3.160 \(\int \sqrt{a-a \cos (x)} \, dx\)

Optimal. Leaf size=16 \[ -\frac{2 a \sin (x)}{\sqrt{a-a \cos (x)}} \]

[Out]

(-2*a*Sin[x])/Sqrt[a - a*Cos[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.011432, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2646} \[ -\frac{2 a \sin (x)}{\sqrt{a-a \cos (x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Cos[x]],x]

[Out]

(-2*a*Sin[x])/Sqrt[a - a*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{a-a \cos (x)} \, dx &=-\frac{2 a \sin (x)}{\sqrt{a-a \cos (x)}}\\ \end{align*}

Mathematica [A]  time = 0.0068481, size = 19, normalized size = 1.19 \[ -2 \cot \left (\frac{x}{2}\right ) \sqrt{a-a \cos (x)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Cos[x]],x]

[Out]

-2*Sqrt[a - a*Cos[x]]*Cot[x/2]

________________________________________________________________________________________

Maple [A]  time = 0.961, size = 25, normalized size = 1.6 \begin{align*} -2\,{\frac{\sin \left ( x/2 \right ) a\cos \left ( x/2 \right ) \sqrt{2}}{\sqrt{a \left ( \sin \left ( x/2 \right ) \right ) ^{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a-a*cos(x))^(1/2),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.94552, size = 31, normalized size = 1.94 \begin{align*} -\frac{2 \, \sqrt{2} \sqrt{a}}{\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((a-a*cos(x))^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.55202, size = 59, normalized size = 3.69 \begin{align*} -\frac{2 \, \sqrt{-a \cos \left (x\right ) + a}{\left (\cos \left (x\right ) + 1\right )}}{\sin \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- a \cos{\left (x \right )} + a}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-a*cos(x) + a), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a \cos \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*cos(x) + a), x)