Optimal. Leaf size=65 \[ -\frac{2 \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)} \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right )}{\sqrt{1-\cos (x)}} \]
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Rubi [A] time = 0.0999325, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4400, 2775, 204} \[ -\frac{2 \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)} \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right )}{\sqrt{1-\cos (x)}} \]
Antiderivative was successfully verified.
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Rule 4400
Rule 2775
Rule 204
Rubi steps
\begin{align*} \int \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \, dx &=\frac{\left (\sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)}\right ) \int \frac{\sqrt{1-\cos (x)}}{\sqrt{a-\cos (x)}} \, dx}{\sqrt{1-\cos (x)}}\\ &=\frac{\left (2 \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right )}{\sqrt{1-\cos (x)}}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sin (x)}{\sqrt{1-\cos (x)} \sqrt{a-\cos (x)}}\right ) \sqrt{\frac{1-\cos (x)}{a-\cos (x)}} \sqrt{a-\cos (x)}}{\sqrt{1-\cos (x)}}\\ \end{align*}
Mathematica [A] time = 0.068058, size = 64, normalized size = 0.98 \[ -\sqrt{2} \csc \left (\frac{x}{2}\right ) \sqrt{\frac{\cos (x)-1}{\cos (x)-a}} \sqrt{\cos (x)-a} \log \left (\sqrt{\cos (x)-a}+\sqrt{2} \cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.5, size = 67, normalized size = 1. \begin{align*} -{\frac{\sqrt{2}\sin \left ( x \right ) }{-1+\cos \left ( x \right ) }\sqrt{{\frac{-1+\cos \left ( x \right ) }{-a+\cos \left ( x \right ) }}}\sqrt{-2\,{\frac{-a+\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-2\,{\frac{-a+\cos \left ( x \right ) }{\cos \left ( x \right ) +1}}}} \right ) } \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 = 1.55248, size = 100, normalized size = 1.54 \begin{align*} -\arctan \left (-\frac{{\left (a - 2 \, \cos \left (x\right ) - 1\right )} \sqrt{-\frac{\cos \left (x\right ) - 1}{a - \cos \left (x\right )}}}{2 \, \sin \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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