Optimal. Leaf size=14 \[ -\frac{2 \sin (x)}{\sqrt{1-\cos (x)}} \]
[Out]
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Rubi [A] time = 0.0151198, 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 \[ -\frac{2 \sin (x)}{\sqrt{1-\cos (x)}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[1 - Cos[x]],x]
[Out]
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Rubi in Sympy [A] time = 0.517319, size = 14, normalized size = 1. \[ - \frac{2 \sin{\left (x \right )}}{\sqrt{- \cos{\left (x \right )} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1-cos(x))**(1/2),x)
[Out]
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Mathematica [A] time = 0.00816821, size = 18, normalized size = 1.29 \[ -2 \sqrt{1-\cos (x)} \cot \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[1 - Cos[x]],x]
[Out]
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Maple [A] time = 0.026, size = 22, normalized size = 1.6 \[ -2\,{\frac{\sin \left ( x/2 \right ) \cos \left ( x/2 \right ) \sqrt{2}}{\sqrt{ \left ( \sin \left ( x/2 \right ) \right ) ^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1-cos(x))^(1/2),x)
[Out]
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Maxima [A] time = 1.55677, size = 27, normalized size = 1.93 \[ -\frac{2 \, \sqrt{2}}{\sqrt{\frac{\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-cos(x) + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218304, size = 24, normalized size = 1.71 \[ -\frac{2 \,{\left (\cos \left (x\right ) + 1\right )} \sqrt{-\cos \left (x\right ) + 1}}{\sin \left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-cos(x) + 1),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{- \cos{\left (x \right )} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1-cos(x))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{-\cos \left (x\right ) + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-cos(x) + 1),x, algorithm="giac")
[Out]