Who needs neural networks?
Let's assume that there is a grammar that describes completely all possible integrands
using some standard character set.
(This is quite plausible).
Now set in motion a program to generate, in alphabetical order, and in size place, all
integrands.
Compute via Fricas, Rubi, Mathematica, Maple, Maxima ... the indefinite integral,
if possible, and store it in a table... [ [integrand_i, result_i] , ....]
Exponentially expensive to compute, and store, but who is counting?
We could also additionally do this: generate all possible integrand answers, and, after
differentiating, store that in the same table. Problem here is that differentiating an
expression does not provide a unique simplified answer. On the other hand,
what if DL or any other program is asked to integrate "x-x" ? Is it supposed to
know that is the same as integrating "0"? On the plus side, we have reduced the
integration problem to a simplification problem. Namely,
for p in table_of_integrands do if simplify(p[1] -input)==0 then p[2] ;
We know that the simplification problem is recursively undecidable, so
there is that problem. Oh, the DL version of integration has the same
flaw, and from the examples posted, where ridiculous constants appear,
it seems that it's truly an in-your-face defect.
Maybe there should be an attempt at the much more fundamental
problem of building a DL that will take any expression and
(a) simplify it
or
(b) just tell you if it is identically zero. [with exponential time and space a
solution to this will also provide a simplifier -- if we agree that the simplest
expression is the shortest alphabetically-ordered lower expression...)
It is possibly worth observing that definite integrals with parameters are vastly
more useful (look in reference books) than indefinite integrals, and so this whole
exercise is perhaps not so interesting to applied mathematicians.
Also note that definite integrals (if all
extra parameters are set) can generally be done very nicely by numerical
quadrature, and with suitable tables and plotting, extra " dimensions" for those
parameters may also be computed.
As for comparing systems , I am reminded of a (true story) from MIT when
Prof. Hubert Dreyfus, a critic of AI who said that computers could never play
chess because it was too difficult, was beaten by a program, MacHack
(
https://en.wikipedia.org/wiki/Richard_Greenblatt_(programmer) ).
Dreyfus said "My brother is a better chess player".
.....
RJF