Integrand size = 15, antiderivative size = 24 \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \]
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Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3400, 3380} \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]
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Rule 3380
Rule 3400
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x} \, dx \\ & = \sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right ) \text {Si}\left (\frac {x}{2}\right ) \]
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\[\int \frac {\sqrt {a -\cos \left (x \right ) a}}{x}d x\]
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Exception generated. \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\int \frac {\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}{x}\, dx \]
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\[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\int { \frac {\sqrt {-a \cos \left (x\right ) + a}}{x} \,d x } \]
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none
Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\sqrt {2} \sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, x\right ) \]
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Timed out. \[ \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx=\int \frac {\sqrt {a-a\,\cos \left (x\right )}}{x} \,d x \]
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