Abstract symbolic matrix exponential

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I'm new to Sage. I was told Sage can do something way better than Mathematica. I tried and didn't see the magic. So I was wondering if I did wrong.

My goal: to compute matrix exponential of 2x2 matrix whose elements are a function of time.

A(t)=[[-a(t), a(t)],[b(t), -b(t)]]

B(s)=integrate of A(t) from 0 to s

exp(B(s)) ===> I was told this can be done in one step like treating B as a constant matrix exp(B*t)

Is this possible? My approach is to solve a system of differential equations.

Thanks.