Integrand size = 19, antiderivative size = 65 \[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=-\frac {2 \arctan \left (\frac {\sin (x)}{\sqrt {1-\cos (x)} \sqrt {a-\cos (x)}}\right ) \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \sqrt {a-\cos (x)}}{\sqrt {1-\cos (x)}} \]
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Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4485, 2854, 210} \[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=-\frac {2 \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \sqrt {a-\cos (x)} \arctan \left (\frac {\sin (x)}{\sqrt {1-\cos (x)} \sqrt {a-\cos (x)}}\right )}{\sqrt {1-\cos (x)}} \]
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Rule 210
Rule 2854
Rule 4485
Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 \arctan \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*}
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=-\sqrt {2} \sqrt {\frac {-1+\cos (x)}{-a+\cos (x)}} \sqrt {-a+\cos (x)} \csc \left (\frac {x}{2}\right ) \log \left (\sqrt {2} \cos \left (\frac {x}{2}\right )+\sqrt {-a+\cos (x)}\right ) \]
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Time = 1.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.97
method | result | size |
default | \(\sqrt {2}\, \sqrt {\frac {\cos \left (x \right )-1}{-a +\cos \left (x \right )}}\, \sqrt {-\frac {2 \left (-a +\cos \left (x \right )\right )}{\cos \left (x \right )+1}}\, \arctan \left (\frac {\sqrt {-\frac {2 \left (-a +\cos \left (x \right )\right )}{\cos \left (x \right )+1}}\, \sqrt {2}}{2}\right ) \left (\cot \left (x \right )+\csc \left (x \right )\right )\) | \(63\) |
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none
Time = 0.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.49 \[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=-\arctan \left (-\frac {{\left (a - 2 \, \cos \left (x\right ) - 1\right )} \sqrt {-\frac {\cos \left (x\right ) - 1}{a - \cos \left (x\right )}}}{2 \, \sin \left (x\right )}\right ) \]
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\[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=\int \sqrt {\frac {1 - \cos {\left (x \right )}}{a - \cos {\left (x \right )}}}\, dx \]
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Exception generated. \[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.71 \[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=2 \, \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {a \tan \left (\frac {1}{2} \, x\right )^{2} + \tan \left (\frac {1}{2} \, x\right )^{2} + a - 1}\right ) \mathrm {sgn}\left (\tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right ) \mathrm {sgn}\left (a - \cos \left (x\right )\right ) \]
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Timed out. \[ \int \sqrt {\frac {1-\cos (x)}{a-\cos (x)}} \, dx=\int \sqrt {-\frac {\cos \left (x\right )-1}{a-\cos \left (x\right )}} \,d x \]
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