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

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

[Out]

ArcSinh[x]

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Rubi [A]  time = 0.00320783, antiderivative size = 2, 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 \[ \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[1 + x^2],x]

[Out]

ArcSinh[x]

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Rubi in Sympy [A]  time = 0.076226, size = 2, normalized size = 1. \[ \operatorname{asinh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(x)

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Mathematica [A]  time = 0.00498694, size = 2, normalized size = 1. \[ \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[1 + x^2],x]

[Out]

ArcSinh[x]

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Maple [A]  time = 0., size = 3, normalized size = 1.5 \[{\it Arcsinh} \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)^(1/2),x)

[Out]

arcsinh(x)

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Maxima [A]  time = 1.54356, size = 3, normalized size = 1.5 \[ \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arcsinh(x)

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Fricas [A]  time = 0.204743, size = 19, normalized size = 9.5 \[ -\log \left (-x + \sqrt{x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(-x + sqrt(x^2 + 1))

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Sympy [A]  time = 0.136914, size = 2, normalized size = 1. \[ \operatorname{asinh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2+1)**(1/2),x)

[Out]

asinh(x)

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GIAC/XCAS [A]  time = 0.200105, size = 19, normalized size = 9.5 \[ -{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(-x + sqrt(x^2 + 1))