Special delimited oscillators that maxima can't solve it.

 I haven't followed all of this, but by a change of variables,
integrate(cos(1/x),x,0,2/%pi)
is integrate(cos(t)/t^2,t, %pi/2, inf);

this latter can presumably be integrated numerically by filon type formula
to any desired accuracy.
See
http://www.cs.berkeley.edu/~fateman/papers/fun-filon.pdf
for a start on this, and references, and maxima code..

RJF