Integrand size = 17, antiderivative size = 16 \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=-\frac {1}{3} (4-\sin (2 x))^{3/2} \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2747, 32} \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=-\frac {1}{3} (4-\sin (2 x))^{3/2} \]
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Rule 32
Rule 2747
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \text {Subst}\left (\int \sqrt {4+x} \, dx,x,-\sin (2 x)\right )\right ) \\ & = -\frac {1}{3} (4-\sin (2 x))^{3/2} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=-\frac {1}{3} (4-\sin (2 x))^{3/2} \]
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Time = 0.17 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {\left (4-\sin \left (2 x \right )\right )^{\frac {3}{2}}}{3}\) | \(13\) |
default | \(-\frac {\left (4-\sin \left (2 x \right )\right )^{\frac {3}{2}}}{3}\) | \(13\) |
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Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=\frac {1}{3} \, {\left (\sin \left (2 \, x\right ) - 4\right )} \sqrt {-\sin \left (2 \, x\right ) + 4} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (12) = 24\).
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.81 \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=\frac {\sqrt {4 - \sin {\left (2 x \right )}} \sin {\left (2 x \right )}}{3} - \frac {4 \sqrt {4 - \sin {\left (2 x \right )}}}{3} \]
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Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=-\frac {1}{3} \, {\left (-\sin \left (2 \, x\right ) + 4\right )}^{\frac {3}{2}} \]
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Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=-\frac {1}{3} \, {\left (-\sin \left (2 \, x\right ) + 4\right )}^{\frac {3}{2}} \]
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Time = 0.11 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \cos (2 x) \sqrt {4-\sin (2 x)} \, dx=-\frac {{\left (4-\sin \left (2\,x\right )\right )}^{3/2}}{3} \]
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