Optimal. Leaf size=10 \[ -\sin ^{-1}\left (1-\frac {x}{2}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 10, 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}\left (1-\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 53
Rule 216
Rule 619
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*}
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Mathematica [A] time = 0.01, size = 14, normalized size = 1.40 \begin {gather*} -2 \sin ^{-1}\left (\sqrt {1-\frac {x}{4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.04, size = 24, normalized size = 2.40 \begin {gather*} 2 i \log \left (\sqrt {4-x}-i \sqrt {x}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 14, normalized size = 1.40 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 8, normalized size = 0.80 \begin {gather*} 2 \, \arcsin \left (\frac {1}{2} \, \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 = 2.70 \begin {gather*} \frac {\sqrt {\left (-x +4\right ) x}\, \arcsin \left (\frac {x}{2}-1\right )}{\sqrt {-x +4}\, \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.99, size = 14, normalized size = 1.40 \begin {gather*} -2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 16, normalized size = 1.60 \begin {gather*} -4\,\mathrm {atan}\left (\frac {\sqrt {4-x}-2}{\sqrt {x}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.99, size = 26, normalized size = 2.60 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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