Hi;

The evaluation numerically of this integral came up in a discussion with anonimnystefy:

This one can resist normal methods quite well. When we say evaluate numerically we typically mean 10 - 12 digits with a programmable calculator and at least 100 when using a CAS.

To start, although any CAS known will gag on this problem they are able to do pieces of so it is to our advantage to split the integral like this.

The first integral is not too difficult and any CAS can do it to 100 digits.

That is about 105 digits or so. This is the hard part:

we notice that asymptotically as x grows large the denominator will act like x^2. The x^2 term will completely drown out the cos(x). That suggests the following chain of moves,

Looking at the RHS we see the same idea works again. As x grows large the denominator acts like x^4 completely drowning out the x^2 cos(x) term. So let's add a cos(x) / x^4 to it, we get

As x grows larger the denominator acts like x^6 completely drowning out the x^4 cos(x) term. So let's subtract a cos(x)^2 / x^6 to it, we get

Two things should be apparent,

1) we can continue the series on the left indefinitely and the RHS is getting smaller and smaller for each term on the left. The RHS will eventually be 0 as we get an infinite series on the LHS.

This becomes:

What have we accomplished? We have replaced a tough integral with a sum of infinite number of hopefully easier to evaluate numerically integrals. The beauty is that although we have an infinite number of integrals we will only need a finite number of them for the accuracy required.
Actually we will only need the first 26 or so terms.

Now I am not proposing this method of evaluating that integral because in practice those 26 integrals can also pose problems and further work is needed. But you wanted to see how the acceleration works.