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 = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {53, 619, 216} \begin {gather*} -\sin ^{-1}(1-2 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 216
Rule 619
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx &=\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 \begin {gather*} -2 \sin ^{-1}\left (\sqrt {1-x}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 8, normalized size = 1.00 \begin {gather*} 2 \sin ^{-1}\left (\sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.26, size = 14, normalized size = 1.75 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.09, size = 6, normalized size = 0.75 \begin {gather*} 2 \, \arcsin \left (\sqrt {x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 27, normalized size = 3.38 \begin {gather*} \frac {\sqrt {\left (-x +1\right ) x}\, \arcsin \left (2 x -1\right )}{\sqrt {-x +1}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.95, size = 14, normalized size = 1.75 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 1}}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 16, normalized size = 2.00 \begin {gather*} -4\,\mathrm {atan}\left (\frac {\sqrt {1-x}-1}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.97, size = 20, normalized size = 2.50 \begin {gather*} \begin {cases} - 2 i \operatorname {acosh}{\left (\sqrt {x} \right )} & \text {for}\: \left |{x}\right | > 1 \\2 \operatorname {asin}{\left (\sqrt {x} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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