Optimal. Leaf size=8 \[ -\sin ^{-1}(1-2 x) \]
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Rubi [A] time = 0.0030581, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {53, 619, 216} \[ -\sin ^{-1}(1-2 x) \]
Antiderivative was successfully verified.
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Rule 53
Rule 619
Rule 216
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*}
Mathematica [A] time = 0.0079737, size = 12, normalized size = 1.5 \[ -2 \sin ^{-1}\left (\sqrt{1-x}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 27, normalized size = 3.4 \begin{align*}{\arcsin \left ( 2\,x-1 \right ) \sqrt{x \left ( 1-x \right ) }{\frac{1}{\sqrt{1-x}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.55005, size = 19, normalized size = 2.38 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.53014, size = 45, normalized size = 5.62 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x + 1}}{\sqrt{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00491, size = 20, normalized size = 2.5 \begin{align*} \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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06984, size = 8, normalized size = 1. \begin{align*} 2 \, \arcsin \left (\sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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