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

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Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, 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]

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fricas [A]  time = 0.58, size = 19, normalized size = 1.19 \[ -\frac {2 \, \sqrt {-a \cos \relax (x) + a} {\left (\cos \relax (x) + 1\right )}}{\sin \relax (x)} \]

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)

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giac [A]  time = 0.41, size = 26, normalized size = 1.62 \[ -2 \, \sqrt {2} {\left (\cos \left (\frac {1}{2} \, x\right ) \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right ) - \mathrm {sgn}\left (\sin \left (\frac {1}{2} \, x\right )\right )\right )} \sqrt {a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a-a*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)))*sqrt(a)

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maple [A]  time = 0.12, size = 25, normalized size = 1.56 \[ -\frac {2 \sin \left (\frac {x}{2}\right ) a \cos \left (\frac {x}{2}\right ) \sqrt {2}}{\sqrt {a \left (\sin ^{2}\left (\frac {x}{2}\right )\right )}} \]

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)

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maxima [A]  time = 1.07, size = 23, normalized size = 1.44 \[ -\frac {2 \, \sqrt {2} \sqrt {a}}{\sqrt {\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.29, size = 34, normalized size = 2.12 \[ -\frac {2\,\sqrt {a}\,\sqrt {1-\cos \relax (x)}\,\left (\cos \relax (x)+1+\sin \relax (x)\,1{}\mathrm {i}\right )}{\sin \relax (x)-\cos \relax (x)\,1{}\mathrm {i}+1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - a*cos(x))^(1/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {- a \cos {\relax (x )} + a}\, dx \]

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)

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