Evaluate definite integral numerically, where the function is indeterminate

I'm trying to evaluate the following integral

Suppose I define a function handle as

f = @(x) x.*cosh(x)./( sinh(x).*(cosh(Phi*x)).^2 );

and evaluate the integral as

I = integral(f,-inf,inf)

the result gives NaN.

This is because the function is indeterminate at -inf, 0 and inf. However, using l'Hopital's rule, one can verify that the function's limits at these points are 0, 1, and 0, respectively, and the integral is indeed finite.

What is the best way to evaluate integrals of this kind numerically in MATLAB?

Answer 1

[Ameer Hamza]

If you have Symbolic toolbox, then you can try

syms x

Phi = 1;
f(x) = x.*cosh(x)./( sinh(x).*(cosh(Phi*x)).^2 );
y = vpaintegral(f, -inf, inf)

Result

y =
2.4674

Alternative solution using integral()

y = integral(@f, -inf, inf)

function y = f(x)
    Phi = 1;
    y = x.*cosh(x)./( sinh(x).*(cosh(Phi*x)).^2 );
    y(isnan(y)) = 0;
end

Answer 2

[Walter Roberson]

Break the integration up into parts that are piecewise numerically integratable, and add the parts together. Do not, however, expect matlab to be able to find the boundary conditions for you. For example it is not enough to integrate from -realmax to - eps(realmin) and the mirror of that, because the hyperbolic expressions are going to overflow to inf by 708 or so for each term and sinh*cosh^2 would overflow about cube root of 708 roughly.