calculus and analysis – Convolution integral taking too long

I am a beginner in Wolfram Mathematica. I am working on a problem in Quantum Optics which concerns obtaining the lineshape. Basically, I have to consider the convolution of a Doppler profile with other exotic profile I have got for my problem.

This then takes me to the following integral

$$int_{-infty} ^{infty}frac{left(2+frac{(delta – frac{v}{lambda})^{2}}{gamma^{2}}+frac{left(delta -delta_{hf}right)^{2}}{gamma^{2}}right)left(delta – frac{v}{lambda}right)e^{-frac{v^{2}}{2v_{th}^{2}}}}{A} dv$$

where $$A$$ is given by

$$A={frac{1}{4} s^2 (alpha +gamma )+frac{1}{2} s left(3 alpha +gamma right) left{-frac{2 delta delta_{hf}}{gamma ^2}+frac{delta_{hf}^2}{gamma ^2}+2 left(frac{left(delta -frac{v}{lambda }right)^2}{gamma ^2}+1right)right}+8 alpha left(frac{left(delta -frac{v}{lambda }right)^2}{gamma ^2}+1right) left(frac{left(delta -delta_{hf}-frac{v}{lambda }right)^2}{gamma ^2}+1right)}$$

When trying to solve it in Mathematica, it takes forever and I do not get any results.

``````Integrate((2+((Delta)-v/(Lambda))^2/(Gamma)^2+((Delta)-v/(Lambda)-(Delta)hf)^2/(Gamma)^2)/(8(Alpha)*(1+((Delta)-v/(Lambda))^2/(Gamma)^2)*(1+((Delta)-v/(Lambda)-(Delta)hf)^2/(Gamma)^2)+(3(Alpha)+(Gamma))/2*(2*(1+((Delta)-v/(Lambda))^2/(Gamma)^2)-2*((Delta)*(Delta)hf)/(Gamma)^2+(Delta)hf^2/(Gamma)^2)*s+1/4 ((Alpha)+(Gamma))s^2)*((Delta)-v/(Lambda))*Exp(-(v^2/(2*vt^2))),{v,-(Infinity),(Infinity)})
``````

Previously, I had a Lorentzian profile that when convoluted with the Doppler profile gives rise to the Voigt profile that can be expressed in terms of the Faddeeva function.I was also unable to calculate it in Mathematica, but luckily I was able to do it using a python implementation of the Faddeeva function.

In what concerns restrictions over the parameters, it can be assumed that $$lambda$$, $$gamma$$, $$delta_{hf}$$, $$s$$ and $$alpha$$ are all greater than zero. I may assign some values for these quantities. In the end, the objective would be to plot the result obtained as a function of $$delta$$.

I wonder if anyone knows how the evaluation of the integral above can be done.

Any references that may address the computation of it are very welcome, too.