Optimal. Leaf size=23 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0086933, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \sqrt{1-x^2} x+\frac{1}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - x^2],x]
[Out]
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Rubi in Sympy [A] time = 0.564831, size = 15, normalized size = 0.65 \[ \frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]
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 = 0.00871474, size = 20, normalized size = 0.87 \[ \frac{1}{2} \left (\sqrt{1-x^2} x+\sin ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - x^2],x]
[Out]
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Maple [A] time = 0.004, size = 18, normalized size = 0.8 \[{\frac{\arcsin \left ( x \right ) }{2}}+{\frac{x}{2}\sqrt{-{x}^{2}+1}} \]
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 = 1.53721, size = 23, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]
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 = 0.204438, size = 109, normalized size = 4.74 \[ -\frac{2 \, x^{3} + 2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )} \arctan \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (x^{3} - 2 \, x\right )} \sqrt{-x^{2} + 1} - 2 \, x}{2 \,{\left (x^{2} + 2 \, \sqrt{-x^{2} + 1} - 2\right )}} \]
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 = 0.220365, size = 15, normalized size = 0.65 \[ \frac{x \sqrt{- x^{2} + 1}}{2} + \frac{\operatorname{asin}{\left (x \right )}}{2} \]
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 = 0.217321, size = 23, normalized size = 1. \[ \frac{1}{2} \, \sqrt{-x^{2} + 1} x + \frac{1}{2} \, \arcsin \left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 1),x, algorithm="giac")
[Out]