# What is the different between t(3) and Ca(0,1) distributions

I did a simulation study for empirical power for normality tests Shapiro-Wilk test and Anderson-Darling test against alternative t(3) and Cauchy distribution with parameters 0 and 1, I know both are symmetric with long-tailed but I have this result (SW test is more powerful than AD test if the alternative is t(3)) and (AD test is more powerful than SW test if the alternative is Ca(0,1)) and I read in many places that the SW test is a more powerful test if the alternative is a symmetric distribution with long-tailed So why do I get different results what is different between t(3) and Ca(0,1)

I do not get important differences at $$n=30$$, and neither at higher samples sizes, where both tests have power ever closer to 1. Can you give a reference?
library(nortest)
> mean(replicate(10000, shapiro.test(rt(n, df=3))$$p.value<.05)) [1] 0.4561 > mean(replicate(10000, ad.test(rt(n, df=3))$$p.value<.05))
> mean(replicate(10000, shapiro.test(rt(n, df=1))$$p.value<.05)) [1] 0.9605 > mean(replicate(10000, ad.test(rt(n, df=1))$$p.value<.05))