Let $a \ge 0$ and $b \ge 0$ be real numbers.
By generalizing the approach from Evaluating $\int_{\mathbb{R}}\frac{\exp(-x^2)}{1+x^2}\,\mathrm{d}x$ . we have derived the following results.
Let $n\ge 1$ be an integer. Then we have: \begin{eqnarray} \int\limits_{-\infty}^\infty \frac{e^{-b x^2}}{a^{2 n}+x^{2 n}} dx= -2\pi \cdot \sum\limits_{p=0}^{n-1} Im\left[ e^{-b [x_p^{(n)}]^2} \cdot \frac{\left( 1+Erf[\imath \sqrt{b} x_p^{(n)}]\right)}{\prod\limits_{l=0,l\neq p}^{2 n-1} (x_p^{(n)}- x_l^{(n)})}\right] \end{eqnarray} where \begin{equation} x_p^{(n)}:=e^{\imath \frac{\pi}{2 n}} e^{\imath \frac{p \pi}{n}} a \end{equation} for $p=0,\cdots 2n-1$.
n = Range[1, 7];
{a, b} = RandomReal[{0, 1}, 2, WorkingPrecision -> 50];
Clear[xx]; xx[p_, n_] := Exp[I Pi/(2 n)] Exp[I Pi/n p] a;
l1 = NIntegrate[
Exp[-b x^2]/(a^(2 #) + x^(2 #)), {x, -Infinity, Infinity},
WorkingPrecision -> 20] & /@ n;
l2 = N[-2 Pi Sum[
Im[ E^(-b xx[p, #]^2) (1 + Erf[I Sqrt[b] xx[p, #]]) 1/
Product[xx[p, #] - xx[l, #], {l, 0, p - 1}] 1/
Product[xx[p, #] - xx[l, #], {l, p + 1, 2 # - 1}]], {p,
0, # - 1}] , 20] & /@ n;
{l1, Re[l2]} // MatrixForm
Now let $n\ge 0$ be an integer. Then we have:
\begin{eqnarray} \int\limits_{-\infty}^\infty \frac{e^{-b x^2}}{a^{2n+1}+x^{2n+1}} dx=-2 \pi \sum\limits_{p=0}^n Im[ e^{-b [x_p^{(n)}]^2} \frac{\left(1+Erf[\imath \sqrt{b} x_p^{(n)}]\right)}{\prod\limits_{l=0,l\neq p}^{2 n-0} (x_p^{(n)}- x_l^{(n)})}](1_{p=n} \cdot \frac{1}{2} + 1_{p\neq n} \cdot 1) \end{eqnarray} where \begin{equation} x_p^{(n)}:=e^{\imath \frac{\pi}{2 n+1}} e^{\imath \frac{p \pi}{n+1/2}} a \end{equation} for $p=0,\cdots 2n$. Here the integral on the left hand side is meant as a principal value.
n = Range[1, 7]; Eps = 10^(-9);
{a, b} = RandomReal[{0, 1}, 2, WorkingPrecision -> 50];
Clear[xx]; xx[p_, n_] := Exp[I Pi/(2 n + 1)] Exp[I 2 Pi/(2 n + 1) p] a;
l1 = (NIntegrate[
Exp[-b x^2]/(a^(2 # + 1) + x^(2 # + 1)), {x, -Infinity, -a -
Eps}, WorkingPrecision -> 20] +
NIntegrate[
Exp[-b x^2]/(a^(2 # + 1) + x^(2 # + 1)), {x, -a + Eps,
Infinity}, WorkingPrecision -> 20]) & /@ n;
l2 = N[-2 Pi Sum[
If[p == #, 1/(2 ),
1] (Im[ (E^(-b xx[p, #]^2) (1 + Erf[I Sqrt[b] xx[p, #]])) 1/
Product[xx[p, #] - xx[l, #], {l, 0, p - 1}] 1/
Product[xx[p, #] - xx[l, #], {l, p + 1, 2 #}]]), {p,
0, #}] & /@ n, 20 ];
{l1, Re[l2]} // MatrixForm
Those results suggest a method for calculating integrals of the kind: \begin{equation} \int\limits_{-\infty}^\infty \frac{e^{-x^2}}{f(x)} dx \end{equation} where $f(x)=\sum\limits_{p=0}^M a_p x^p$ is some polynomial whose value at the origin is non-zero. It is not hard to see that the result will be expressed through zeros of $f(x)$ that lie in the upper complex plane. Now by taking the limit $M\rightarrow \infty$ one could also extend this approach to analytic functions and derive interesting results. I was wondering if anyone is aware if this line of attack was already pursued and what results were achieved.
