Optimal. Leaf size=24 \[ \text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]
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Rubi [A] time = 0.09, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3319, 3299} \[ \text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3319
Rubi steps
\begin {align*} \int \frac {\sqrt {a-a \cos (x)}}{x} \, dx &=\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*}
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Mathematica [A] time = 0.02, size = 24, normalized size = 1.00 \[ \text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 16, normalized size = 0.67 \[ \sqrt {2} \sqrt {a} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a -a \cos \relax (x )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a \cos \relax (x) + a}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {a-a\,\cos \relax (x)}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- a \left (\cos {\relax (x )} - 1\right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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