How Can You Integrate (1-a-cos x)^(-1/2) in Terms of Elliptic Integrals?

[c299792458]

Homework Statement

how does one integrate (1-a-cos x)^-1/2, where a is an arbitrary constant?

Homework Equations

as above

The Attempt at a Solution

Thought of writing cos x as 1-2*(sin(\frac{x}{2}))²
then the integrand simplifies to [2*(sin(\frac{x}{2}))² - a]^-1/2...
But then...? Is there a more elegant way of integrating this?

 

[Dickfore]

The solution involves the elliptic integral of the first kind:

http://www.wolframalpha.com/input/?i=Integrate[(1+-+a+-+Cos[x])^(-1/2),x]

 

[Ray Vickson]

Homework Statement

how does one integrate (1-a-cos x)^-1/2, where a is an arbitrary constant?

Homework Equations

as above

The Attempt at a Solution

Thought of writing cos x as 1-2*(sin(\frac{x}{2}))²
then the integrand simplifies to [2*(sin(\frac{x}{2}))² - a]^-1/2...
But then...? Is there a more elegant way of integrating this?

It is a non-elementary integral. Maple 9.5 gets:
J:=int(1/sqrt(b-cos(x)),x):simplify(J,symbolic);

- 2*EllipticF(cos(x/2),sqrt(2/(b+1)) )/sqrt(b+1),

where EllipticF is the incomplete elliptic integral of the first kind.

RGV