Optimal. Leaf size=13 \[ 2 \tan (x) \sqrt{\cos (x) \cot (x)} \]
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Rubi [A] time = 0.0381709, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4400, 2589} \[ 2 \tan (x) \sqrt{\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
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Rule 4400
Rule 2589
Rubi steps
\begin{align*} \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*}
Mathematica [A] time = 0.0687166, size = 13, normalized size = 1. \[ 2 \tan (x) \sqrt{\cos (x) \cot (x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.132, size = 20, normalized size = 1.5 \begin{align*} 2\,{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }\sqrt{{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{\sin \left ( x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.79664, size = 254, normalized size = 19.54 \begin{align*} \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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38723, size = 53, normalized size = 4.08 \begin{align*} \frac{2 \, \sqrt{\frac{\cos \left (x\right )^{2}}{\sin \left (x\right )}} \sin \left (x\right )}{\cos \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\cos{\left (x \right )} \cot{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14641, size = 16, normalized size = 1.23 \begin{align*} 2 \, \mathrm{sgn}\left (\cos \left (x\right )\right ) \mathrm{sgn}\left (\sin \left (x\right )\right ) \sqrt{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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