IIRC

  Midpoint
  v(n+1) = v(n) + a(n)*dt
  x(n+1) = x + v(n)*dt + 0.5*a(n)*dt^2

  Velocity Verlet
  v(n+1) = v(n) + 0.5*[a(n)+a(n+1)]*dt
  x(n+1) = x + v(n)*dt + 0.25*[a(n)+a(n+1)]*dt^2

With fixed acceleration a(n)=a(n+1) so the two methods are equivalent.