Thanks again, @John Fredsted! I was quite slow with my respons in the other thread, sorry for that. I think I've gotten the help I need.

Really appreciate the time and effort! 

sol := R(t, q, p) = _F1(p^2+2*q^2+4*q, (1/2)*sqrt(2)*arctan((q+1)*sqrt(2)*(1/sqrt(p^2)))+t)

Yes! I get _F1, just as you said, but I don't understand what the solution is. Why is there a comma in the function _F1? 

@John Fredsted 

H := (1/2)*p1^2+q1^2+2*q1;
hamilton_eqs(H);

I used the hamilton_eqs(H) in maple and got diff(p1(t), t) = -2*q1(t)-2, but I will recalculate and check my eqn! 

Many thanks! :) 

@John Fredsted