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 = 0.0169694, antiderivative size = 36, normalized size of antiderivative = 1., 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} \[ \frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right )-\frac{1}{2 \sqrt{2 x-x^2}} \]
Antiderivative was successfully verified.
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Rule 687
Rule 688
Rule 207
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*}
Mathematica [A] time = 0.0203524, size = 42, normalized size = 1.17 \[ -\frac{2 \sqrt{x-2} \sqrt{x} \tan ^{-1}\left (\sqrt{\frac{x-2}{x}}\right )+1}{2 \sqrt{-(x-2) x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.048, size = 29, normalized size = 0.8 \begin{align*} -{\frac{1}{2}{\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}}}+{\frac{1}{2}{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( -1+x \right ) ^{2}+1}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12103, size = 61, normalized size = 1.69 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.01464, size = 169, normalized size = 4.69 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35855, size = 66, normalized size = 1.83 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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