Can the Integral of 1/sqrt(a^2-x^2) be Applied to Complex Numbers?

[bcyang]

Homework Statement

The problem occurred when solving $$x'' - \frac{1}{x^2} = 0$$.
You can think of this as if there is a mass in the origin (M) and a small particle (m << M) is being pulled by this mass.

Daniel helped me to solve this diff. eq. and we are at

Homework Equations

$$\frac{1}{2} (x')^2 + \frac{1}{x} = C$$ where C is a constant.

The Attempt at a Solution

I asked Mathematica to solve $$\int \frac{dx}{2\sqrt{C-1/x}}$$. It gives me some very complicated formula which isn't too handy. At first, this problem seemed to me a trivial exercise, but now I realize that this may not be an easy one. I hope somebody can help. Thank you very much in advance!

 

[bob1182006]

I know there's a formula to solve integrals of the form 1/sqrt(a^2-x^2) but I'm not sure if holds for complex numbers.

if integration is about the same for complex numbers then you can try getting it in the form of the derivative of arcsin x.