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calculus and analysis - How to augment the realm of functions Mathematica thinks it knows how to integrate symbolically



My question involves extending the functionality of Integrate over specific integrals in the most generic manner.




Specifically, is it possible to "hack into" Integrate so as a to add to the "grammar" of analytic integration a new family of integrals which the user would have provided (in the spirit of ProbabilityDistribution which allows the user to include his own PDF into the existing tools).


The idea is that Mathematica would be able to identify when such integrals pop up and replace it by the given definition. The specific example I have in mind involves the integral of MultinormalDistribution over square subregions of the integration domain. That is to say I would like to attach to either MultiNormalDistribution, or Expectation, (or Integrate) the following rule


Expectation[Boole[(x>a)&&(y>b)], {x,y} ~Distributed~
    MultinormalDistribution[{0,0},{{1,c},{c,1}}]] = FF[a,b,c]

(where FF is a given function of a, b, and c which could be tabulated).


@JM suggested using TagSet or TagSetDelayed, but the difficulty is to decide to which function it should be associated with, and with what efficiency, given that it would be unwise to temper with build in symbols like Integrate?


Attempt


As a simpler toy problem I tried


Unprotect[Integrate]; 

Integrate /: Integrate[f[x], {x, a_, b_}] = F[a, b] 
Protect[Integrate];

for which Integrate[f[x],{x,1,2}] returns F[1,2]


But for instance Integrate[2 f[x], {x, 1, 2}] is not evaluated, which suggests that my new (possibly admittedly dangerous) definition has not been taken as a global rule for integration.


More relevantly, I tried


Unprotect[Expectation];
Expectation[ Boole[x_ > a_ && y_ > b_], {x_, y_}~Distributed~
    MultinormalDistribution[{0, 0}, {{1, c_}, {c_, 1}}]] = FF[a, b, c]
Protect[Expectation];


so that


Expectation[ 
 Boole[x > 1 && y > 2], {x, y}~Distributed~
  MultinormalDistribution[{0, 0}, {{1, 1/3}, {1/3, 1}}]]

returns FF[1,2,1/3]


(after a short while).


Questions





  1. What is the best way to write down the definition? Assign it to MultinormalDistribution, Expectation, Integrate?




  2. How to make sure it is "integrated" by Mathematica so that when attempting, e.g. to compute the expectation of d*Boole[x>a && y>b] it would return d*FF[a,b,c] (or the expectation of u+Boole[x>a && y>b] the Expectation of u + FF[a,b,c])? More specifically, how to make sure Mathematica tries to transform a given integral into a form which could match the new definition?





UPDATE




The package Rubi4 effectively extends the number of functions mathematica can integrate, though not following the path envisaged in this post.


Mathematica graphics




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