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 = 0.087634, antiderivative size = 24, normalized size of antiderivative = 1., 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 3319
Rule 3299
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*}
Mathematica [A] time = 0.020777, size = 24, normalized size = 1. \[ \text{Si}\left (\frac{x}{2}\right ) \csc \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.137, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\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}\,{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}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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