In my course, I have to prove formula below $$I=\int_0^\infty \frac{e^{\cos(ax)}\cos\left(\sin (ax)+bx\right)}{c^2+x^2}dx =\frac{\pi}{2c}\exp\left(e^{-ac}-bc\right)$$ for $a,b,c>0.$
I know that this integral can be easily solved with complex analysis using $$f(z)=\frac{1}{2} \ \mathbb{R} \left(\int_{-\infty}^\infty \frac{\exp\left(e^{iaz}+ibz\right)}{c^2+z^2}dz\right)$$ but right now I am in a course dealing with real analysis. I tried to use parametrization integral method $$I'(a)=-\int_0^\infty \frac{xe^{\cos(ax)}\sin(\sin(ax)+(a+b)x)}{c^2+x^2}dx $$ but it doesn't look easier to handle. I tried to differentiate it again, but I just got a horrible form. An idea came to mind to differentiate with respect to parameter $b$ and set a differential equation $$I''(b)+x^2I(b)=0$$ plugging this ODE to W|A, I got $$I(b)=c_1\cos(bx^2)+c_2\sin(bx^2)$$ It's definitely wrong! After seeing Samrat's answer, I tried to plug in again to W|A and I got $$I(b)=c_1 D_{-1/2}((i+1)b)+c_2 D_{-1/2}((i-1)b)$$
where $D_n(z)$ is the parabolic cylinder function but I have no idea what does that mean.
Any idea? Thanks in advance.