While studying the use of the trapezoidal rule for numerically evaluating the complementary error function $\mathrm{erfc}(z)$, the following integrals showed up when I was trying to derive expressions for the truncation error:
$$\int_0^\pi \exp\left(-z^2\tan^2\frac{u}{2}\right)\cos(2mu) \mathrm du$$
where $z$ is positive and $m$ is a positive integer.
Evaluating a bunch of these integrals in Mathematica, I gather that these integrals follow the pattern
$$\pi z^2\exp(z^2)\mathrm{erfc}(z)R_n(z)-2\sqrt{\pi}z S_n(z)$$
where $R_n(z)$ and $S_n(z)$ are polynomials.
Are there any closed forms for these two polynomials?