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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 No thats not true, there are many functions that are continuous over the region being integrated but can not be symbolically integrated. You are correct however that it can always be numerically integrated.
@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
@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.
