Integration help, Kepler's problem Lagrangian dynamics

[black_hole]

Homework Statement

Carry out the integration ψ = ∫[M(dr/r²)] / √(2m(E-U(r)) - (M²/r²))

E = energy, U = potential, M = angular momentum

using the substitution: u = 1/r for U = -α/r

Homework Equations

The Attempt at a Solution

This is as far as I've gotten: -∫ (Mdu) / √(2m(E + αu) - (M²u²))
I have no idea how to take this integral by hand which seems to be what the question is implying. Wolfram gives me something crazy looking.

My book gives the answer as ψ = arccos( (M/r - mα/M) / √(2mE + m²α²/M²) ) ?

 

[TSny]

Try "completing the square" of the expression inside the square root in the denominator. Factor out the coefficient of u² from the square root beforehand.