Question: How can inert summation-integral expressions be converted into sums of inert integrals, and how can x-independent terms be extracted outside an inert integral in Maple?

I am currently working with inert integrals and summations in Maple, and I encountered a difficulty when manipulating symbolic expressions. I would appreciate any suggestions or recommended approaches.

Specifically, I would like to achieve the following:

1. Convert an inert expression in the form of a “summation inside an integral” into an equivalent “sum of integrals” form.
   In other words, transform expressions of the type:
   Int(i=1∑n​fi​(x,t),x=a..b)

   into
   ∑​Int(fi​(x,t),x=a..b)

   while preserving the inert (Int) representation rather than evaluating the integral directly.

   I have tried commands such as value, expand, convert, and related manipulation methods, but I have not yet found a suitable solution.
    

2. The integrand contains two variables, where xxx is the integration variable.
   How can terms that are independent of xxx be automatically extracted outside the inert integral? For example, if the integrand contains coefficients or functions depending only on ttt, I would like Maple to rewrite
   Int(g(t)f(x),x=a..b)

   as g(t)Int(f(x),x=a..b)

   without forcing evaluation of the inert integral.

   Any recommendations, built-in commands, or pattern-based approaches would be greatly appreciated.
   question525.mw