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 = 0.0683295, antiderivative size = 31, normalized size of antiderivative = 1., 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 4400
Rule 2598
Rule 2589
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*}
Mathematica [A] time = 0.0384839, 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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Maple [A] time = 0.117, size = 26, normalized size = 0.8 \begin{align*}{\frac{ \left ( 2\, \left ( \cos \left ( x \right ) \right ) ^{2}-8 \right ) \sin \left ( x \right ) }{3\, \left ( \cos \left ( x \right ) \right ) ^{3}} \left ({\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{\sin \left ( x \right ) }} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.8546, size = 424, normalized size = 13.68 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39589, size = 66, normalized size = 2.13 \begin{align*} \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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12894, size = 26, normalized size = 0.84 \begin{align*} -\frac{2}{3} \,{\left (\sin \left (x\right )^{\frac{3}{2}} + \frac{3}{\sqrt{\sin \left (x\right )}}\right )} \mathrm{sgn}\left (\cos \left (x\right )\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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