Optimal. Leaf size=31 \[ \frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sec (x) \sqrt {\cos (x) \cot (x)} \]
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Rubi [A] time = 0.07, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4400, 2598, 2589} \[ \frac {2}{3} \cos (x) \sqrt {\cos (x) \cot (x)}-\frac {8}{3} \sec (x) \sqrt {\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
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Rule 2589
Rule 2598
Rule 4400
Rubi steps
\begin {align*} \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*}
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Mathematica [A] time = 0.04, size = 21, normalized size = 0.68 \[ \frac {2}{3} \left (\cos ^2(x)-4\right ) \sec (x) \sqrt {\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 23, normalized size = 0.74 \[ \frac {2 \, {\left (\cos \relax (x)^{2} - 4\right )} \sqrt {\frac {\cos \relax (x)^{2}}{\sin \relax (x)}}}{3 \, \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 19, normalized size = 0.61 \[ -\frac {2}{3} \, {\left (\sin \relax (x)^{\frac {3}{2}} + \frac {3}{\sqrt {\sin \relax (x)}}\right )} \mathrm {sgn}\left (\cos \relax (x)\right ) \mathrm {sgn}\left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 26, normalized size = 0.84 \[ \frac {2 \left (\cos ^{2}\relax (x )-4\right ) \left (\frac {\cos ^{2}\relax (x )}{\sin \relax (x )}\right )^{\frac {3}{2}} \sin \relax (x )}{3 \cos \relax (x )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 314, normalized size = 10.13 \[ \frac {{\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}} {\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 \relax (x), \cos \relax (x) - 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 \relax (x), \cos \relax (x) - 1\right )\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 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 \relax (x), \cos \relax (x) - 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 \relax (x), \cos \relax (x) - 1\right )\right )\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right )\right )\right )}}{6 \, {\left (\cos \relax (x)^{4} + \sin \relax (x)^{4} + 2 \, {\left (\cos \relax (x)^{2} + 1\right )} \sin \relax (x)^{2} - 2 \, \cos \relax (x)^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (\cos \relax (x)\,\mathrm {cot}\relax (x)\right )}^{3/2} \,d x \]
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 \[ \int \left (\cos {\relax (x )} \cot {\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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