Integrand size = 15, antiderivative size = 7 \[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\arcsin \left (\frac {\sin (x)}{2}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3269, 222} \[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\arcsin \left (\frac {\sin (x)}{2}\right ) \]
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Rule 222
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {4-x^2}} \, dx,x,\sin (x)\right ) \\ & = \arcsin \left (\frac {\sin (x)}{2}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.00 \[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\arcsin \left (\frac {\sin (x)}{2}\right ) \]
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Time = 0.58 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\arcsin \left (\frac {\sin \left (x \right )}{2}\right )\) | \(6\) |
default | \(\arcsin \left (\frac {\sin \left (x \right )}{2}\right )\) | \(6\) |
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Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (5) = 10\).
Time = 0.25 (sec) , antiderivative size = 53, normalized size of antiderivative = 7.57 \[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\frac {1}{2} \, \arctan \left (\frac {\sqrt {\cos \left (x\right )^{2} + 3} {\left (\cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 4 \, \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{4} + 6 \, \cos \left (x\right )^{2} - 3}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right )}\right ) \]
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\[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\int \frac {\cos {\left (x \right )}}{\sqrt {- \left (\sin {\left (x \right )} - 2\right ) \left (\sin {\left (x \right )} + 2\right )}}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\arcsin \left (\frac {1}{2} \, \sin \left (x\right )\right ) \]
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none
Time = 0.27 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\arcsin \left (\frac {1}{2} \, \sin \left (x\right )\right ) \]
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Time = 26.85 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \frac {\cos (x)}{\sqrt {4-\sin ^2(x)}} \, dx=\mathrm {asin}\left (\frac {\sin \left (x\right )}{2}\right ) \]
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