Optimal. Leaf size=16 \[ -\frac{\cos (2 x)}{\sqrt{\sin (2 x)+1}} \]
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Rubi [A] time = 0.0086676, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2646} \[ -\frac{\cos (2 x)}{\sqrt{\sin (2 x)+1}} \]
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 = 0.0138993, size = 25, normalized size = 1.56 \[ \frac{\sqrt{\sin (2 x)+1} (\sin (x)-\cos (x))}{\sin (x)+\cos (x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 22, normalized size = 1.4 \begin{align*}{\frac{-1+\sin \left ( 2\,x \right ) }{\cos \left ( 2\,x \right ) }\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 = 1.80544, size = 99, normalized size = 6.19 \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{\sin{\left (2 x \right )} + 1}\, 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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