Optimal. Leaf size=18 \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]
[Out]
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Rubi [A] time = 0.0330142, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095 \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((2 - 2*x)*Sqrt[2*x - x^2]),x]
[Out]
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Rubi in Sympy [A] time = 6.14732, size = 12, normalized size = 0.67 \[ \frac{\operatorname{atanh}{\left (\sqrt{- x^{2} + 2 x} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(2-2*x)/(-x**2+2*x)**(1/2),x)
[Out]
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Mathematica [B] time = 0.0325672, size = 59, normalized size = 3.28 \[ -\frac{\sqrt{x-2} \sqrt{x} \left (\tan ^{-1}\left (\frac{\sqrt{x}-2}{\sqrt{x-2}}\right )+\tan ^{-1}\left (\frac{\sqrt{x}+2}{\sqrt{x-2}}\right )\right )}{2 \sqrt{-(x-2) x}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((2 - 2*x)*Sqrt[2*x - x^2]),x]
[Out]
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Maple [A] time = 0.007, size = 15, normalized size = 0.8 \[{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(2-2*x)/(-x^2+2*x)^(1/2),x)
[Out]
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Maxima [A] time = 0.687985, size = 42, normalized size = 2.33 \[ \frac{1}{2} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac{2}{{\left | x - 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/2/(sqrt(-x^2 + 2*x)*(x - 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.221766, size = 59, normalized size = 3.28 \[ \frac{1}{2} \, \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - \frac{1}{2} \, \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/2/(sqrt(-x^2 + 2*x)*(x - 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{\int \frac{1}{x \sqrt{- x^{2} + 2 x} - \sqrt{- x^{2} + 2 x}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(2-2*x)/(-x**2+2*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.220043, size = 35, normalized size = 1.94 \[ -\frac{1}{2} \,{\rm ln}\left (-\frac{2 \,{\left (\sqrt{-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-1/2/(sqrt(-x^2 + 2*x)*(x - 1)),x, algorithm="giac")
[Out]