Optimal. Leaf size=26 \[ -2 \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right ) \]
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Rubi [A] time = 0.0776947, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2775, 204} \[ -2 \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right ) \]
Antiderivative was successfully verified.
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Rule 2775
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt{1-\cos (x)}}{\sqrt{a-\cos (x)}} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right )\\ \end{align*}
Mathematica [C] time = 0.0674033, size = 47, normalized size = 1.81 \[ i \sqrt{2-2 \cos (x)} \csc \left (\frac{x}{2}\right ) \log \left (\sqrt{a-\cos (x)}+i \sqrt{2} \cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.346, size = 67, normalized size = 2.6 \begin{align*} -{\frac{1}{\sin \left ( x \right ) \left ( -1+\cos \left ( x \right ) \right ) } \left ( 2-2\,\cos \left ( x \right ) \right ) ^{{\frac{3}{2}}}\sqrt{a-\cos \left ( x \right ) }\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-2\,{\frac{-a+\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}} \right ){\frac{1}{\sqrt{-2\,{\frac{-a+\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6862, size = 103, normalized size = 3.96 \begin{align*} \arctan \left (\frac{{\left (a - 2 \, \cos \left (x\right ) - 1\right )} \sqrt{-\cos \left (x\right ) + 1}}{2 \, \sqrt{a - \cos \left (x\right )} \sin \left (x\right )}\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 \frac{\sqrt{1 - \cos{\left (x \right )}}}{\sqrt{a - \cos{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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