9 \, x^{2}\right )} \log \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \,{\left (2 \, x^{2} - 9\right )} \sqrt{-4 \, x^{2} + 9} - 81}{6 \,{\left (2 \, x^{4} + 3 \, \sqrt{-4 \, x^{2} + 9} x^{2} - 9 \, x^{2}\right )}} \]

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*(36*x^2 - 4*(2*x^4 + 3*sqrt(-4*x^2 + 9)*x^2 - 9*x^2)*log((sqrt(-4*x^2 + 9) -
3)/x) - 3*(2*x^2 - 9)*sqrt(-4*x^2 + 9) - 81)/(2*x^4 + 3*sqrt(-4*x^2 + 9)*x^2 -
9*x^2)

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Sympy [A]  time = 6.33424, size = 99, normalized size = 2.54 \[ \begin{cases} \frac{2 \operatorname{acosh}{\left (\frac{3}{2 x} \right )}}{3} + \frac{1}{x \sqrt{-1 + \frac{9}{4 x^{2}}}} - \frac{9}{4 x^{3} \sqrt{-1 + \frac{9}{4 x^{2}}}} & \text{for}\: \frac{9 \left |{\frac{1}{x^{2}}}\right |}{4} > 1 \\- \frac{2 i \operatorname{asin}{\left (\frac{3}{2 x} \right )}}{3} - \frac{i}{x \sqrt{1 - \frac{9}{4 x^{2}}}} + \frac{9 i}{4 x^{3} \sqrt{1 - \frac{9}{4 x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208098, size = 61, normalized size = 1.56 \[ -\frac{\sqrt{-4 \, x^{2} + 9}}{2 \, x^{2}} + \frac{1}{3} \,{\rm ln}\left (\sqrt{-4 \, x^{2} + 9} + 3\right ) - \frac{1}{3} \,{\rm ln}\left (-\sqrt{-4 \, x^{2} + 9} + 3\right ) \]

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