For some non-negative numbers $m_1$, $m_2$, $E$ I define a function $$ f\left(\boldsymbol{q},E\right)=\frac{1}{2\omega_{1}\omega_{2}}\frac{1}{\omega_{1}+\omega_{2}+E}+\frac{1}{4\omega_{1}\omega_{2}}\frac{1}{\omega_{1}-\omega_{2}-E}+\frac{1}{4\omega_{1}\omega_{2}}\frac{1}{\omega_{2}-\omega_{1}-E} $$ where $$ \omega_{1,2}\left(\boldsymbol{q}\right)=\sqrt{m_{1,2}^{2}+\boldsymbol{q}^{2}} $$ Note that $f$ actually depends on $\boldsymbol{q}^2$.
I want to evaluate at least numerically the integral over all $\mathbb{R}^3$ $$ I\left(E\right)=\int\frac{d^{3}q}{\left(2\pi\right)^{3}}f\left(\boldsymbol{q},E\right)e^{i\boldsymbol{q}\cdot\boldsymbol{n}} $$ where $\boldsymbol{n}$ in a non-negative integer 3-vector.
I've tried NIntegrate using e.g. $m_1=0.2$, $m_2=0.7$, $E=0.8$ and $\boldsymbol{n}=\left(1,1,1\right)$ . The calculation takes very long and I just abort. I've tried changing the input numbers, adding some methods as options. At most, I get warnings like NIntegrate::slwcon. I believe Mathematica runs into trouble because of the oscillating exponential.
With a simpler version, where $f$ is just $$ f\left(\boldsymbol{q}\right)=\frac{1}{2\sqrt{m^{2}+\boldsymbol{q}^{2}}} $$ similar issues appear. When it works, after long time, it's quite different from the analytical result.
Are you aware of some magical methods/options combination that would speed up the process and give a reliable result? Of course, if you know some analytical converging expression for the integral, I'd be happy to know.