Optimal. Leaf size=16 \[ -\frac{2 a \sin (x)}{\sqrt{a-a \cos (x)}} \]
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Rubi [A] time = 0.011432, antiderivative size = 16, normalized size of antiderivative = 1., 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*}
Mathematica [A] time = 0.0068481, 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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Maple [A] time = 0.961, size = 25, normalized size = 1.6 \begin{align*} -2\,{\frac{\sin \left ( x/2 \right ) a\cos \left ( x/2 \right ) \sqrt{2}}{\sqrt{a \left ( \sin \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 = 1.94552, size = 31, normalized size = 1.94 \begin{align*} -\frac{2 \, \sqrt{2} \sqrt{a}}{\sqrt{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55202, size = 59, normalized size = 3.69 \begin{align*} -\frac{2 \, \sqrt{-a \cos \left (x\right ) + a}{\left (\cos \left (x\right ) + 1\right )}}{\sin \left (x\right )} \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{- a \cos{\left (x \right )} + a}\, 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{-a \cos \left (x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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