Integrand size = 11, antiderivative size = 13 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=2 \sqrt {\cos (x) \cot (x)} \tan (x) \]
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Time = 0.06 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4482, 4485, 2669} \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=2 \tan (x) \sqrt {\cos (x) \cot (x)} \]
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Rule 2669
Rule 4482
Rule 4485
Rubi steps \begin{align*} \text {integral}& = \int \sqrt {\cos (x) \cot (x)} \, dx \\ & = \frac {\sqrt {\cos (x) \cot (x)} \int \sqrt {\cos (x)} \sqrt {\cot (x)} \, dx}{\sqrt {\cos (x)} \sqrt {\cot (x)}} \\ & = 2 \sqrt {\cos (x) \cot (x)} \tan (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=2 \sqrt {\cos (x) \cot (x)} \tan (x) \]
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Time = 1.78 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
default | \(2 \sqrt {\cot \left (x \right ) \cos \left (x \right )}\, \tan \left (x \right )\) | \(12\) |
risch | \(-\frac {i \sqrt {2}\, \sqrt {\frac {i \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-i x}}{{\mathrm e}^{2 i x}-1}}\, \left ({\mathrm e}^{2 i x}-1\right )}{{\mathrm e}^{2 i x}+1}\) | \(51\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\frac {2 \, \sqrt {\frac {\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right )} \]
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\[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\int \sqrt {- \sin {\left (x \right )} + \csc {\left (x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (11) = 22\).
Time = 0.41 (sec) , antiderivative size = 188, normalized size of antiderivative = 14.46 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\frac {{\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) - {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right ) - {\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right ) + {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) - 1\right )\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right )\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}} {\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )}^{\frac {1}{4}}} \]
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\[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\int { \sqrt {\csc \left (x\right ) - \sin \left (x\right )} \,d x } \]
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Time = 27.75 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.15 \[ \int \sqrt {\csc (x)-\sin (x)} \, dx=\frac {2\,\left |\cos \left (x\right )\right |}{\cos \left (x\right )\,\sqrt {\frac {1}{\sin \left (x\right )}}} \]
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