3.269 \(\int \frac{1}{\sqrt{4+x^2}} \, dx\)

Optimal. Leaf size=6 \[ \sinh ^{-1}\left (\frac{x}{2}\right ) \]

[Out]

ArcSinh[x/2]

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Rubi [A]  time = 0.00359629, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \sinh ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[4 + x^2],x]

[Out]

ArcSinh[x/2]

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Rubi in Sympy [A]  time = 0.5122, size = 3, normalized size = 0.5 \[ \operatorname{asinh}{\left (\frac{x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(x/2)

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Mathematica [A]  time = 0.00512773, size = 6, normalized size = 1. \[ \sinh ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[4 + x^2],x]

[Out]

ArcSinh[x/2]

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Maple [A]  time = 0.002, size = 5, normalized size = 0.8 \[{\it Arcsinh} \left ({\frac{x}{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+4)^(1/2),x)

[Out]

arcsinh(1/2*x)

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Maxima [A]  time = 1.49965, size = 5, normalized size = 0.83 \[ \operatorname{arsinh}\left (\frac{1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + 4),x, algorithm="maxima")

[Out]

arcsinh(1/2*x)

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Fricas [A]  time = 0.201026, size = 19, normalized size = 3.17 \[ -\log \left (-x + \sqrt{x^{2} + 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + 4),x, algorithm="fricas")

[Out]

-log(-x + sqrt(x^2 + 4))

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Sympy [A]  time = 0.14089, size = 3, normalized size = 0.5 \[ \operatorname{asinh}{\left (\frac{x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+4)**(1/2),x)

[Out]

asinh(x/2)

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GIAC/XCAS [A]  time = 0.203885, size = 19, normalized size = 3.17 \[ -{\rm ln}\left (-x + \sqrt{x^{2} + 4}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt(x^2 + 4),x, algorithm="giac")

[Out]

-ln(-x + sqrt(x^2 + 4))