Optimal. Leaf size=44 \[ \frac {1}{2} \text {Ci}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}-\frac {\sqrt {a-a \cos (x)}}{x} \]
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Rubi [A] time = 0.09, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3319, 3297, 3302} \[ \frac {1}{2} \text {CosIntegral}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}-\frac {\sqrt {a-a \cos (x)}}{x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3302
Rule 3319
Rubi steps
\begin {align*} \int \frac {\sqrt {a-a \cos (x)}}{x^2} \, dx &=\left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x^2} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{x}+\frac {1}{2} \left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{x}+\frac {1}{2} \sqrt {a-a \cos (x)} \text {Ci}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 34, normalized size = 0.77 \[ \frac {\sqrt {a-a \cos (x)} \left (x \text {Ci}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )-2\right )}{2 x} \]
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 = 2.65, size = 34, normalized size = 0.77 \[ \frac {\sqrt {2} {\left (x \operatorname {Ci}\left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - 2 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a -a \cos \relax (x )}}{x^{2}}\, dx \]
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 \[ \int \frac {\sqrt {-a \cos \relax (x) + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a-a\,\cos \relax (x)}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {- a \left (\cos {\relax (x )} - 1\right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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