Optimal. Leaf size=14 \[ -\frac{2 \sin (x)}{\sqrt{1-\cos (x)}} \]
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Rubi [A] time = 0.0086146, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2646} \[ -\frac{2 \sin (x)}{\sqrt{1-\cos (x)}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{1-\cos (x)} \, dx &=-\frac{2 \sin (x)}{\sqrt{1-\cos (x)}}\\ \end{align*}
Mathematica [A] time = 0.0077716, size = 18, normalized size = 1.29 \[ -2 \sqrt{1-\cos (x)} \cot \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 22, normalized size = 1.6 \begin{align*} -2\,{\frac{\sin \left ( x/2 \right ) \cos \left ( x/2 \right ) \sqrt{2}}{\sqrt{ \left ( \sin \left ( x/2 \right ) \right ) ^{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46405, size = 27, normalized size = 1.93 \begin{align*} -\frac{2 \, \sqrt{2}}{\sqrt{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62238, size = 57, normalized size = 4.07 \begin{align*} -\frac{2 \,{\left (\cos \left (x\right ) + 1\right )} \sqrt{-\cos \left (x\right ) + 1}}{\sin \left (x\right )} \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 \sqrt{1 - \cos{\left (x \right )}}\, 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 \sqrt{-\cos \left (x\right ) + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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