how to speed up symbolic integration or turn this to numerical integration

Hello, I am new to matlab and I am trying to use int to integrate my symbolic matrix but it is taking too much time to execute.

Here is my code:

%% form shape functions and its matrix 
syms m
Ni=sym(length(n)); %n is pre-defined vector and depends on user input
N=sym(zeros(2,length(n)));
for ii=1:length(n)
    Ni(ii)=1;
    for j=1:length(n)
        if j~=ii
            Ni(ii)=Ni(ii) * ((m - n(j)) / (n(ii) - n(j)));
        end
       
    end
    N(1,2*ii - 1)=Ni(ii);
    N(2,2*ii)=Ni(ii);
end

%% Jacobian 
J=l/2;
%% form matrices B1 and B2
b1=[1 0;0 0;0 1];
b2=(1/J) * [0 0;0 1;1 0];

B1=b1 * N;
B2=b2 * diff(N,m);

%% form SBFEM coefficient matrices 
E0=double(int(B1' * D * B1 * J , m , -1 , 1));
E1=double(int(B2' * D * B1 * J , m , -1 , 1));
E2=double(int(B2' * D * B2 * J , m , -1 , 1));

Would appreciate if someone can help :)

[Walter Roberson]

We cannot test, as n and l and D are not defined.

Answer 1

[John D'Errico]

As a suggestion, it is not at all uncommon to use a Gauss-Legendre quadrature for these things. Why? The kernel should be polynomial in m. Therefore as long as you use the correct order Gauss-Legendre quadrature, the result will be exact. It is already set up on the correct interval for Gauss-Legendre, so there is no need to transform the weights and nodes to match. Remember that for p nodes, a Gauss-Legendre rule will be order 2*p-1. The beauty is, Gauss-Legendre will run like blazes here compared to a symbolic computation.

https://en.wikipedia.org/wiki/Gauss–Legendre_quadrature

help sym.legendreP

  LEGENDREP  Legendre polynomials.
     Y = LEGENDREP(N, X) is the N-th Legendre polynomial.

    Documentation for sym/legendreP
       doc sym/legendreP

The nodes and weights can be derived from the Legendre polynomials.

I recall there are multiplye tools on the FEX that will compute the nodes and weights for such a rule. I see the chebfun toolbox has one (legpts).

format long g
[nodes,W] = legpts(5)
nodes =
-0.906179845938664
-0.538469310105683
0
0.538469310105683
0.906179845938664

W =
Columns 1 through 3
0.236926885056189         0.478628670499366         0.568888888888889
Columns 4 through 5
0.478628670499366         0.236926885056189

But they are also pretty easy to generate too. The wiki page I show should have what you need. As well, they actually have a table of the nodes and weights for the first few rules, and you don't need a high order for it to be exact, as I suggest above.

[Weilin Piao]

Hello, thank you so much for your answer!