$$\int_{0}^{2\pi}\cos x\times \exp\left [ \frac{a}{\cos x}\right]\,dx$$
where $a$ is complex parameter. (Integral does not converge if $a$ is Real)
I tried to solve it by two methods.
- The first is direct replacement $z=e^{ix}$. After that the integral goes to $$-\frac{i}{2}\oint_{\left| z \right| < 1}\frac{1+z^2}{z^2}\exp{\left[\frac{2az}{z^2+1}\right]}\,dz$$
- Before switching to complex numbers, you can use substitution $t=1/\cos x$ to reduce this integral to the form: $$2\int_{0}^{+\infty }\frac{e^{at}}{t^2\sqrt{t^2-1}}\,dt$$ And after that we can go to complex plane.
But because of the complex contours, integrating in both cases is a difficult task for me.
As noted in comments, the integral converges only if $a$ is purely imaginary (otherwise zeros of $\cos x$ represent non-integrable singularities). The substitution $t=1/\cos x$, done carefully, leads to $$f(a)=2\int_1^\infty\frac{e^{at}-e^{-at}}{t^2\sqrt{t^2-1}}\,dt,$$ with $f''(i\lambda)=2\pi i J_0(\lambda)$ for $\lambda>0$, according to this integral for the Bessel function.
This may give a way to confirm (or refute) the CAS result given in comments.