From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral:
$$\prod_a^b f(x)^{dx}=\exp\left(\int_a^b\ln f(x)dx\right).$$
This works if $f(a)\cdot f(b)=f(b)\cdot f(a)$, i.e. product of $f$'s is commutative. But what if $f$ is e.g. matrix-valued or quaternion-valued? We then no longer can as easily take logarithm of the product, find the sum and exponentiate back.
So, we have to use some techniques of computing product integrals directly, without resorting to usual integrals.
Are there any such techniques known — like analogs of integration by parts etc.? Are there any tables of product integrals of standard functions?
Or maybe there are some nice tricks to work around the uncommutativity of $f$?