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

Optimal. Leaf size=30 \[ \sqrt{9-4 x^2}-3 \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0505154, antiderivative size = 30, 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 \[ \sqrt{9-4 x^2}-3 \tanh ^{-1}\left (\frac{1}{3} \sqrt{9-4 x^2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[9 - 4*x^2]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.6659, size = 24, normalized size = 0.8 \[ \sqrt{- 4 x^{2} + 9} - 3 \operatorname{atanh}{\left (\frac{\sqrt{- 4 x^{2} + 9}}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0128454, size = 32, normalized size = 1.07 \[ \sqrt{9-4 x^2}-3 \log \left (\sqrt{9-4 x^2}+3\right )+3 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[9 - 4*x^2]/x,x]

[Out]

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

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Maple [A]  time = 0.006, size = 25, normalized size = 0.8 \[ \sqrt{-4\,{x}^{2}+9}-3\,{\it Artanh} \left ( 3\,{\frac{1}{\sqrt{-4\,{x}^{2}+9}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-4*x^2+9)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 1.48048, size = 47, normalized size = 1.57 \[ \sqrt{-4 \, x^{2} + 9} - 3 \, \log \left (\frac{6 \, \sqrt{-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac{18}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-4*x^2 + 9) - 3*log(6*sqrt(-4*x^2 + 9)/abs(x) + 18/abs(x))

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Fricas [A]  time = 0.225252, size = 68, normalized size = 2.27 \[ -\frac{4 \, x^{2} - 3 \,{\left (\sqrt{-4 \, x^{2} + 9} - 3\right )} \log \left (\frac{\sqrt{-4 \, x^{2} + 9} - 3}{x}\right )}{\sqrt{-4 \, x^{2} + 9} - 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.52126, size = 76, normalized size = 2.53 \[ \begin{cases} i \sqrt{4 x^{2} - 9} - 3 \log{\left (x \right )} + \frac{3 \log{\left (x^{2} \right )}}{2} + 3 i \operatorname{asin}{\left (\frac{3}{2 x} \right )} & \text{for}\: \frac{4 \left |{x^{2}}\right |}{9} > 1 \\\sqrt{- 4 x^{2} + 9} + \frac{3 \log{\left (x^{2} \right )}}{2} - 3 \log{\left (\sqrt{- \frac{4 x^{2}}{9} + 1} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217346, size = 54, normalized size = 1.8 \[ \sqrt{-4 \, x^{2} + 9} - \frac{3}{2} \,{\rm ln}\left (\sqrt{-4 \, x^{2} + 9} + 3\right ) + \frac{3}{2} \,{\rm ln}\left (-\sqrt{-4 \, x^{2} + 9} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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