Integrand size = 11, antiderivative size = 91 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {3 \arctan \left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \text {arctanh}\left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \]
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Time = 0.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {4482, 4485, 2677, 2679, 2681, 2645, 335, 304, 209, 212} \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=-\frac {3 \sin (x) \arctan \left (\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \sin (x) \text {arctanh}\left (\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}} \]
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 2645
Rule 2677
Rule 2679
Rule 2681
Rule 4482
Rule 4485
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(\sin (x) \tan (x))^{5/2}} \, dx \\ & = \frac {\left (\sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(x) \tan ^{\frac {5}{2}}(x)} \, dx}{\sqrt {\sin (x) \tan (x)}} \\ & = -\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {\left (3 \sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(x) \sqrt {\tan (x)}} \, dx}{8 \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {\left (3 \sqrt {\sin (x)} \sqrt {\tan (x)}\right ) \int \frac {1}{\sqrt {\sin (x)} \sqrt {\tan (x)}} \, dx}{32 \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {(3 \sin (x)) \int \sqrt {\cos (x)} \csc (x) \, dx}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}+\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {\sqrt {x}}{1-x^2} \, dx,x,\cos (x)\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}+\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\sqrt {\cos (x)}\right )}{16 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}+\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}-\frac {(3 \sin (x)) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\cos (x)}\right )}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ & = \frac {3 \cot (x)}{16 \sqrt {\sin (x) \tan (x)}}-\frac {\cot (x) \csc ^2(x)}{4 \sqrt {\sin (x) \tan (x)}}-\frac {3 \arctan \left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}}+\frac {3 \text {arctanh}\left (\sqrt {\cos (x)}\right ) \sin (x)}{32 \sqrt {\cos (x)} \sqrt {\sin (x) \tan (x)}} \\ \end{align*}
Time = 0.73 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=-\frac {\cot (x) \left (3 \arctan \left (\sqrt [4]{\cos ^2(x)}\right ) \cos (x)-3 \text {arctanh}\left (\sqrt [4]{\cos ^2(x)}\right ) \cos (x)+\cos ^2(x)^{3/4} (5+3 \cos (2 x)) \cot (x) \csc ^3(x)\right ) \sqrt {\sin (x) \tan (x)}}{32 \cos ^2(x)^{3/4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(67)=134\).
Time = 1.23 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.14
method | result | size |
default | \(-\frac {12 \cot \left (x \right )^{3} \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-3 \cot \left (x \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )-3 \cot \left (x \right ) \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )+3 \csc \left (x \right ) \arctan \left (\frac {1}{2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\right )+3 \csc \left (x \right ) \ln \left (\frac {2 \cos \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}+2 \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}-\cos \left (x \right )+1}{\cos \left (x \right )+1}\right )+4 \csc \left (x \right )^{2} \cot \left (x \right ) \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}{64 \sqrt {\sin \left (x \right ) \tan \left (x \right )}\, \sqrt {-\frac {\cos \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}}\) | \(195\) |
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Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (67) = 134\).
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\frac {3 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \arctan \left (\frac {2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) + 3 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \log \left (\frac {{\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) + 2 \, \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}} \cos \left (x\right )}{{\left (\cos \left (x\right ) - 1\right )} \sin \left (x\right )}\right ) \sin \left (x\right ) - 4 \, {\left (3 \, \cos \left (x\right )^{4} + \cos \left (x\right )^{2}\right )} \sqrt {-\frac {\cos \left (x\right )^{2} - 1}{\cos \left (x\right )}}}{64 \, {\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )} \]
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\[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\int \frac {1}{\left (- \cos {\left (x \right )} + \sec {\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\int { \frac {1}{{\left (-\cos \left (x\right ) + \sec \left (x\right )\right )}^{\frac {5}{2}}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (67) = 134\).
Time = 0.38 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\frac {{\left (\frac {4 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2}} + 1\right )} \tan \left (\frac {1}{2} \, x\right )^{4}}{256 \, {\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{2}} - \frac {1}{64} \, \sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} {\left (\tan \left (\frac {1}{2} \, x\right )^{2} - 2\right )} - \frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{64 \, \tan \left (\frac {1}{2} \, x\right )^{2}} - \frac {{\left (\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1\right )}^{2}}{256 \, \tan \left (\frac {1}{2} \, x\right )^{4}} - \frac {3}{64} \, \arcsin \left (\tan \left (\frac {1}{2} \, x\right )^{2}\right ) + \frac {3}{64} \, \log \left (-\frac {\sqrt {-\tan \left (\frac {1}{2} \, x\right )^{4} + 1} - 1}{\tan \left (\frac {1}{2} \, x\right )^{2}}\right ) \]
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Timed out. \[ \int \frac {1}{(-\cos (x)+\sec (x))^{5/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\cos \left (x\right )}-\cos \left (x\right )\right )}^{5/2}} \,d x \]
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