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

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

[Out]

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

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Rubi [A]  time = 0.0388239, 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 \[ \sqrt{9 x^2-4}-2 \tan ^{-1}\left (\frac{1}{2} \sqrt{9 x^2-4}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[-4 + 9*x^2]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.57937, size = 24, normalized size = 0.8 \[ \sqrt{9 x^{2} - 4} - 2 \operatorname{atan}{\left (\frac{\sqrt{9 x^{2} - 4}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0132835, size = 28, normalized size = 0.93 \[ \sqrt{9 x^2-4}+2 \tan ^{-1}\left (\frac{2}{\sqrt{9 x^2-4}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[-4 + 9*x^2]/x,x]

[Out]

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

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Maple [A]  time = 0.01, size = 25, normalized size = 0.8 \[ \sqrt{9\,{x}^{2}-4}+2\,\arctan \left ( 2\,{\frac{1}{\sqrt{9\,{x}^{2}-4}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((9*x^2-4)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 1.4975, size = 26, normalized size = 0.87 \[ \sqrt{9 \, x^{2} - 4} + 2 \, \arcsin \left (\frac{2}{3 \,{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.206904, size = 96, normalized size = 3.2 \[ -\frac{9 \, x^{2} + 4 \,{\left (3 \, x - \sqrt{9 \, x^{2} - 4}\right )} \arctan \left (-\frac{3}{2} \, x + \frac{1}{2} \, \sqrt{9 \, x^{2} - 4}\right ) - 3 \, \sqrt{9 \, x^{2} - 4} x - 4}{3 \, x - \sqrt{9 \, x^{2} - 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.24578, size = 94, normalized size = 3.13 \[ \begin{cases} - \frac{3 i x}{\sqrt{-1 + \frac{4}{9 x^{2}}}} - 2 i \operatorname{acosh}{\left (\frac{2}{3 x} \right )} + \frac{4 i}{3 x \sqrt{-1 + \frac{4}{9 x^{2}}}} & \text{for}\: \frac{4 \left |{\frac{1}{x^{2}}}\right |}{9} > 1 \\\frac{3 x}{\sqrt{1 - \frac{4}{9 x^{2}}}} + 2 \operatorname{asin}{\left (\frac{2}{3 x} \right )} - \frac{4}{3 x \sqrt{1 - \frac{4}{9 x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.205206, size = 32, normalized size = 1.07 \[ \sqrt{9 \, x^{2} - 4} - 2 \, \arctan \left (\frac{1}{2} \, \sqrt{9 \, x^{2} - 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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