3.238 \(\int \sqrt{1-\cos (x)} \, dx\)

Optimal. Leaf size=14 \[ -\frac{2 \sin (x)}{\sqrt{1-\cos (x)}} \]

[Out]

(-2*Sin[x])/Sqrt[1 - Cos[x]]

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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]

(-2*Sin[x])/Sqrt[1 - Cos[x]]

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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]

-2*sin(x)/sqrt(-cos(x) + 1)

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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]

-2*Sqrt[1 - Cos[x]]*Cot[x/2]

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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]

-2*sin(1/2*x)*cos(1/2*x)*2^(1/2)/(sin(1/2*x)^2)^(1/2)

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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]

-2*sqrt(2)/sqrt(sin(x)^2/(cos(x) + 1)^2 + 1)

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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]

-2*(cos(x) + 1)*sqrt(-cos(x) + 1)/sin(x)

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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]

Integral(sqrt(-cos(x) + 1), x)

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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]

integrate(sqrt(-cos(x) + 1), x)