I think both! There are parts of mathematics that just look like truths that have been waiting the whole time to be discovered. On the other hand, people invent problem solving techniques which definitely feel more like inventions than discoveries. Then in the middle, there are made-up mathematical structures introduced to bridge between two “clearly discovered” canonical objects, but this made-up structure certainly has the invented flavour.

So I think it is a continuum, and really fantastic mathematics will feature ideas from all the way along the spectrum: “discoveries” for the beauty, “inventions” for the problem solving, and the “in between” for the subtlety and art.

How does math software such as Mathematica solve integrals?

Like this: https://en.wikipedia.org/wiki/Risch_algorithm

Are maps invented or discovered?

Eh,this is just due to linguistic ambiguity. The syntax of Mathematics (as a language) is used to describe relationships and properties that are discovered.

It is discovered. Two alien species could effectively communicate with math alone (after exchanging notation translate keys).

I agree with your logic but not your premise. Two alien species could effectively communicate if they happened to agree on a shared set of fundamental axioms. The axiom of choice is somewhat contentious here on earth, since it underlies the Banach-Tarski paradox, and it's not clear at all that a sophisticated alien society would have ever accepted the axiom of choice into their mathematics.

Is there a list of generally accepted root mathematical axioms?

This is an interesting thought experiment. The space of all consistent and potentially useful mathematical constructs is gigantic, so I think there would be a good chance that two alien species would share almost no mathematical constructs, and would require decades or hundreds of years to discover - so in this sense, there is a large element of invention to mathematics as a human endeavor.

Even for physics, there are often many mathematical theories that can be used to model the same physical observations (talking about equivalent structures, not about competing theories). For example, many problems can be described equivalently using vectors, complex numbers, or linear algebra. There is a good chance that there are many (perhaps infinitely many) other systems that we haven't thought about that could be used equivalently.

So, while I agree that ultimately the structures in mathematics exist independent of our use of them, so we are only discovering pre-existing structures, I would also say that new mathematical theories are developed using a process that is more similar to invention than to discovery (i.e. you can't explore the space of mathematic theories to discover new ones, as it is infinite in every direction - you can only explore the properties of a structure you essentially invent for yourself).

A flat head screwdriver was designed to insert and remove screws. But it can also be used to open a paint can! OMG! Is the flat head screwdriver invented or discovered?

Our mathematical system is an invented human language. We know that all symbolic systems with sufficient complexity are equivalent (see Turing machines.) Finding an arbitrary one to be useful and flexible is not evidence of magic.

That thing to which all symbolic systems with sufficient complexity are equivalent, discovered or invented?

(Yours is a "turtles all the way down" argument, I think.)