Optimal. Leaf size=15 \[ \frac {2 a \sin (x)}{\sqrt {a \cos (x)+a}} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2646} \[ \frac {2 a \sin (x)}{\sqrt {a \cos (x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin {align*} \int \sqrt {a+a \cos (x)} \, dx &=\frac {2 a \sin (x)}{\sqrt {a+a \cos (x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.20 \[ 2 \tan \left (\frac {x}{2}\right ) \sqrt {a (\cos (x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 18, normalized size = 1.20 \[ \frac {2 \, \sqrt {a \cos \relax (x) + a} \sin \relax (x)}{\cos \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.37, size = 17, normalized size = 1.13 \[ 2 \, \sqrt {2} \sqrt {a} \mathrm {sgn}\left (\cos \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 25, normalized size = 1.67 \[ \frac {2 a \cos \left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right ) \sqrt {2}}{\sqrt {a \left (\cos ^{2}\left (\frac {x}{2}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.48, size = 12, normalized size = 0.80 \[ 2 \, \sqrt {2} \sqrt {a} \sin \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 34, normalized size = 2.27 \[ \frac {2\,\sqrt {a}\,\sqrt {\cos \relax (x)+1}\,\left (\cos \relax (x)-1+\sin \relax (x)\,1{}\mathrm {i}\right )}{\cos \relax (x)\,1{}\mathrm {i}-\sin \relax (x)+1{}\mathrm {i}} \]
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 \[ \int \sqrt {a \cos {\relax (x )} + a}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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