Integrand size = 13, antiderivative size = 23 \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=-2 \sqrt {1+\sin (x)}+\frac {2}{3} (1+\sin (x))^{3/2} \]
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Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2912, 45} \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=\frac {2}{3} (\sin (x)+1)^{3/2}-2 \sqrt {\sin (x)+1} \]
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Rule 45
Rule 2912
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x}{\sqrt {1+x}} \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \left (-\frac {1}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,\sin (x)\right ) \\ & = -2 \sqrt {1+\sin (x)}+\frac {2}{3} (1+\sin (x))^{3/2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=\frac {2 \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2 (-2+\sin (x))}{3 \sqrt {1+\sin (x)}} \]
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Time = 0.64 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {2 \left (1+\sin \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\sin \left (x \right )}\) | \(18\) |
default | \(\frac {2 \left (1+\sin \left (x \right )\right )^{\frac {3}{2}}}{3}-2 \sqrt {1+\sin \left (x \right )}\) | \(18\) |
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none
Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=\frac {2}{3} \, \sqrt {\sin \left (x\right ) + 1} {\left (\sin \left (x\right ) - 2\right )} \]
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Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=\frac {2 \sqrt {\sin {\left (x \right )} + 1} \sin {\left (x \right )}}{3} - \frac {4 \sqrt {\sin {\left (x \right )} + 1}}{3} \]
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none
Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=\frac {2}{3} \, {\left (\sin \left (x\right ) + 1\right )}^{\frac {3}{2}} - 2 \, \sqrt {\sin \left (x\right ) + 1} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (17) = 34\).
Time = 0.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.83 \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=\frac {2 \, {\left (2 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )^{3} - 3 \, \sqrt {2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )}}{3 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )} \]
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Time = 0.14 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52 \[ \int \frac {\cos (x) \sin (x)}{\sqrt {1+\sin (x)}} \, dx=\frac {2\,\sqrt {\sin \left (x\right )+1}\,\left (\sin \left (x\right )-2\right )}{3} \]
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