Inverse Laplace transform

Question

For this expression: $ (1 - Exp[-Sqrt[1+s] x])/(1+s)$

How to make the Inverse Laplace Transform analytically?

InverseLaplaceTransform[(1 - Exp[-Sqrt[1+s] x])/(1+s),s,t] does not directly give solution

Answer 1

I'm not a mathematician so this may be "smoke and mirrors."

You are looking for the inverse Laplace transform of

g[s_] = (1 - Exp[-Sqrt[1 + s] x])/(1 + s);

Looking at the simpler problem

InverseLaplaceTransform[g[s - 1], s, t] // Simplify[#, x > 0] &

1 - Erfc[x/(2*Sqrt[t])]

With x > 0, let

f[t_] = E^-t (1 - Erfc[x/(2*Sqrt[t])]);

g[s] == LaplaceTransform[f[t], t, s] // Simplify[#, x > 0] &

True

So f[t] is the inverse Laplace transform of g[s]