Optimal. Leaf size=46 \[ -\frac {3}{2} \sqrt {2 x-x^2}-\frac {1}{2} x \sqrt {2 x-x^2}-\frac {3}{2} \sin ^{-1}(1-x) \]
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Rubi [A]
time = 0.01, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {684, 654, 633,
222} \begin {gather*} -\frac {3}{2} \text {ArcSin}(1-x)-\frac {1}{2} \sqrt {2 x-x^2} x-\frac {3}{2} \sqrt {2 x-x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 633
Rule 654
Rule 684
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {2 x-x^2}} \, dx &=-\frac {1}{2} x \sqrt {2 x-x^2}+\frac {3}{2} \int \frac {x}{\sqrt {2 x-x^2}} \, dx\\ &=-\frac {3}{2} \sqrt {2 x-x^2}-\frac {1}{2} x \sqrt {2 x-x^2}+\frac {3}{2} \int \frac {1}{\sqrt {2 x-x^2}} \, dx\\ &=-\frac {3}{2} \sqrt {2 x-x^2}-\frac {1}{2} x \sqrt {2 x-x^2}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{4}}} \, dx,x,2-2 x\right )\\ &=-\frac {3}{2} \sqrt {2 x-x^2}-\frac {1}{2} x \sqrt {2 x-x^2}-\frac {3}{2} \sin ^{-1}(1-x)\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 49, normalized size = 1.07 \begin {gather*} \frac {x \left (-6+x+x^2\right )+6 \sqrt {-2+x} \sqrt {x} \tanh ^{-1}\left (\frac {1}{\sqrt {\frac {-2+x}{x}}}\right )}{2 \sqrt {-((-2+x) x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.40, size = 35, normalized size = 0.76
method | result | size |
risch | \(\frac {\left (3+x \right ) x \left (x -2\right )}{2 \sqrt {-x \left (x -2\right )}}+\frac {3 \arcsin \left (x -1\right )}{2}\) | \(25\) |
default | \(\frac {3 \arcsin \left (x -1\right )}{2}-\frac {3 \sqrt {-x^{2}+2 x}}{2}-\frac {x \sqrt {-x^{2}+2 x}}{2}\) | \(35\) |
meijerg | \(-\frac {4 i \left (-\frac {i \sqrt {\pi }\, \sqrt {x}\, \sqrt {2}\, \left (5 x +15\right ) \sqrt {1-\frac {x}{2}}}{40}+\frac {3 i \sqrt {\pi }\, \arcsin \left (\frac {\sqrt {2}\, \sqrt {x}}{2}\right )}{4}\right )}{\sqrt {\pi }}\) | \(47\) |
trager | \(\left (-\frac {3}{2}-\frac {x}{2}\right ) \sqrt {-x^{2}+2 x}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+2 x}+x -1\right )}{2}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 36, normalized size = 0.78 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} x - \frac {3}{2} \, \sqrt {-x^{2} + 2 \, x} - \frac {3}{2} \, \arcsin \left (-x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.48, size = 35, normalized size = 0.76 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} {\left (x + 3\right )} - 3 \, \arctan \left (\frac {\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*} \int \frac {x^{2}}{\sqrt {- x \left (x - 2\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.84, size = 23, normalized size = 0.50 \begin {gather*} -\frac {1}{2} \, \sqrt {-x^{2} + 2 \, x} {\left (x + 3\right )} + \frac {3}{2} \, \arcsin \left (x - 1\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 {x^2}{\sqrt {2\,x-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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