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 = {701, 702, 213}
\begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 701
Rule 702
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 \text {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*}
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Mathematica [A]
time = 0.12, size = 57, normalized size = 1.08 \begin {gather*} \frac {-1-6 x+3 x^2+6 (-2+x)^{3/2} x^{3/2} \tan ^{-1}\left (1+\sqrt {-2+x} \sqrt {x}-x\right )}{6 (-((-2+x) x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 42, normalized size = 0.79
method | result | size |
default | \(-\frac {1}{6 \left (-\left (x -1\right )^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{2 \sqrt {-\left (x -1\right )^{2}+1}}+\frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2}\) | \(42\) |
trager | \(\frac {\left (3 x^{2}-6 x -1\right ) \sqrt {-x^{2}+2 x}}{6 \left (x -2\right )^{2} x^{2}}-\frac {\ln \left (\frac {\sqrt {-x^{2}+2 x}-1}{x -1}\right )}{2}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 58, normalized size = 1.09 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs.
\(2 (41) = 82\).
time = 1.86, size = 112, normalized size = 2.11 \begin {gather*} \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 {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^{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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.48, size = 57, normalized size = 1.08 \begin {gather*} \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 {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 {1}{\left (2\,x-2\right )\,{\left (2\,x-x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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