Optimal. Leaf size=21 \[ \frac{1}{2} \sqrt{x^2+1} x+\frac{1}{2} \sinh ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.00867122, 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]
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Rubi in Sympy [A] time = 0.580111, 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]
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Mathematica [A] time = 0.0067718, 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]
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Maple [A] time = 0.002, 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]
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Maxima [A] time = 0.775904, 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]
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Fricas [A] time = 0.262929, 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]
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Sympy [A] time = 0.447945, 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]
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GIAC/XCAS [A] time = 0.267376, 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]