3.138 \(\int \frac{1}{x \sqrt{3+x^2}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+3}}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0290964, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+3}}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[3 + x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 1.82175, size = 24, normalized size = 1.04 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{x^{2} + 3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0140943, size = 29, normalized size = 1.26 \[ \frac{\log (x)-\log \left (\sqrt{3} \sqrt{x^2+3}+3\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 18, normalized size = 0.8 \[ -{\frac{\sqrt{3}}{3}{\it Artanh} \left ({\sqrt{3}{\frac{1}{\sqrt{{x}^{2}+3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(x^2+3)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.48, size = 19, normalized size = 0.83 \[ -\frac{1}{3} \, \sqrt{3} \operatorname{arsinh}\left (\frac{\sqrt{3}}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*sqrt(3)*arcsinh(sqrt(3)/abs(x))

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Fricas [A]  time = 0.207199, size = 70, normalized size = 3.04 \[ \frac{1}{3} \, \sqrt{3} \log \left (\frac{\sqrt{3}{\left (x^{2} + 3\right )} - \sqrt{x^{2} + 3}{\left (\sqrt{3} x + 3\right )} + 3 \, x}{x^{2} - \sqrt{x^{2} + 3} x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.77199, size = 15, normalized size = 0.65 \[ - \frac{\sqrt{3} \operatorname{asinh}{\left (\frac{\sqrt{3}}{x} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(3)*asinh(sqrt(3)/x)/3

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GIAC/XCAS [A]  time = 0.206418, size = 50, normalized size = 2.17 \[ -\frac{1}{6} \, \sqrt{3}{\rm ln}\left (\sqrt{3} + \sqrt{x^{2} + 3}\right ) + \frac{1}{6} \, \sqrt{3}{\rm ln}\left (-\sqrt{3} + \sqrt{x^{2} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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