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

Optimal. Leaf size=21 \[ \frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x) \]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

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Rubi [A]  time = 0.00627967, antiderivative size = 21, 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 \[ \frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2])/2 + ArcSinh[x]/2

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Rubi in Sympy [A]  time = 0.558377, size = 15, normalized size = 0.71 \[ \frac{x \sqrt{x^{2} + 1}}{2} + \frac{\operatorname{asinh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**2 + 1)/2 + asinh(x)/2

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Mathematica [A]  time = 0.00793974, size = 18, normalized size = 0.86 \[ \frac{1}{2} \left (\sqrt{x^2+1} x+\sinh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x^2],x]

[Out]

(x*Sqrt[1 + x^2] + ArcSinh[x])/2

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \[{\frac{{\it Arcsinh} \left ( x \right ) }{2}}+{\frac{x}{2}\sqrt{{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.51876, size = 20, normalized size = 0.95 \[ \frac{1}{2} \, \sqrt{x^{2} + 1} x + \frac{1}{2} \, \operatorname{arsinh}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(x^2 + 1)*x + 1/2*arcsinh(x)

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Fricas [A]  time = 0.198284, size = 105, normalized size = 5. \[ -\frac{2 \, x^{4} + 2 \, x^{2} +{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )} \log \left (-x + \sqrt{x^{2} + 1}\right ) -{\left (2 \, x^{3} + x\right )} \sqrt{x^{2} + 1}}{2 \,{\left (2 \, x^{2} - 2 \, \sqrt{x^{2} + 1} x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.242164, size = 15, normalized size = 0.71 \[ \frac{x \sqrt{x^{2} + 1}}{2} + \frac{\operatorname{asinh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2+1)**(1/2),x)

[Out]

x*sqrt(x**2 + 1)/2 + asinh(x)/2

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GIAC/XCAS [A]  time = 0.20612, size = 34, normalized size = 1.62 \[ \frac{1}{2} \, \sqrt{x^{2} + 1} x - \frac{1}{2} \,{\rm ln}\left (-x + \sqrt{x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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