Optimal. Leaf size=13 \[ 2 \tan (x) \sqrt {\cos (x) \cot (x)} \]
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Rubi [A] time = 0.05, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4397, 4400, 2589} \[ 2 \tan (x) \sqrt {\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
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Rule 2589
Rule 4397
Rule 4400
Rubi steps
\begin {align*} \int \sqrt {\csc (x)-\sin (x)} \, dx &=\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*}
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Mathematica [A] time = 0.02, size = 13, normalized size = 1.00 \[ 2 \tan (x) \sqrt {\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 19, normalized size = 1.46 \[ \frac {2 \, \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}} \sin \relax (x)}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\csc \relax (x) - \sin \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.23, size = 20, normalized size = 1.54 \[ \frac {2 \sin \relax (x ) \sqrt {\frac {\cos ^{2}\relax (x )}{\sin \relax (x )}}}{\cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 188, normalized size = 14.46 \[ \frac {{\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right ) - {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )\right ) - {\left ({\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) - \sin \left (\frac {3}{2} \, x\right ) - \sin \left (\frac {1}{2} \, x\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right ) + {\left (\cos \left (\frac {3}{2} \, x\right ) - \cos \left (\frac {1}{2} \, x\right ) + \sin \left (\frac {3}{2} \, x\right ) + \sin \left (\frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) - 1\right )\right )\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )\right )}{{\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right )}^{\frac {1}{4}} {\left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right )}^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.48, size = 15, normalized size = 1.15 \[ \frac {2\,\left |\cos \relax (x)\right |}{\cos \relax (x)\,\sqrt {\frac {1}{\sin \relax (x)}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- \sin {\relax (x )} + \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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