Optimal. Leaf size=18 \[ \frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {688, 207} \[ \frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 688
Rubi steps
\begin {align*} \int \frac {1}{(2-2 x) \sqrt {2 x-x^2}} \, dx &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sqrt {2 x-x^2}\right )\right )\\ &=\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )\\ \end {align*}
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Mathematica [B] time = 0.01, size = 38, normalized size = 2.11 \[ -\frac {\sqrt {x-2} \sqrt {x} \tan ^{-1}\left (\frac {\sqrt {x-2}}{\sqrt {x}}\right )}{\sqrt {-((x-2) x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.79, size = 44, normalized size = 2.44 \[ \frac {1}{2} \, \log \left (\frac {x + \sqrt {-x^{2} + 2 \, x}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - \sqrt {-x^{2} + 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 26, normalized size = 1.44 \[ -\frac {1}{2} \, \log \left (-\frac {2 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 15, normalized size = 0.83 \[ \frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 31, normalized size = 1.72 \[ \frac {1}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac {2}{{\left | x - 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ -\int \frac {1}{\left (2\,x-2\right )\,\sqrt {2\,x-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {1}{x \sqrt {- x^{2} + 2 x} - \sqrt {- x^{2} + 2 x}}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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