Integration help, Kepler's problem Lagrangian dynamics

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

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) ) ???