First, include in the dsolve command the option output=listprocedure. This creates a list of procedures that include R(theta)=proc(theta)... and mu(theta)=proc...
Access the procedures that generate the values of R and mu as follows. Use names other than R and mu. For example:
Then, write just Int(... using RR and M in place of R and mu. Apply evalf to this unevaluated integral. When i did this, I immediately obtained 46087.3499759295 as the value of the integral.
In other words, assigning the result of the dsolve command to F as was originally done does not access the solutions for R and mu in any way useful for computing. It is only useful in odeplot, the command to graph numeric solutions.
Then, writing Int(... = evalf(int(... isn't optimal. Use of Int provides the unevaluated integral. Use of int causes Maple to try to obtain an exact solution, after which, (when that fails) Maple tries to implement a numeric solution. That's wasteful. Since the integration must be carried out numerically, write just Int, the unevaluated integral, and apply evalf to that.
Finally, the use of eval is a handy strategy in Maple when trying to access the left-hand side of an equation. Suppose the equation is x=5 and it is included in a long list L of other such equations. To extract the value of x and assign it to a different name, use X:=eval(x,L) which evaluates the name x with the equation in the list L, thereby picking out the value 5 and assigning it to the new name X.