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 = 0.0241723, antiderivative size = 53, normalized size of antiderivative = 1., 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} \[ -\frac{1}{6} \left (2 x-x^2\right )^{3/2}-\frac{1}{2} \sqrt{2 x-x^2}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 685
Rule 688
Rule 207
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*}
Mathematica [A] time = 0.0600379, size = 48, normalized size = 0.91 \[ \frac{1}{6} \sqrt{-(x-2) x} \left (x^2-2 x+\frac{6 \tan ^{-1}\left (\sqrt{\frac{x-2}{x}}\right )}{\sqrt{x-2} \sqrt{x}}-3\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.052, size = 42, normalized size = 0.8 \begin{align*} -{\frac{1}{6} \left ( - \left ( -1+x \right ) ^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{2}\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 = 1.67784, size = 78, normalized size = 1.47 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12967, size = 150, normalized size = 2.83 \begin{align*} \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{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{2 x \sqrt{- x^{2} + 2 x}}{x - 1}\, dx + \int - \frac{x^{2} \sqrt{- x^{2} + 2 x}}{x - 1}\, dx}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31522, size = 63, normalized size = 1.19 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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