Problems evaluating symbolic integral

I'm having issues evaluating the definite integral of a symbolic expression using MATLAB's Symbolic Toolbox:

syms t n
assume(n, 'integer')
expression = sin(3*t)^2 * cos(n*t);
value = int(expression, t, -pi, pi);

This returns that the integral is 0 everywhere, despite the fact that there should be a special case where the integral is non-zero at n = 6. When I don't make the assumption that 'n' is an integer, I get an expression that is singular at n = 6. Is there some way that I can adapt my code so that it returns the expected answer? For reference, I go on to take the sum of the integral expressions for increasing values of n using symsum(), and would like to avoid the case where I need to explicitly define these singular values of 'n'. I am trying to avoid evaluating this integral numerically, if at all possible.

[David Goodmanson]

Hi Joshua,

Certainly value = 0 for integer n is a bad result.

But for unrestricted n, the result at n = 6 is not singular. The result is

value =  -(36*sin(pi*n))/(n*(n^2 - 36)) = -(36*sin(pi*n))/(n^3 - 36n))

Both the numerator and denominator are zero at n = 6, and by l'hopital's rule the result is

-36*pi*cos(pi*6) / (3n^2 - 36)  | n= 6     =  -pi/2 

which is correct.

[Joshua Wilson]

Hi David,

Thanks for your response. This makes sense, I missed the fact that both the numerator and denominator being 0 would allow this to be handled. I suppose I can use the limit() function to evaluate the expressions properly.