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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?

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