Optimal. Leaf size=44 \[ -\frac {\sqrt {a-a \cos (x)}}{x}+\frac {1}{2} \sqrt {a-a \cos (x)} \text {CosIntegral}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \]
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Rubi [A]
time = 0.07, 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 = {3400, 3378,
3383} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3383
Rule 3400
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 = 0.02, size = 34, normalized size = 0.77 \begin {gather*} \frac {\sqrt {a-a \cos (x)} \left (-2+x \text {CosIntegral}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )\right )}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a -a \cos \left (x \right )}}{x^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- a \left (\cos {\left (x \right )} - 1\right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 34, normalized size = 0.77 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {a-a\,\cos \left (x\right )}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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