Optimal. Leaf size=70 \[ -\frac {1}{8} \text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}}{4 x} \]
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Rubi [A] time = 0.11, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3319, 3297, 3299} \[ -\frac {1}{8} \text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\cot \left (\frac {x}{2}\right ) \sqrt {a-a \cos (x)}}{4 x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3319
Rubi steps
\begin {align*} \int \frac {\sqrt {a-a \cos (x)}}{x^3} \, dx &=\left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x^3} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{2 x^2}+\frac {1}{4} \left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\cos \left (\frac {x}{2}\right )}{x^2} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )}{4 x}-\frac {1}{8} \left (\sqrt {a-a \cos (x)} \csc \left (\frac {x}{2}\right )\right ) \int \frac {\sin \left (\frac {x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a-a \cos (x)}}{2 x^2}-\frac {\sqrt {a-a \cos (x)} \cot \left (\frac {x}{2}\right )}{4 x}-\frac {1}{8} \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 = 0.06, size = 45, normalized size = 0.64 \[ -\frac {\sqrt {a-a \cos (x)} \left (x^2 \text {Si}\left (\frac {x}{2}\right ) \csc \left (\frac {x}{2}\right )+2 x \cot \left (\frac {x}{2}\right )+4\right )}{8 x^2} \]
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 = 0.45, size = 48, normalized size = 0.69 \[ -\frac {\sqrt {2} {\left (x^{2} \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \operatorname {Si}\left (\frac {1}{2} \, x\right ) + 2 \, x \cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) + 4 \, \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) \sin \left (\frac {1}{2} \, x\right )\right )} \sqrt {a}}{8 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a -a \cos \relax (x )}}{x^{3}}\, dx \]
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 \[ \int \frac {\sqrt {-a \cos \relax (x) + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a-a\,\cos \relax (x)}}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {- a \left (\cos {\relax (x )} - 1\right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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