Find inverse Laplace transform for the function $$ F(p)=\sqrt{\frac{p+1}{p^2+3p+1}}\, {\rm coth}\left( \sqrt{\frac{p^2+3p+1}{p+1}}\right). $$ In the book Prudnikov & ... I have found two formulas for $$ \frac{1}{p} {\rm coth}(p) $$ and for $$ F(\sqrt{p}) $$
but I still can find the solution. Any ideas to do it?