Optimal. Leaf size=16 \[ -\frac {2 a \sin (x)}{\sqrt {a-a \cos (x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2646} \[ -\frac {2 a \sin (x)}{\sqrt {a-a \cos (x)}} \]
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 = 19, normalized size = 1.19 \[ -2 \cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 19, normalized size = 1.19 \[ -\frac {2 \, \sqrt {-a \cos \relax (x) + a} {\left (\cos \relax (x) + 1\right )}}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 26, normalized size = 1.62 \[ -2 \, \sqrt {2} {\left (\cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 25, normalized size = 1.56 \[ -\frac {2 \sin \left (\frac {x}{2}\right ) a \cos \left (\frac {x}{2}\right ) \sqrt {2}}{\sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.07, size = 23, normalized size = 1.44 \[ -\frac {2 \, \sqrt {2} \sqrt {a}}{\sqrt {\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 34, normalized size = 2.12 \[ -\frac {2\,\sqrt {a}\,\sqrt {1-\cos \relax (x)}\,\left (\cos \relax (x)+1+\sin \relax (x)\,1{}\mathrm {i}\right )}{\sin \relax (x)-\cos \relax (x)\,1{}\mathrm {i}+1{}\mathrm {i}} \]
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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