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]]

________________________________________________________________________________________

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.

[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*}

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.

[In]

Integrate[Sqrt[1 - Cos[x]],x]

[Out]

-2*Sqrt[1 - Cos[x]]*Cot[x/2]

________________________________________________________________________________________

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.

[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)

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

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

________________________________________________________________________________________

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.

[In]

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

[Out]

Integral(sqrt(1 - cos(x)), x)

________________________________________________________________________________________

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.

[In]

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

[Out]

integrate(sqrt(-cos(x) + 1), x)