Optimal. Leaf size=12 \[ -\frac{2 \cos (x)}{\sqrt{\sin (x)+1}} \]
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Rubi [A] time = 0.0063663, 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} \[ -\frac{2 \cos (x)}{\sqrt{\sin (x)+1}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{1+\sin (x)} \, dx &=-\frac{2 \cos (x)}{\sqrt{1+\sin (x)}}\\ \end{align*}
Mathematica [B] time = 0.011964, size = 40, normalized size = 3.33 \[ \frac{2 \sqrt{\sin (x)+1} \left (\sin \left (\frac{x}{2}\right )-\cos \left (\frac{x}{2}\right )\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 17, normalized size = 1.4 \begin{align*} 2\,{\frac{ \left ( -1+\sin \left ( x \right ) \right ) \sqrt{1+\sin \left ( x \right ) }}{\cos \left ( 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 (x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65851, size = 88, normalized size = 7.33 \begin{align*} -\frac{2 \,{\left (\cos \left (x\right ) - \sin \left (x\right ) + 1\right )} \sqrt{\sin \left (x\right ) + 1}}{\cos \left (x\right ) + \sin \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{\sin{\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{\sin \left (x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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