Optimal. Leaf size=51 \[ -\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {x}{2}-\frac {1}{2} \log (1-x) \]
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Rubi [A] time = 0.11, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {6742, 685, 688, 207} \[ -\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {x}{2}-\frac {1}{2} \log (1-x) \]
Antiderivative was successfully verified.
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Rule 207
Rule 685
Rule 688
Rule 6742
Rubi steps
\begin {align*} \int \frac {x}{-x+\sqrt {2 x-x^2}} \, dx &=\int \left (-\frac {1}{2}-\frac {1}{2 (-1+x)}+\frac {\sqrt {2 x-x^2}}{2 (1-x)}\right ) \, dx\\ &=-\frac {x}{2}-\frac {1}{2} \log (1-x)+\frac {1}{2} \int \frac {\sqrt {2 x-x^2}}{1-x} \, dx\\ &=-\frac {x}{2}-\frac {1}{2} \sqrt {2 x-x^2}-\frac {1}{2} \log (1-x)+\frac {1}{2} \int \frac {1}{(1-x) \sqrt {2 x-x^2}} \, dx\\ &=-\frac {x}{2}-\frac {1}{2} \sqrt {2 x-x^2}-\frac {1}{2} \log (1-x)-2 \operatorname {Subst}\left (\int \frac {1}{-4+4 x^2} \, dx,x,\sqrt {2 x-x^2}\right )\\ &=-\frac {x}{2}-\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {1}{2} \log (1-x)\\ \end {align*}
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Mathematica [A] time = 0.05, size = 39, normalized size = 0.76 \[ \frac {1}{2} \left (-x-\sqrt {-((x-2) x)}-\log (1-x)+\tanh ^{-1}\left (\sqrt {-((x-2) x)}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 66, normalized size = 1.29 \[ -\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \log \left (x - 1\right ) + \frac {1}{2} \, \log \left (\frac {x + \sqrt {-x^{2} + 2 \, x}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - \sqrt {-x^{2} + 2 \, x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 50, normalized size = 0.98 \[ -\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {1}{2} \, \log \left (-\frac {2 \, {\left (\sqrt {-x^{2} + 2 \, x} - 1\right )}}{{\left | -2 \, x + 2 \right |}}\right ) - \frac {1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 38, normalized size = 0.75 \[ -\frac {x}{2}+\frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2}-\frac {\ln \left (x -1\right )}{2}-\frac {\sqrt {-\left (x -1\right )^{2}+1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {x}{x - \sqrt {-x^{2} + 2 \, x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int -\frac {x}{x-\sqrt {2\,x-x^2}} \,d x \]
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 \[ - \int \frac {x}{x - \sqrt {- x^{2} + 2 x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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