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 = {687, 688, 207} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )-\frac {1}{2 \sqrt {2 x-x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 687
Rule 688
Rubi steps
\begin {align*} \int \frac {1}{(2-2 x) \left (2 x-x^2\right )^{3/2}} \, 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 = 0.02, size = 42, normalized size = 1.17 \begin {gather*} -\frac {2 \sqrt {x-2} \sqrt {x} \tan ^{-1}\left (\sqrt {\frac {x-2}{x}}\right )+1}{2 \sqrt {-((x-2) x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.23, size = 44, normalized size = 1.22 \begin {gather*} \frac {\sqrt {2 x-x^2}}{2 (x-2) x}+\tanh ^{-1}\left (\frac {\sqrt {2 x-x^2}}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 79, normalized size = 2.19 \begin {gather*} \frac {{\left (x^{2} - 2 \, x\right )} \log \left (\frac {x + \sqrt {-x^{2} + 2 \, x}}{x}\right ) - {\left (x^{2} - 2 \, x\right )} \log \left (-\frac {x - \sqrt {-x^{2} + 2 \, x}}{x}\right ) + \sqrt {-x^{2} + 2 \, x}}{2 \, {\left (x^{2} - 2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 49, normalized size = 1.36 \begin {gather*} \frac {\sqrt {-x^{2} + 2 \, x}}{2 \, {\left (x^{2} - 2 \, x\right )}} - \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 = 29, normalized size = 0.81 \begin {gather*} \frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2}-\frac {1}{2 \sqrt {-\left (x -1\right )^{2}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 45, normalized size = 1.25 \begin {gather*} -\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 = -0.03 \begin {gather*} -\int \frac {1}{\left (2\,x-2\right )\,{\left (2\,x-x^2\right )}^{3/2}} \,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 {1}{- x^{3} \sqrt {- x^{2} + 2 x} + 3 x^{2} \sqrt {- x^{2} + 2 x} - 2 x \sqrt {- x^{2} + 2 x}}\, dx}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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