Inverse Lapace transform of $F(s)=\frac{I_0(r\sqrt{n^2+s})}{s I_0(\sqrt{n^2+s})}$

Question

While trying to solve this diffusion-reaction problem, I encountered the following Laplace inversion problem: \begin{align} \mathcal{L}^{-1}_s\left\{\frac{I_0(r\sqrt{n^2+s})}{s I_0(\sqrt{n^2+s})} \right\}\,, \end{align} with $0<r<1$, with $n$ a constant, $I_0$ a modified Bessel function of the first kind, and $s$ being the Laplace variable conjugate to time $t$. Can anyone help me solve this?

FYI: numerical Laplace inversions for $n=2$ and $t=10^{-4},10^{-2}$ and $1$ are shown below:

code:

import numpy as np
import matplotlib.pyplot as plt
from mpmath import *

def F(s,r,n):
    return mp.besseli(0,mp.sqrt(n**2+s)*r)/(s*mp.besseli(0,mp.sqrt(n**2+s)))

mp.dps = 20
def f(t,r,n):
    ft = lambda s: F(s,r,n)
    return invertlaplace(ft,t,method='talbot')

r  = np.linspace(0,1,200)
n  = 2

fig, ax = plt.subplots()
ax.plot(r,[f(1e-4,i,n) for i in r],'r',lw=1,label='$t=10^{-4}$')
ax.plot(r,[f(1e-2,i,n) for i in r],'g',lw=1,label='$t=10^{-2}$')
ax.plot(r,[f(1,i,n) for i in r],'b',lw=1,label='$t=1$')
ax.set(xlabel=r'$r$', ylabel=r'$f(r,t)$') 
ax.legend()