Optimal. Leaf size=14 \[ -\frac {2 \sin (x)}{\sqrt {1-\cos (x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.42, size = 18, normalized size = 1.29 \[ -\frac {2 \, {\left (\cos \relax (x) + 1\right )} \sqrt {-\cos \relax (x) + 1}}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.07, size = 23, normalized size = 1.64 \[ -2 \, \sqrt {2} {\left (\cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 22, normalized size = 1.57 \[ -\frac {4 \sqrt {2}\, \cos \left (\frac {x}{2}\right ) \sin \left (\frac {x}{2}\right )}{\sqrt {-2 \cos \relax (x )+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 20, normalized size = 1.43 \[ -\frac {2 \, \sqrt {2}}{\sqrt {\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 12, normalized size = 0.86 \[ -\frac {2\,\sin \relax (x)}{\sqrt {1-\cos \relax (x)}} \]
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 \[ \int \sqrt {1 - \cos {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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