Optimal. Leaf size=12 \[ -\frac {2 \cos (x)}{\sqrt {\sin (x)+1}} \]
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Rubi [A] time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, 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*}
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Mathematica [B] time = 0.01, 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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fricas [B] time = 0.42, size = 24, normalized size = 2.00 \[ -\frac {2 \, {\left (\cos \relax (x) - \sin \relax (x) + 1\right )} \sqrt {\sin \relax (x) + 1}}{\cos \relax (x) + \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.16, size = 22, normalized size = 1.83 \[ 2 \, \sqrt {2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 17, normalized size = 1.42 \[ \frac {2 \left (\sin \relax (x )-1\right ) \sqrt {\sin \relax (x )+1}}{\cos \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sin \relax (x) + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 16, normalized size = 1.33 \[ \frac {2\,\left (\sin \relax (x)-1\right )\,\sqrt {\sin \relax (x)+1}}{\cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sin {\relax (x )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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