Integrand size = 13, antiderivative size = 70 \[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=-\frac {4 \cos ^5(x) \sin (x)}{9 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}-\frac {8 \cos ^3(x) \sin ^3(x)}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}+\frac {4 \cos (x) \sin ^5(x)}{7 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \]
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Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6851, 276} \[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=\frac {4 \sin ^5(x) \cos (x)}{7 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}}-\frac {4 \sin (x) \cos ^5(x)}{9 \sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}}-\frac {8 \sin ^3(x) \cos ^3(x)}{\sqrt [4]{\sin ^{13}(x) \cos ^{11}(x)}} \]
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Rule 276
Rule 6851
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\sqrt [4]{\frac {x^{13}}{\left (1+x^2\right )^{12}}} \left (1+x^2\right )} \, dx,x,\tan (x)\right ) \\ & = \frac {\left (\cos ^6(x) \tan ^{\frac {13}{4}}(x)\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^{13/4}} \, dx,x,\tan (x)\right )}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \\ & = \frac {\left (\cos ^6(x) \tan ^{\frac {13}{4}}(x)\right ) \text {Subst}\left (\int \left (\frac {1}{x^{13/4}}+\frac {2}{x^{5/4}}+x^{3/4}\right ) \, dx,x,\tan (x)\right )}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \\ & = -\frac {4 \cos ^5(x) \sin (x)}{9 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}-\frac {8 \cos ^3(x) \sin ^3(x)}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}}+\frac {4 \cos (x) \sin ^5(x)}{7 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.50 \[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=-\frac {4 \cos (x) (15+8 \cos (2 x)-16 \cos (4 x)) \sin (x)}{63 \sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \]
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\[\int \frac {1}{{\left (\left (\cos ^{11}\left (x \right )\right ) \left (\sin ^{13}\left (x \right )\right )\right )}^{\frac {1}{4}}}d x\]
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Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.44 \[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=\frac {4 \, {\left (128 \, \cos \left (x\right )^{4} - 144 \, \cos \left (x\right )^{2} + 9\right )} \left ({\left (\cos \left (x\right )^{23} - 6 \, \cos \left (x\right )^{21} + 15 \, \cos \left (x\right )^{19} - 20 \, \cos \left (x\right )^{17} + 15 \, \cos \left (x\right )^{15} - 6 \, \cos \left (x\right )^{13} + \cos \left (x\right )^{11}\right )} \sin \left (x\right )\right )^{\frac {3}{4}}}{63 \, {\left (\cos \left (x\right )^{22} - 6 \, \cos \left (x\right )^{20} + 15 \, \cos \left (x\right )^{18} - 20 \, \cos \left (x\right )^{16} + 15 \, \cos \left (x\right )^{14} - 6 \, \cos \left (x\right )^{12} + \cos \left (x\right )^{10}\right )}} \]
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Timed out. \[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=\frac {4}{23} \, \tan \left (x\right )^{\frac {23}{4}} + \frac {8}{15} \, \tan \left (x\right )^{\frac {15}{4}} + \frac {4}{7} \, \tan \left (x\right )^{\frac {7}{4}} - \frac {4 \, {\left (35 \, \tan \left (x\right )^{7} + 161 \, \tan \left (x\right )^{5} + 345 \, \tan \left (x\right )^{3} - 805 \, \tan \left (x\right )\right )}}{805 \, \tan \left (x\right )^{\frac {5}{4}}} + \frac {4 \, {\left (21 \, \tan \left (x\right )^{7} + 135 \, \tan \left (x\right )^{5} - 945 \, \tan \left (x\right )^{3} - 35 \, \tan \left (x\right )\right )}}{315 \, \tan \left (x\right )^{\frac {13}{4}}} \]
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\[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=\int { \frac {1}{\left (\cos \left (x\right )^{11} \sin \left (x\right )^{13}\right )^{\frac {1}{4}}} \,d x } \]
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Time = 3.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\sqrt [4]{\cos ^{11}(x) \sin ^{13}(x)}} \, dx=-\frac {2^{3/4}\,\left (-32\,{\cos \left (2\,x\right )}^2+8\,\cos \left (2\,x\right )+31\right )\,{\left (924\,\sin \left (2\,x\right )-132\,\sin \left (4\,x\right )-660\,\sin \left (6\,x\right )+165\,\sin \left (8\,x\right )+330\,\sin \left (10\,x\right )-110\,\sin \left (12\,x\right )-110\,\sin \left (14\,x\right )+44\,\sin \left (16\,x\right )+22\,\sin \left (18\,x\right )-10\,\sin \left (20\,x\right )-2\,\sin \left (22\,x\right )+\sin \left (24\,x\right )\right )}^{3/4}}{2016\,{\left (\cos \left (2\,x\right )-1\right )}^6\,{\left (\cos \left (2\,x\right )+1\right )}^5} \]
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