Optimal. Leaf size=33 \[ -\frac{1}{2} \sqrt{2 x-x^2} (1-x)-\frac{1}{2} \sin ^{-1}(1-x) \]
[Out]
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Rubi [A] time = 0.017401, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{2} \sqrt{2 x-x^2} (1-x)-\frac{1}{2} \sin ^{-1}(1-x) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[2*x - x^2],x]
[Out]
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Rubi in Sympy [A] time = 0.798937, size = 22, normalized size = 0.67 \[ - \frac{\left (- 2 x + 2\right ) \sqrt{- x^{2} + 2 x}}{4} + \frac{\operatorname{asin}{\left (x - 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-x**2+2*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0408164, size = 45, normalized size = 1.36 \[ \frac{1}{2} \sqrt{-(x-2) x} \left (x-\frac{2 \log \left (\sqrt{x-2}+\sqrt{x}\right )}{\sqrt{x-2} \sqrt{x}}-1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[2*x - x^2],x]
[Out]
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Maple [A] time = 0.006, size = 26, normalized size = 0.8 \[ -{\frac{-2\,x+2}{4}\sqrt{-{x}^{2}+2\,x}}+{\frac{\arcsin \left ( -1+x \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-x^2+2*x)^(1/2),x)
[Out]
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Maxima [A] time = 1.53843, size = 49, normalized size = 1.48 \[ \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} x - \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x} - \frac{1}{2} \, \arcsin \left (-x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 2*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.203513, size = 47, normalized size = 1.42 \[ \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x - 1\right )} - \arctan \left (\frac{\sqrt{-x^{2} + 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 2*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- x^{2} + 2 x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x**2+2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.213038, size = 31, normalized size = 0.94 \[ \frac{1}{2} \, \sqrt{-x^{2} + 2 \, x}{\left (x - 1\right )} + \frac{1}{2} \, \arcsin \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-x^2 + 2*x),x, algorithm="giac")
[Out]