Optimal. Leaf size=53 \[ -\frac{1}{2 \sqrt{2 x-x^2}}-\frac{1}{6 \left (2 x-x^2\right )^{3/2}}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]
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Rubi [A] time = 0.0244244, 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 = {687, 688, 207} \[ -\frac{1}{2 \sqrt{2 x-x^2}}-\frac{1}{6 \left (2 x-x^2\right )^{3/2}}+\frac{1}{2} \tanh ^{-1}\left (\sqrt{2 x-x^2}\right ) \]
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 )^{5/2}} \, dx &=-\frac{1}{6 \left (2 x-x^2\right )^{3/2}}+\int \frac{1}{(2-2 x) \left (2 x-x^2\right )^{3/2}} \, dx\\ &=-\frac{1}{6 \left (2 x-x^2\right )^{3/2}}-\frac{1}{2 \sqrt{2 x-x^2}}+\int \frac{1}{(2-2 x) \sqrt{2 x-x^2}} \, dx\\ &=-\frac{1}{6 \left (2 x-x^2\right )^{3/2}}-\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}{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{align*}
Mathematica [A] time = 0.0427255, size = 50, normalized size = 0.94 \[ \frac{3 x^2+6 (x-2)^{3/2} x^{3/2} \tan ^{-1}\left (\sqrt{\frac{x-2}{x}}\right )-6 x-1}{6 (-(x-2) x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.049, 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}{\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 = 1.07213, size = 78, normalized size = 1.47 \begin{align*} -\frac{1}{2 \, \sqrt{-x^{2} + 2 \, x}} - \frac{1}{6 \,{\left (-x^{2} + 2 \, x\right )}^{\frac{3}{2}}} + \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 = 1.88967, size = 239, normalized size = 4.51 \begin{align*} \frac{3 \,{\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} \log \left (\frac{x + \sqrt{-x^{2} + 2 \, x}}{x}\right ) - 3 \,{\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )} \log \left (-\frac{x - \sqrt{-x^{2} + 2 \, x}}{x}\right ) +{\left (3 \, x^{2} - 6 \, x - 1\right )} \sqrt{-x^{2} + 2 \, x}}{6 \,{\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\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^{5} \sqrt{- x^{2} + 2 x} - 5 x^{4} \sqrt{- x^{2} + 2 x} + 8 x^{3} \sqrt{- x^{2} + 2 x} - 4 x^{2} \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 = 1.40564, size = 77, normalized size = 1.45 \begin{align*} \frac{{\left (3 \,{\left (x - 2\right )} x - 1\right )} \sqrt{-x^{2} + 2 \, x}}{6 \,{\left (x^{2} - 2 \, x\right )}^{2}} - \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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