Integrand size = 13, antiderivative size = 61 \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=-\frac {1}{16} \arcsin (\cos (x)-\sin (x))+\frac {1}{16} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}} \]
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Time = 0.07 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {4379, 4393, 4390} \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=-\frac {1}{16} \arcsin (\cos (x)-\sin (x))+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{16} \log \left (\sin (x)+\sqrt {\sin (2 x)}+\cos (x)\right ) \]
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Rule 4379
Rule 4390
Rule 4393
Rubi steps \begin{align*} \text {integral}& = \frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {1}{4} \int \frac {\sin ^3(x)}{\sin ^{\frac {3}{2}}(2 x)} \, dx \\ & = \frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{16} \int \csc (x) \sqrt {\sin (2 x)} \, dx \\ & = \frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}}+\frac {1}{8} \int \frac {\cos (x)}{\sqrt {\sin (2 x)}} \, dx \\ & = -\frac {1}{16} \arcsin (\cos (x)-\sin (x))+\frac {1}{16} \log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )+\frac {\sin ^5(x)}{5 \sin ^{\frac {5}{2}}(2 x)}-\frac {\sin (x)}{4 \sqrt {\sin (2 x)}} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.82 \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\frac {1}{80} \left (5 \left (-\arcsin (\cos (x)-\sin (x))+\log \left (\cos (x)+\sin (x)+\sqrt {\sin (2 x)}\right )\right )+2 \sec (x) \left (-6+\sec ^2(x)\right ) \sqrt {\sin (2 x)}\right ) \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 6.09 (sec) , antiderivative size = 510, normalized size of antiderivative = 8.36
method | result | size |
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 (5 \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 ^{14}\left (\frac {x}{2}\right )\right )+35 \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 ^{12}\left (\frac {x}{2}\right )\right )+10 \left (\tan ^{15}\left (\frac {x}{2}\right )\right )+105 \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 ^{10}\left (\frac {x}{2}\right )\right )+66 \left (\tan ^{13}\left (\frac {x}{2}\right )\right )+175 \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 ^{8}\left (\frac {x}{2}\right )\right )-1014 \left (\tan ^{11}\left (\frac {x}{2}\right )\right )+175 \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 ^{6}\left (\frac {x}{2}\right )\right )+2002 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )+105 \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 ^{4}\left (\frac {x}{2}\right )\right )-2002 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )+35 \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 )+1014 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )+5 \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 )-66 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )-10 \tan \left (\frac {x}{2}\right )\right )}{2688 \sqrt {\tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )}\, \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{7} \sqrt {\tan ^{3}\left (\frac {x}{2}\right )-\tan \left (\frac {x}{2}\right )}}\) | \(510\) |
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Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (47) = 94\).
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.97 \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\frac {10 \, \arctan \left (-\frac {\sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} {\left (\cos \left (x\right ) - \sin \left (x\right )\right )} + \cos \left (x\right ) \sin \left (x\right )}{\cos \left (x\right )^{2} + 2 \, \cos \left (x\right ) \sin \left (x\right ) - 1}\right ) \cos \left (x\right )^{3} - 10 \, \arctan \left (-\frac {2 \, \sqrt {2} \sqrt {\cos \left (x\right ) \sin \left (x\right )} - \cos \left (x\right ) - \sin \left (x\right )}{\cos \left (x\right ) - \sin \left (x\right )}\right ) \cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{3} \log \left (-32 \, \cos \left (x\right )^{4} + 4 \, \sqrt {2} {\left (4 \, \cos \left (x\right )^{3} - {\left (4 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right ) - 5 \, \cos \left (x\right )\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )} + 32 \, \cos \left (x\right )^{2} + 16 \, \cos \left (x\right ) \sin \left (x\right ) + 1\right ) - 48 \, \cos \left (x\right )^{3} - 8 \, \sqrt {2} {\left (6 \, \cos \left (x\right )^{2} - 1\right )} \sqrt {\cos \left (x\right ) \sin \left (x\right )}}{320 \, \cos \left (x\right )^{3}} \]
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Timed out. \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\text {Timed out} \]
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\[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int { \frac {\sin \left (x\right )^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int { \frac {\sin \left (x\right )^{7}}{\sin \left (2 \, x\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sin ^7(x)}{\sin ^{\frac {7}{2}}(2 x)} \, dx=\int \frac {{\sin \left (x\right )}^7}{{\sin \left (2\,x\right )}^{7/2}} \,d x \]
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