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

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

[Out]

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

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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.

[In]

Int[Sqrt[1 - Cos[x]],x]

[Out]

(-2*Sin[x])/Sqrt[1 - Cos[x]]

Rule 2646

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

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.

[In]

Integrate[Sqrt[1 - Cos[x]],x]

[Out]

-2*Sqrt[1 - Cos[x]]*Cot[x/2]

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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.

[In]

integrate((1-cos(x))^(1/2),x, algorithm="fricas")

[Out]

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

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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.

[In]

integrate((1-cos(x))^(1/2),x, algorithm="giac")

[Out]

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

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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.

[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 = 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.

[In]

integrate((1-cos(x))^(1/2),x, algorithm="maxima")

[Out]

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

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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.

[In]

int((1 - cos(x))^(1/2),x)

[Out]

-(2*sin(x))/(1 - cos(x))^(1/2)

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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.

[In]

integrate((1-cos(x))**(1/2),x)

[Out]

Integral(sqrt(1 - cos(x)), x)

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