Integrand size = 11, antiderivative size = 31 \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sqrt {\cos (x) \cot (x)} \sec (x) \]
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Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4482, 4485, 2678, 2669} \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sec (x) \sqrt {\cos (x) \cot (x)} \]
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Rule 2669
Rule 2678
Rule 4482
Rule 4485
Rubi steps \begin{align*} \text {integral}& = \int (\cos (x) \cot (x))^{3/2} \, dx \\ & = \frac {\sqrt {\cos (x) \cot (x)} \int \cos ^{\frac {3}{2}}(x) \cot ^{\frac {3}{2}}(x) \, dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}} \\ & = \frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}+\frac {\left (4 \sqrt {\cos (x) \cot (x)}\right ) \int \frac {\cot ^{\frac {3}{2}}(x)}{\sqrt {\cos (x)}} \, dx}{3 \sqrt {\cos (x)} \sqrt {\cot (x)}} \\ & = \frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sqrt {\cos (x) \cot (x)} \sec (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.68 \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2}{3} \left (-4+\cos ^2(x)\right ) \sqrt {\cos (x) \cot (x)} \sec (x) \]
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Time = 1.78 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55
method | result | size |
default | \(\frac {2 \sqrt {\cot \left (x \right ) \cos \left (x \right )}\, \left (\cos \left (x \right )-4 \sec \left (x \right )\right )}{3}\) | \(17\) |
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none
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.74 \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {2 \, {\left (\cos \left (x\right )^{2} - 4\right )} \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}}}{3 \, \cos \left (x\right )} \]
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\[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\int \left (- \sin {\left (x \right )} + \csc {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (23) = 46\).
Time = 0.39 (sec) , antiderivative size = 314, normalized size of antiderivative = 10.13 \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\frac {{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}} {\left ({\left ({\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {9}{2} \, x\right ) + 15 \, \sin \left (\frac {5}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) + {\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {9}{2} \, x\right ) - 15 \, \sin \left (\frac {5}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right ) + {\left ({\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {9}{2} \, x\right ) - 15 \, \sin \left (\frac {5}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) - {\left (\cos \left (\frac {9}{2} \, x\right ) - 15 \, \cos \left (\frac {5}{2} \, x\right ) - \cos \left (\frac {3}{2} \, x\right ) + 15 \, \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {9}{2} \, x\right ) + 15 \, \sin \left (\frac {5}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - 15 \, \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right )\right )}}{6 \, {\left (\cos \left (x\right )^{4} + \sin \left (x\right )^{4} + 2 \, {\left (\cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )^{2} - 2 \, \cos \left (x\right )^{2} + 1\right )}} \]
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\[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\int { {\left (\csc \left (x\right ) - \sin \left (x\right )\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (\csc (x)-\sin (x))^{3/2} \, dx=\int {\left (\frac {1}{\sin \left (x\right )}-\sin \left (x\right )\right )}^{3/2} \,d x \]
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