Integrand size = 11, antiderivative size = 118 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}-\frac {5 \arctan \left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}} \]
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Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used = {4482, 4485, 2677, 2679, 2681, 2644, 335, 218, 212, 209} \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=-\frac {5 \sqrt {-\sin (x)} \cot (x) \arctan \left (\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sqrt {-\sin (x)} \cot (x) \text {arctanh}\left (\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}+\frac {\tan ^2(x) \sec ^3(x)}{6 \sqrt {\cos (x) \cot (x)}} \]
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 2644
Rule 2677
Rule 2679
Rule 2681
Rule 4482
Rule 4485
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(\cos (x) \cot (x))^{7/2}} \, dx \\ & = \frac {\left (\sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {1}{\cos ^{\frac {7}{2}}(x) \cot ^{\frac {7}{2}}(x)} \, dx}{\sqrt {\cos (x) \cot (x)}} \\ & = \frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {1}{\cos ^{\frac {7}{2}}(x) \cot ^{\frac {3}{2}}(x)} \, dx}{12 \sqrt {\cos (x) \cot (x)}} \\ & = -\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}+\frac {\left (5 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {\sqrt {\cot (x)}}{\cos ^{\frac {7}{2}}(x)} \, dx}{96 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}+\frac {\left (5 \sqrt {\cos (x)} \sqrt {\cot (x)}\right ) \int \frac {\sqrt {\cot (x)}}{\cos ^{\frac {3}{2}}(x)} \, dx}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}+\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \int \frac {\sec (x)}{\sqrt {-\sin (x)}} \, dx}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-x^2\right )} \, dx,x,-\sin (x)\right )}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\sqrt {-\sin (x)}\right )}{64 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}}-\frac {\left (5 \cot (x) \sqrt {-\sin (x)}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-\sin (x)}\right )}{128 \sqrt {\cos (x) \cot (x)}} \\ & = \frac {5 \sec (x)}{192 \sqrt {\cos (x) \cot (x)}}-\frac {5 \sec ^3(x)}{48 \sqrt {\cos (x) \cot (x)}}-\frac {5 \arctan \left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}-\frac {5 \text {arctanh}\left (\sqrt {-\sin (x)}\right ) \cot (x) \sqrt {-\sin (x)}}{128 \sqrt {\cos (x) \cot (x)}}+\frac {\sec ^3(x) \tan ^2(x)}{6 \sqrt {\cos (x) \cot (x)}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\frac {15 \arctan \left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+15 \text {arctanh}\left (\sqrt [4]{\sin ^2(x)}\right ) \cos (x)+2 \sec (x) \left (5-52 \sec ^2(x)+32 \sec ^4(x)\right ) \sqrt [4]{\sin ^2(x)}}{384 \sqrt {\cos (x) \cot (x)} \sqrt [4]{\sin ^2(x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(185\) vs. \(2(90)=180\).
Time = 16.42 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.58
method | result | size |
default | \(\frac {\tan \left (x \right )^{2} \sec \left (x \right )^{3} \left (15 \cos \left (x \right )^{6} \arctan \left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )-15 \cos \left (x \right )^{6} \operatorname {arctanh}\left (\sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )-10 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{5}-10 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{4}+104 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{3}+104 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )^{2}-64 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \cos \left (x \right )-64 \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\right )}{384 \left (\cos \left (x \right )-1\right ) \left (\cos \left (x \right )+1\right )^{2} \sqrt {\frac {\sin \left (x \right )}{\left (\cos \left (x \right )+1\right )^{2}}}\, \sqrt {\cot \left (x \right ) \cos \left (x \right )}}\) | \(186\) |
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Time = 0.30 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=-\frac {30 \, \arctan \left (\frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right ) \sin \left (x\right ) - \cos \left (x\right )}\right ) \cos \left (x\right )^{7} - 15 \, \cos \left (x\right )^{7} \log \left (\frac {\cos \left (x\right )^{3} - 5 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} + 6 \, \cos \left (x\right ) + 4\right )} \sin \left (x\right ) + 4 \, {\left (\cos \left (x\right )^{2} - {\left (\cos \left (x\right ) + 1\right )} \sin \left (x\right ) - 1\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} - 2 \, \cos \left (x\right ) + 4}{\cos \left (x\right )^{3} + 3 \, \cos \left (x\right )^{2} - {\left (\cos \left (x\right )^{2} - 2 \, \cos \left (x\right ) - 4\right )} \sin \left (x\right ) - 2 \, \cos \left (x\right ) - 4}\right ) - 8 \, {\left (5 \, \cos \left (x\right )^{4} - 52 \, \cos \left (x\right )^{2} + 32\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{1536 \, \cos \left (x\right )^{7}} \]
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Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int { \frac {1}{{\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(\csc (x)-\sin (x))^{7/2}} \, dx=\int \frac {1}{{\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{7/2}} \,d x \]
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