Let $X\sim N(\theta,1)$ and $\pi(\theta)\sim \mathrm{Cauchy}(0,1)$ find a 90% credible set for $\theta$
To find the credible set I need to find the distribution of $f(\theta\mid x)$, but $$f(\theta\mid x)\propto \pi(\theta)f(x\mid\theta)$$ $$\propto \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}(x-\theta)^2}\frac{1}{\pi(\theta^2+1)}$$ $$\propto e^{-\frac{1}{2}(x-\theta)^2}\frac{1}{(\theta^2+1)}$$
I tried to do some manipulations but I get nothing.
I would like to know if exists a analytical derivation, or it just can be obtained computationally through a software