Integrand size = 15, antiderivative size = 10 \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=-\arcsin \left (1-\frac {x}{2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {55, 633, 222} \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=-\arcsin \left (1-\frac {x}{2}\right ) \]
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Rule 55
Rule 222
Rule 633
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {4 x-x^2}} \, dx \\ & = -\left (\frac {1}{4} \text {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*}
Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(10)=20\).
Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 4.00 \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=-\frac {2 \sqrt {-4+x} \sqrt {x} \log \left (\sqrt {-4+x}-\sqrt {x}\right )}{\sqrt {-((-4+x) x)}} \]
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Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.90
method | result | size |
meijerg | \(2 \arcsin \left (\frac {\sqrt {x}}{2}\right )\) | \(9\) |
default | \(\frac {\sqrt {\left (4-x \right ) x}\, \arcsin \left (-1+\frac {x}{2}\right )}{\sqrt {4-x}\, \sqrt {x}}\) | \(27\) |
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Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=-2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 2.40 \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=\begin {cases} - 2 i \operatorname {acosh}{\left (\frac {\sqrt {x}}{2} \right )} & \text {for}\: \left |{x}\right | > 4 \\2 \operatorname {asin}{\left (\frac {\sqrt {x}}{2} \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (6) = 12\).
Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40 \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=-2 \, \arctan \left (\frac {\sqrt {-x + 4}}{\sqrt {x}}\right ) \]
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none
Time = 0.29 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=2 \, \arcsin \left (\frac {1}{2} \, \sqrt {x}\right ) \]
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Time = 0.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {1}{\sqrt {4-x} \sqrt {x}} \, dx=-4\,\mathrm {atan}\left (\frac {\sqrt {4-x}-2}{\sqrt {x}}\right ) \]
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