I'd like to solve the following equations: $$\int_{0}^{\infty} \frac{J_{0}(z)+J_{2}(z)}{z+a_{0}+a_{1}\left(\operatorname{coth}\left(a_{2}z\right)-1\right)} dz$$ where $J_{0}$ and $J_{2}$ are Bessel functions of first kind. $a_{0},\,a_{1},\,a_{2}$ are real positive constant.
I find some interesting identity shown as: $$ \int_{0}^{\infty} \frac{x^{\nu} J_{\nu}(a x)}{x+k} d x=\frac{\pi k^{\nu}}{2 \cos \nu \pi}\left[\mathbf{H}_{-\nu}(a k)-Y_{-\nu}(a k)\right] \quad\left[-\frac{1}{2}<\operatorname{Re} \nu<\frac{3}{2}, \quad a>0, \quad|\arg k|<\pi\right] $$ where $\mathbf{H}_{-\nu}$ is Struve function and $Y_{-\nu}$ is Bessel function of second kind. But I do not know how to use this.
Any ideas on how to tackle the first integral? Thanks a lot.