Optimal. Leaf size=8 \[ -\sin ^{-1}(1-2 x) \]
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Rubi [A] time = 0.00, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {619, 216} \[ -\sin ^{-1}(1-2 x) \]
Antiderivative was successfully verified.
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Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {x-x^2}} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )\\ &=-\sin ^{-1}(1-2 x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 12, normalized size = 1.50 \[ -2 \sin ^{-1}\left (\sqrt {1-x}\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.08, size = 18, normalized size = 2.25 \[ -2 \tan ^{-1}\left (\frac {\sqrt {x-x^2}}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 16, normalized size = 2.00 \[ -2 \, \arctan \left (\frac {\sqrt {-x^{2} + x}}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.86, size = 6, normalized size = 0.75 \[ \arcsin \left (2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 7, normalized size = 0.88
method | result | size |
default | \(\arcsin \left (-1+2 x \right )\) | \(7\) |
meijerg | \(2 \arcsin \left (\sqrt {x}\right )\) | \(7\) |
trager | \(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {-x^{2}+x}+\RootOf \left (\textit {\_Z}^{2}+1\right )\right )\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 6, normalized size = 0.75 \[ \arcsin \left (2 \, x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 6, normalized size = 0.75 \[ \mathrm {asin}\left (2\,x-1\right ) \]
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 \[ \int \frac {1}{\sqrt {- x^{2} + x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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