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