Optimal. Leaf size=36 \[ \frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {1}{2} \sqrt {2 x-x^2} \]
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Rubi [A] time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {685, 688, 207} \[ \frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {1}{2} \sqrt {2 x-x^2} \]
Antiderivative was successfully verified.
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Rule 207
Rule 685
Rule 688
Rubi steps
\begin {align*} \int \frac {\sqrt {2 x-x^2}}{2-2 x} \, dx &=-\frac {1}{2} \sqrt {2 x-x^2}+\int \frac {1}{(2-2 x) \sqrt {2 x-x^2}} \, dx\\ &=-\frac {1}{2} \sqrt {2 x-x^2}-4 \operatorname {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sqrt {2 x-x^2}\right )\\ &=-\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.28 \[ \frac {(x-2) x-2 \sqrt {x-2} \sqrt {x} \tan ^{-1}\left (\sqrt {\frac {x-2}{x}}\right )}{2 \sqrt {-((x-2) x)}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.37, size = 57, normalized size = 1.58 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} + \frac {1}{2} \, \log \left (\frac {x + \sqrt {-x^{2} + 2 \, x}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - \sqrt {-x^{2} + 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 40, normalized size = 1.11 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \log \left (-\frac {2 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 0.81 \[ \frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2}-\frac {\sqrt {-\left (x -1\right )^{2}+1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.99, size = 45, normalized size = 1.25 \[ -\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} + \frac {1}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 2 \, x}}{{\left | x - 1 \right |}} + \frac {2}{{\left | x - 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ -\int \frac {\sqrt {2\,x-x^2}}{2\,x-2} \,d x \]
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 \[ - \frac {\int \frac {\sqrt {- x^{2} + 2 x}}{x - 1}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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