$r_1=\Lambda+(a+b)^*b$
$r_2=(b+aa^*b)^*$
$r_3=b+$$$+aa^*b+(b+$$$+aa^*b)(b+aa^*b)*(b+$$$+aa^*b)$
For this FA, which i think of as accepting "$\Lambda$ or anything ending in $b$", i came up with $r_1$, then i used FSM2Regex which gave me $r_3$ which i simplified into $r_2$ using regex simplifier.
now $r_2$ seems better than $r_1$. should i go with it? if so, what is lacking in $r_1$? i can't see the difference b/w $(a+b)^*b$ from $r_1$ AND $(aa^*b)^*$ from $r_2$. is one the subset of other? are they overlapping or disjoint?
Inputting any of the three RE's back into FSM2Regex gives RIDICULOUSLY complex TG's esp cuz for reason that is beyond me, it also draws $ (null string) transitions. even $r_3$ doesn't give back the FA i used to generate it. so question is : are $r_1$ and $r_2$ different? how? what should i go with?
sidenote 1: any other good/ more flexible tools like FSM2Regex?

sidenote 2: why don't backticks escape within $'s i.e. math mode ? i have to get out of math mode. is it intentional? reason?