3.150 \(\int \frac{\sqrt{3-x^2}}{x} \, dx\)

Optimal. Leaf size=37 \[ \sqrt{3-x^2}-\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0473773, antiderivative size = 37, 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 \[ \sqrt{3-x^2}-\sqrt{3} \tanh ^{-1}\left (\frac{\sqrt{3-x^2}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 - x^2]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.37152, size = 29, normalized size = 0.78 \[ \sqrt{- x^{2} + 3} - \sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{- x^{2} + 3}}{3} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0214785, size = 41, normalized size = 1.11 \[ \sqrt{3-x^2}-\sqrt{3} \log \left (\sqrt{9-3 x^2}+3\right )+\sqrt{3} \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 - x^2]/x,x]

[Out]

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

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \[ \sqrt{-{x}^{2}+3}-\sqrt{3}{\it Artanh} \left ({\sqrt{3}{\frac{1}{\sqrt{-{x}^{2}+3}}}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+3)^(1/2)/x,x)

[Out]

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

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Maxima [A]  time = 1.51444, size = 55, normalized size = 1.49 \[ -\sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{-x^{2} + 3}}{{\left | x \right |}} + \frac{6}{{\left | x \right |}}\right ) + \sqrt{-x^{2} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216663, size = 54, normalized size = 1.46 \[ \frac{1}{2} \, \sqrt{3} \log \left (-\frac{x^{2} + 2 \, \sqrt{3} \sqrt{-x^{2} + 3} - 6}{x^{2}}\right ) + \sqrt{-x^{2} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(3)*log(-(x^2 + 2*sqrt(3)*sqrt(-x^2 + 3) - 6)/x^2) + sqrt(-x^2 + 3)

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Sympy [A]  time = 2.22713, size = 88, normalized size = 2.38 \[ \begin{cases} i \sqrt{x^{2} - 3} - \sqrt{3} \log{\left (x \right )} + \frac{\sqrt{3} \log{\left (x^{2} \right )}}{2} + \sqrt{3} i \operatorname{asin}{\left (\frac{\sqrt{3}}{x} \right )} & \text{for}\: \frac{\left |{x^{2}}\right |}{3} > 1 \\\sqrt{- x^{2} + 3} + \frac{\sqrt{3} \log{\left (x^{2} \right )}}{2} - \sqrt{3} \log{\left (\sqrt{- \frac{x^{2}}{3} + 1} + 1 \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+3)**(1/2)/x,x)

[Out]

Piecewise((I*sqrt(x**2 - 3) - sqrt(3)*log(x) + sqrt(3)*log(x**2)/2 + sqrt(3)*I*a
sin(sqrt(3)/x), Abs(x**2)/3 > 1), (sqrt(-x**2 + 3) + sqrt(3)*log(x**2)/2 - sqrt(
3)*log(sqrt(-x**2/3 + 1) + 1), True))

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GIAC/XCAS [A]  time = 0.226267, size = 63, normalized size = 1.7 \[ \frac{1}{2} \, \sqrt{3}{\rm ln}\left (\frac{\sqrt{3} - \sqrt{-x^{2} + 3}}{\sqrt{3} + \sqrt{-x^{2} + 3}}\right ) + \sqrt{-x^{2} + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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