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Well back to work for me and an endless see of integrals... Anyway last night might have been a dead end but I think I have a good solution to try today
This is Mur 
@mur2501 Looks a lot like my notebook right now.
@freemo
what kind of integrals you deal with?
@mur2501 the integrals themselves arent all that fancy... the issue is that i have a particularly complex graph that has speicifc properties but can be represented by dozens of different actual equations... i need to find an equation that gives the intended graph with the desired properties but is actually integratable (no taylor series or numerical integration).... Most forms I try are not integratable
The only difficult part here is that im kinda working in reverse. Instead of given an equation and asked to integrate it I know what sort of result I need and need to figure out the equation.
@freemo
As long as the function is continuos it is integratable in the formal sense. I have no idea what exactly you said here but if you are going in reverse then you should be differentiating.
@mur2501 the reason i need closed/form symbolic integration is because the integral will be used to compute real world costs. As such precision will have to be high enough that it can require quite a few terms to represent it accurately enough.
On top of that the problem is computationally sensative. That is, I need a solution where every single calculation has a significant cost to it. So a symbolic/closed form representation will be much easier to compute numerical values for than a large summation operation.
@freemo
Yes I know but I just don't understand your fixture on symbolic integration
https://www.quora.com/Why-cant-the-integrals-of-some-functions-like-dfrac-sin-x-x-be-found?share=1