Any comment, idea or innovation to calculate this parametric implicit integral?

Any comment, idea or innovation to calculate this parametric implicit integral?

Note M, II, JJ are arbitrary positive integers (0<M, II, JJ<11).

F must be a function of Pm at the final!

clear
M = 3;
JJ = 5;
II = 5;
W = rand(II, JJ, M);
V = rand(II, JJ, M);
p = sym('p',[1 M]);
syms x y
w = sym('0');
v = sym('0');
L = sym('0');
for m=1:M
    for i=1:II
        for j=1:JJ
            w =w+W(i, j, m)*legendreP(i-1, x)*legendreP(j-1, y)*p(m);
            v =v+V(i, j, m)*legendreP(i-1, x)*legendreP(j-1, y)*p(m);
           L = L+(legendreP(i-1, x)*legendreP(j-1, y))^2;
        end
    end
end
H = 1+tanh(w-v);
F = int(int(H*L,x,[-1 , 1]), y,[-1, 1])

F = 

[Walter Roberson]

You should be specifying the variables of integration. Your expression includes x, y, and all the M p variables.

In practice the priority would be given to integrating on x and then y, but it would be better code to not rely on that.

[Walter Roberson]

I am not sure why you call it an implicit integral?

It looks to me as if it is double integral of tanh of a multinomial in x and y.

I do not think it is the multinomial that is the primary problem; I think it is the tanh that is the primary problem.

[Mehdi]

by implicit I meant that pm could not be seperated (factorized) from integrand to get out of the integral.

[Walter Roberson]

In mathematics, an implicit function describes one of the variables implicitly, by having the function equal something particular, and having the variable be "whatever set of values is needed for the equation to work out".

"implicit" is not used to describe integrals for which there is no known closed form.

You would potentially have an implicit equation if you had a known value for the integral and needed to solve for one of the p_* variables that made the equation true.

[Mehdi]

Any other suggestion to find F?

[Mehdi]

i am thiniking to replace 1+tanh(w-v) with other simpler continuous functions which help to solve this integral, since it acts like step function. What do you think?

Approximate solutions are also welcome.

[Walter Roberson]

I am not clear as to what you are requesting?

The integral does not appear to be implicit, just not closed form.

If you are asking for a way to find a closed form expression for it, then I doubt that is possible.

You can use techniques such as taylor series, but that gets messy quickly and is going to be pretty inaccurate.