Integrand size = 16, antiderivative size = 19 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=2 \text {arctanh}\left (\frac {\sin (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3339, 634, 212} \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=2 \text {arctanh}\left (\frac {\sin (x)}{\sqrt {\sin ^2(x)+2 \sin (x)}}\right ) \]
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Rule 212
Rule 634
Rule 3339
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt {2 x+x^2}} \, dx,x,\sin (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sin (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}}\right ) \\ & = 2 \text {arctanh}\left (\frac {\sin (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(40\) vs. \(2(19)=38\).
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.11 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\frac {2 \text {arcsinh}\left (\frac {\sqrt {\sin (x)}}{\sqrt {2}}\right ) \sqrt {\sin (x)} \sqrt {2+\sin (x)}}{\sqrt {\sin (x) (2+\sin (x))}} \]
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Time = 1.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\ln \left (1+\sin \left (x \right )+\sqrt {2 \sin \left (x \right )+\sin \left (x \right )^{2}}\right )\) | \(17\) |
default | \(\ln \left (1+\sin \left (x \right )+\sqrt {2 \sin \left (x \right )+\sin \left (x \right )^{2}}\right )\) | \(17\) |
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Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (17) = 34\).
Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\frac {1}{2} \, \log \left (-2 \, \cos \left (x\right )^{2} + 2 \, \sqrt {-\cos \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1} {\left (\sin \left (x\right ) + 1\right )} + 4 \, \sin \left (x\right ) + 3\right ) \]
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\[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\int \frac {\cos {\left (x \right )}}{\sqrt {\left (\sin {\left (x \right )} + 2\right ) \sin {\left (x \right )}}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\log \left (2 \, \sqrt {\sin \left (x\right )^{2} + 2 \, \sin \left (x\right )} + 2 \, \sin \left (x\right ) + 2\right ) \]
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none
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=-\log \left (-\sqrt {\sin \left (x\right )^{2} + 2 \, \sin \left (x\right )} + \sin \left (x\right ) + 1\right ) \]
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Time = 26.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {\cos (x)}{\sqrt {2 \sin (x)+\sin ^2(x)}} \, dx=\ln \left (\sin \left (x\right )+\sqrt {\sin \left (x\right )\,\left (\sin \left (x\right )+2\right )}+1\right ) \]
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