Optimal. Leaf size=53 \[ -\frac {1}{6} \left (2 x-x^2\right )^{3/2}-\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {685, 688, 207} \begin {gather*} -\frac {1}{6} \left (2 x-x^2\right )^{3/2}-\frac {1}{2} \sqrt {2 x-x^2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 685
Rule 688
Rubi steps
\begin {align*} \int \frac {\left (2 x-x^2\right )^{3/2}}{2-2 x} \, dx &=-\frac {1}{6} \left (2 x-x^2\right )^{3/2}+\int \frac {\sqrt {2 x-x^2}}{2-2 x} \, dx\\ &=-\frac {1}{2} \sqrt {2 x-x^2}-\frac {1}{6} \left (2 x-x^2\right )^{3/2}+\int \frac {1}{(2-2 x) \sqrt {2 x-x^2}} \, dx\\ &=-\frac {1}{2} \sqrt {2 x-x^2}-\frac {1}{6} \left (2 x-x^2\right )^{3/2}-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}{6} \left (2 x-x^2\right )^{3/2}+\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 48, normalized size = 0.91 \begin {gather*} \frac {1}{6} \sqrt {-((x-2) x)} \left (x^2-2 x+\frac {6 \tan ^{-1}\left (\sqrt {\frac {x-2}{x}}\right )}{\sqrt {x-2} \sqrt {x}}-3\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.25, size = 44, normalized size = 0.83 \begin {gather*} \frac {1}{6} \sqrt {2 x-x^2} \left (x^2-2 x-3\right )+\tanh ^{-1}\left (\frac {\sqrt {2 x-x^2}}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 65, normalized size = 1.23 \begin {gather*} \frac {1}{6} \, {\left (x^{2} - 2 \, x - 3\right )} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 47, normalized size = 0.89 \begin {gather*} \frac {1}{6} \, {\left ({\left (x - 2\right )} x - 3\right )} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 42, normalized size = 0.79 \begin {gather*} \frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2}-\frac {\left (-\left (x -1\right )^{2}+1\right )^{\frac {3}{2}}}{6}-\frac {\sqrt {-\left (x -1\right )^{2}+1}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.00, size = 58, normalized size = 1.09 \begin {gather*} -\frac {1}{6} \, {\left (-x^{2} + 2 \, x\right )}^{\frac {3}{2}} - \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 ) \end {gather*}
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 \begin {gather*} -\int \frac {{\left (2\,x-x^2\right )}^{3/2}}{2\,x-2} \,d x \end {gather*}
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 \begin {gather*} - \frac {\int \frac {2 x \sqrt {- x^{2} + 2 x}}{x - 1}\, dx + \int \left (- \frac {x^{2} \sqrt {- x^{2} + 2 x}}{x - 1}\right )\, dx}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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