Optimal. Leaf size=17 \[ \frac{\cos (2 x)}{\sqrt{1-\sin (2 x)}} \]
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Rubi [A] time = 0.0119692, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2646} \[ \frac{\cos (2 x)}{\sqrt{1-\sin (2 x)}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{1-\sin (2 x)} \, dx &=\frac{\cos (2 x)}{\sqrt{1-\sin (2 x)}}\\ \end{align*}
Mathematica [A] time = 0.0135429, size = 27, normalized size = 1.59 \[ \frac{\sqrt{1-\sin (2 x)} (\sin (x)+\cos (x))}{\cos (x)-\sin (x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 31, normalized size = 1.8 \begin{align*} -{\frac{ \left ( -1+\sin \left ( 2\,x \right ) \right ) \left ( 1+\sin \left ( 2\,x \right ) \right ) }{\cos \left ( 2\,x \right ) }{\frac{1}{\sqrt{1-\sin \left ( 2\,x \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\sin \left (2 \, x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81164, size = 99, normalized size = 5.82 \begin{align*} \frac{{\left (\cos \left (2 \, x\right ) + \sin \left (2 \, x\right ) + 1\right )} \sqrt{-\sin \left (2 \, x\right ) + 1}}{\cos \left (2 \, x\right ) - \sin \left (2 \, x\right ) + 1} \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{1 - \sin{\left (2 x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-\sin \left (2 \, x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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