3.261 \(\int \sqrt{3+x^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{1}{2} \sqrt{x^2+3} x+\frac{3}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.00890737, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{2} \sqrt{x^2+3} x+\frac{3}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[3 + x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 0.592704, size = 24, normalized size = 0.89 \[ \frac{x \sqrt{x^{2} + 3}}{2} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{3} x}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**2 + 3)/2 + 3*asinh(sqrt(3)*x/3)/2

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Mathematica [A]  time = 0.00983404, size = 27, normalized size = 1. \[ \frac{1}{2} \sqrt{x^2+3} x+\frac{3}{2} \sinh ^{-1}\left (\frac{x}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[3 + x^2],x]

[Out]

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

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Maple [A]  time = 0.003, size = 21, normalized size = 0.8 \[{\frac{3}{2}{\it Arcsinh} \left ({\frac{x\sqrt{3}}{3}} \right ) }+{\frac{x}{2}\sqrt{{x}^{2}+3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+3)^(1/2),x)

[Out]

3/2*arcsinh(1/3*x*3^(1/2))+1/2*x*(x^2+3)^(1/2)

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Maxima [A]  time = 1.47996, size = 27, normalized size = 1. \[ \frac{1}{2} \, \sqrt{x^{2} + 3} x + \frac{3}{2} \, \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3} x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x^2 + 3)*x + 3/2*arcsinh(1/3*sqrt(3)*x)

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Fricas [A]  time = 0.215627, size = 109, normalized size = 4.04 \[ -\frac{2 \, x^{4} + 6 \, x^{2} + 3 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 3} x + 3\right )} \log \left (-x + \sqrt{x^{2} + 3}\right ) -{\left (2 \, x^{3} + 3 \, x\right )} \sqrt{x^{2} + 3}}{2 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 3} x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*x^4 + 6*x^2 + 3*(2*x^2 - 2*sqrt(x^2 + 3)*x + 3)*log(-x + sqrt(x^2 + 3))
- (2*x^3 + 3*x)*sqrt(x^2 + 3))/(2*x^2 - 2*sqrt(x^2 + 3)*x + 3)

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Sympy [A]  time = 0.235009, size = 24, normalized size = 0.89 \[ \frac{x \sqrt{x^{2} + 3}}{2} + \frac{3 \operatorname{asinh}{\left (\frac{\sqrt{3} x}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+3)**(1/2),x)

[Out]

x*sqrt(x**2 + 3)/2 + 3*asinh(sqrt(3)*x/3)/2

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GIAC/XCAS [A]  time = 0.207324, size = 34, normalized size = 1.26 \[ \frac{1}{2} \, \sqrt{x^{2} + 3} x - \frac{3}{2} \,{\rm ln}\left (-x + \sqrt{x^{2} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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