Optimal. Leaf size=12 \[ \frac{2 \sin (x)}{\sqrt{\cos (x)+1}} \]
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Rubi [A] time = 0.007023, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2646} \[ \frac{2 \sin (x)}{\sqrt{\cos (x)+1}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{1+\cos (x)} \, dx &=\frac{2 \sin (x)}{\sqrt{1+\cos (x)}}\\ \end{align*}
Mathematica [A] time = 0.0062206, size = 16, normalized size = 1.33 \[ 2 \sqrt{\cos (x)+1} \tan \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 22, normalized size = 1.8 \begin{align*} 2\,{\frac{\cos \left ( x/2 \right ) \sin \left ( x/2 \right ) \sqrt{2}}{\sqrt{ \left ( \cos \left ( x/2 \right ) \right ) ^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.589, size = 12, normalized size = 1. \begin{align*} 2 \, \sqrt{2} \sin \left (\frac{1}{2} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49828, size = 36, normalized size = 3. \begin{align*} \frac{2 \, \sin \left (x\right )}{\sqrt{\cos \left (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{\cos{\left (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{\cos \left (x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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