Optimal. Leaf size=10 \[ -\sin ^{-1}\left (1-\frac{x}{2}\right ) \]
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Rubi [A] time = 0.0074331, antiderivative size = 10, 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} \[ -\sin ^{-1}\left (1-\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 53
Rule 619
Rule 216
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{4-x} \sqrt{x}} \, dx &=\int \frac{1}{\sqrt{4 x-x^2}} \, dx\\ &=-\left (\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{16}}} \, dx,x,4-2 x\right )\right )\\ &=-\sin ^{-1}\left (1-\frac{x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0099107, size = 14, normalized size = 1.4 \[ -2 \sin ^{-1}\left (\sqrt{1-\frac{x}{4}}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 27, normalized size = 2.7 \begin{align*}{\sqrt{ \left ( -x+4 \right ) x}\arcsin \left ( -1+{\frac{x}{2}} \right ){\frac{1}{\sqrt{-x+4}}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.42676, size = 19, normalized size = 1.9 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x + 4}}{\sqrt{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02496, size = 45, normalized size = 4.5 \begin{align*} -2 \, \arctan \left (\frac{\sqrt{-x + 4}}{\sqrt{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00913, size = 26, normalized size = 2.6 \begin{align*} \begin{cases} - 2 i \operatorname{acosh}{\left (\frac{\sqrt{x}}{2} \right )} & \text{for}\: \frac{\left |{x}\right |}{4} > 1 \\2 \operatorname{asin}{\left (\frac{\sqrt{x}}{2} \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06964, size = 11, normalized size = 1.1 \begin{align*} 2 \, \arcsin \left (\frac{1}{2} \, \sqrt{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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