Optimal. Leaf size=18 \[ \frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {702, 213}
\begin {gather*} \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 702
Rubi steps
\begin {align*} \int \frac {1}{(2-2 x) \sqrt {2 x-x^2}} \, dx &=-\left (4 \text {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sqrt {2 x-x^2}\right )\right )\\ &=\frac {1}{2} \tanh ^{-1}\left (\sqrt {2 x-x^2}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(18)=36\).
time = 0.05, size = 43, normalized size = 2.39 \begin {gather*} -\frac {\sqrt {-2+x} \sqrt {x} \tan ^{-1}\left (1+\sqrt {-2+x} \sqrt {x}-x\right )}{\sqrt {-((-2+x) x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 15, normalized size = 0.83
method | result | size |
default | \(\frac {\arctanh \left (\frac {1}{\sqrt {-\left (x -1\right )^{2}+1}}\right )}{2}\) | \(15\) |
trager | \(-\frac {\ln \left (\frac {\sqrt {-x^{2}+2 x}-1}{x -1}\right )}{2}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 31 vs.
\(2 (14) = 28\).
time = 0.28, size = 31, normalized size = 1.72 \begin {gather*} \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. 44 vs.
\(2 (14) = 28\).
time = 1.70, size = 44, normalized size = 2.44 \begin {gather*} \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 {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 \sqrt {- x^{2} + 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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Giac [A]
time = 1.97, size = 26, normalized size = 1.44 \begin {gather*} -\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 = -0.06 \begin {gather*} -\int \frac {1}{\left (2\,x-2\right )\,\sqrt {2\,x-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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