Optimal. Leaf size=44 \[ \frac{1}{2} \text{CosIntegral}\left (\frac{x}{2}\right ) \csc \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)}-\frac{\sqrt{a-a \cos (x)}}{x} \]
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Rubi [A] time = 0.0917802, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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 3319
Rule 3297
Rule 3302
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*}
Mathematica [A] time = 0.0264103, size = 34, normalized size = 0.77 \[ \frac{\sqrt{a-a \cos (x)} \left (x \text{CosIntegral}\left (\frac{x}{2}\right ) \csc \left (\frac{x}{2}\right )-2\right )}{2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.096, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}}\sqrt{a-a\cos \left ( x \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \cos \left (x\right ) + a}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- a \left (\cos{\left (x \right )} - 1\right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a \cos \left (x\right ) + a}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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