Integrand size = 16, antiderivative size = 47 \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=-\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\frac {1}{2} \cos (x) \sqrt {\sin (2 x)}+\frac {1}{2} \sin (x) \sqrt {\sin (2 x)} \]
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Time = 0.07 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4486, 4386, 4391, 4387, 4390} \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\frac {1}{2} \sin (x) \sqrt {\sin (2 x)}+\frac {1}{2} \sqrt {\sin (2 x)} \cos (x)-\frac {1}{2} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \]
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Rule 4386
Rule 4387
Rule 4390
Rule 4391
Rule 4486
Rubi steps \begin{align*} \text {integral}& = \int \left (\cos (x) \sqrt {\sin (2 x)}-\sin (x) \sqrt {\sin (2 x)}\right ) \, dx \\ & = \int \cos (x) \sqrt {\sin (2 x)} \, dx-\int \sin (x) \sqrt {\sin (2 x)} \, dx \\ & = \frac {1}{2} \cos (x) \sqrt {\sin (2 x)}+\frac {1}{2} \sin (x) \sqrt {\sin (2 x)}-\frac {1}{2} \int \frac {\cos (x)}{\sqrt {\sin (2 x)}} \, dx+\frac {1}{2} \int \frac {\sin (x)}{\sqrt {\sin (2 x)}} \, dx \\ & = -\frac {1}{2} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\frac {1}{2} \cos (x) \sqrt {\sin (2 x)}+\frac {1}{2} \sin (x) \sqrt {\sin (2 x)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\frac {1}{2} \left (-\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\cos (x) \sqrt {\sin (2 x)}+\sin (x) \sqrt {\sin (2 x)}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 5.27 (sec) , antiderivative size = 396, normalized size of antiderivative = 8.43
method | result | size |
parts | \(\frac {2 \sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (2 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}+2 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}}-\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )+2 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-2 \tan \left (\frac {x}{2}\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\) | \(396\) |
default | \(\frac {\sqrt {-\frac {\tan \left (\frac {x}{2}\right )}{\tan ^{2}\left (\frac {x}{2}\right )-1}}\, \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right ) \left (4 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-3 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+4 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, E\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1+\tan \left (\frac {x}{2}\right )}\, \sqrt {-2 \tan \left (\frac {x}{2}\right )+2}\, \sqrt {-\tan \left (\frac {x}{2}\right )}\, F\left (\sqrt {1+\tan \left (\frac {x}{2}\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}+4 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-2 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \left (\tan ^{3}\left (\frac {x}{2}\right )\right )+4 \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \tan \left (\frac {x}{2}\right )\right )}{\sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \sqrt {\left (1+\tan \left (\frac {x}{2}\right )\right ) \left (\tan \left (\frac {x}{2}\right )-1\right ) \tan \left (\frac {x}{2}\right )}\, \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}\) | \(442\) |
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Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (35) = 70\).
Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.62 \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\frac {1}{2} \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} + \frac {1}{8} \, \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) \]
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Timed out. \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\text {Timed out} \]
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\[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\int { {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} \sqrt {\sin \left (2 \, x\right )} \,d x } \]
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\[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\int { {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} \sqrt {\sin \left (2 \, x\right )} \,d x } \]
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Timed out. \[ \int (\cos (x)-\sin (x)) \sqrt {\sin (2 x)} \, dx=\int \sqrt {\sin \left (2\,x\right )}\,\left (\cos \left (x\right )-\sin \left (x\right )\right ) \,d x \]
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