@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.

@mur2501 To put it another way, it wont have a closed form expression once integrated (numerical integration or taylor series or whatever would likely result in an infinite number of terms)

@freemo
Yes though if you aim for unlimited precision then maybe change of perspective can help in symbolic answers

@mur2501 I dont need unlimited precision. But there is a certain minimal level of acceptable precision. The more difficult aspect is delivering the level of needed precision while requiring minimal level of computation to get there.

@freemo
Ahh I see computation limitations though even with a symbolic answer that could be a problem as you can't expect the answer would come out to be a simple equation.
One thing you can do is try to find the numerical integration answers with very high precision and then find a fitting function for the results that approximates it to your required want and is not computationally costly

@mur2501 yea closed form is not garunteed to be computationally simple. But they do stand a better chance at at it. Sadly thats why its a tricky problem, i cant avoid at least some degree of trial and error. I've had some closed forms I had to reject as well.

Finding a fitting function could work. In a sensethe problem is already pretty much just me manually trying to find an efficient fitting function working backwards from a known graph.