We take two tcl access scripts, presuming presence of [Maxima] and [PostgreSQL] on a system or in the latter case on some system in the network:

 proc domaxima { {m} } {
    set t "display2d:false;\n$m;"
    # return [string range [exec maxima << $t | tail -2  ] 6 end-7]
    return [string range [exec maxima -q << $t ] 32 end-7]
    #return [exec maxima --real-quiet << $t ]
 }

 proc dosql {s} {
  global db
   if {[catch {pg_exec $db $s} reply]} {
      puts "sql error : $reply,[pg_result $reply -error]"
      return "Error"
   }
   if {[pg_result $reply -status] == "PGRES_COMMAND_OK"} {
      return {}
   }
   if {[pg_result $reply -status] != "PGRES_TUPLES_OK"} {
      puts "sql error: [pg_result $reply -error]"
      return "Error"
   }
   return [pg_result $reply -llist]
 #   pg_result $reply -clear
   return 
 }

 # You could use another with similar results, it appears:
 package require pgintcl
 # connect to your sql server:
 set db [pg_connect -conninfo [list host = 192.168.0.1 user = zxy dbname = somedb password = 0x987654]]

 # Now create a table for formulas
 dosql "create table formula (f varchar(256))"

Now we can try to make a nice database with formulas:

 dosql "insert into formula values ([pg_quote "x^2"])"
 dosql "insert into formula values ([pg_quote "x^3"])"

But we can also use maxima to help us, for instance the optimizer will reorder terms and at request also simplify so we can keep formulas unique for instance:

 dosql "insert into formula values ([pg_quote [domaxima "x^4"]])"
 dosql "insert into formula values ([pg_quote [domaxima "x^3"]])"

Now lets look at the nice lil'  list of math work:

 foreach i [dosql "select distinct f from formula"] {puts |$i}
  |x^2
  |x^3
  |x^4

A more interesting example, a list of repeated derivatives of a products of two functions, a well known formula for case 1:

 dosql "create table derivatives (ref varchar(64), f varchar(10000), df varchar(10000))"
 set i 1; set fo "q(x)*r(x)"; 
 dosql "insert into derivatives values ( [pg_quote "prod$i"], [pg_quote [domaxima $fo]], [pg_quote [domaxima "diff($fo)"]] )"
 incr i; 
 set fo "diff($fo)";
 dosql "insert into derivatives values ( [pg_quote "prod$i"], [pg_quote [domaxima $fo]], [pg_quote [domaxima "diff($fo)"]] )"
 incr i; 
 set fo "diff($fo)";
 dosql "insert into derivatives values ( [pg_quote "prod$i"], [pg_quote [domaxima $fo]], [pg_quote [domaxima "diff($fo)"]] )"
 incr i; 
 set fo "diff($fo)";
 dosql "insert into derivatives values ( [pg_quote "prod$i"], [pg_quote [domaxima $fo]], [pg_quote [domaxima "diff($fo)"]] )"

Of course that could be easily put in a loop, but it's to prove the principle. Now:
 % foreach i [dosql "select * from derivatives"] {puts $i}
 prod1 q(x)*r(x) (q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*del(x)
 prod2 (q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*del(x) {((q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                            +r(x)*'diff(q(x),x,2))
       *del(x)
       +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,1))
       *del(x)}
 prod3 {((q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                            +r(x)*'diff(q(x),x,2))
       *del(x)
       +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,1))
       *del(x)} {(((q(x)*'diff(r(x),x,3)+3*'diff(q(x),x,1)*'diff(r(x),x,2)
                             +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                             +r(x)*'diff(q(x),x,3))
       *del(x)
       +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,2)
       +2*(q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                               +r(x)*'diff(q(x),x,2))*'diff(del(x),x,1))
       *del(x)
       +'diff(del(x),x,1)*((q(x)*'diff(r(x),x,2)
                          +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                          +r(x)*'diff(q(x),x,2))
                          *del(x)
                          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                           *'diff(del(x),x,1)))
       *del(x)}
 prod4 {(((q(x)*'diff(r(x),x,3)+3*'diff(q(x),x,1)*'diff(r(x),x,2)
                             +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                             +r(x)*'diff(q(x),x,3))
       *del(x)
       +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,2)
       +2*(q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                               +r(x)*'diff(q(x),x,2))*'diff(del(x),x,1))
       *del(x)
       +'diff(del(x),x,1)*((q(x)*'diff(r(x),x,2)
                          +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                          +r(x)*'diff(q(x),x,2))
                          *del(x)
                          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                           *'diff(del(x),x,1)))
       *del(x)} {((((q(x)*'diff(r(x),x,4)+4*'diff(q(x),x,1)*'diff(r(x),x,3)
                              +6*'diff(q(x),x,2)*'diff(r(x),x,2)
                              +4*'diff(q(x),x,3)*'diff(r(x),x,1)
                              +r(x)*'diff(q(x),x,4))
       *del(x)
       +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,3)
       +3*(q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                               +r(x)*'diff(q(x),x,2))*'diff(del(x),x,2)
       +3*(q(x)*'diff(r(x),x,3)+3*'diff(q(x),x,1)*'diff(r(x),x,2)
                               +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                               +r(x)*'diff(q(x),x,3))*'diff(del(x),x,1))
       *del(x)
       +2*'diff(del(x),x,1)
         *((q(x)*'diff(r(x),x,3)+3*'diff(q(x),x,1)*'diff(r(x),x,2)
                                +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                                +r(x)*'diff(q(x),x,3))
          *del(x)
          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,2)
          +2*(q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                                  +r(x)*'diff(q(x),x,2))*'diff(del(x),x,1))
       +'diff(del(x),x,2)*((q(x)*'diff(r(x),x,2)
                          +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                          +r(x)*'diff(q(x),x,2))
                          *del(x)
                          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                           *'diff(del(x),x,1)))
       *del(x)
       +'diff(del(x),x,1)*(((q(x)*'diff(r(x),x,3)
                          +3*'diff(q(x),x,1)*'diff(r(x),x,2)
                          +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                          +r(x)*'diff(q(x),x,3))
                          *del(x)
                          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                           *'diff(del(x),x,2)
                          +2*(q(x)*'diff(r(x),x,2)
                             +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                             +r(x)*'diff(q(x),x,2))*'diff(del(x),x,1))
                          *del(x)
                          +'diff(del(x),x,1)
                           *((q(x)*'diff(r(x),x,2)
                            +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                            +r(x)*'diff(q(x),x,2))
                            *del(x)
                            +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                             *'diff(del(x),x,1))))
       *del(x)}

More intelligently we can now look up in our little database what the general form is of a certain derivative of a product of functions, and what the original was from that derivative (the integral except for a constant), for instance:

 (Tcl) 69 % dosql "select ref, f from derivatives where df = [pg_quote "(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*del(x)"]"
 {prod1 q(x)*r(x)}

Or case 4:
 dosql "select ref from derivatives where df = [pg_quote [domaxima "((((q(x)*'diff(r(x),x,4)+4*'diff(q(x),x,1)*'diff(r(x),x,3)
                              +6*'diff(q(x),x,2)*'diff(r(x),x,2)
                              +4*'diff(q(x),x,3)*'diff(r(x),x,1)
                              +r(x)*'diff(q(x),x,4))
       *del(x)
       +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,3)
       +3*(q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                               +r(x)*'diff(q(x),x,2))*'diff(del(x),x,2)
       +3*(q(x)*'diff(r(x),x,3)+3*'diff(q(x),x,1)*'diff(r(x),x,2)
                               +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                               +r(x)*'diff(q(x),x,3))*'diff(del(x),x,1))
       *del(x)
       +2*'diff(del(x),x,1)
         *((q(x)*'diff(r(x),x,3)+3*'diff(q(x),x,1)*'diff(r(x),x,2)
                                +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                                +r(x)*'diff(q(x),x,3))
          *del(x)
          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))*'diff(del(x),x,2)
          +2*(q(x)*'diff(r(x),x,2)+2*'diff(q(x),x,1)*'diff(r(x),x,1)
                                  +r(x)*'diff(q(x),x,2))*'diff(del(x),x,1))
       +'diff(del(x),x,2)*((q(x)*'diff(r(x),x,2)
                          +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                          +r(x)*'diff(q(x),x,2))
                          *del(x)
                          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                           *'diff(del(x),x,1)))
       *del(x)
       +'diff(del(x),x,1)*(((q(x)*'diff(r(x),x,3)
                          +3*'diff(q(x),x,1)*'diff(r(x),x,2)
                          +3*'diff(q(x),x,2)*'diff(r(x),x,1)
                          +r(x)*'diff(q(x),x,3))
                          *del(x)
                          +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                           *'diff(del(x),x,2)
                          +2*(q(x)*'diff(r(x),x,2)
                             +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                             +r(x)*'diff(q(x),x,2))*'diff(del(x),x,1))
                          *del(x)
                          +'diff(del(x),x,1)
                           *((q(x)*'diff(r(x),x,2)
                            +2*'diff(q(x),x,1)*'diff(r(x),x,1)
                            +r(x)*'diff(q(x),x,2))
                            *del(x)
                            +(q(x)*'diff(r(x),x,1)+r(x)*'diff(q(x),x,1))
                             *'diff(del(x),x,1))))
       *del(x)"]]"
 prod4

It finds the correct f for df, with the help of an additional 'domaxima' it doesn't matter what the formatting of the text of the formula is. Try for yourself.

----
To remember manipulations easier adn centrally, I've made a history table which can contain all succesfully executed sql commands with timestamp on the sql server:

 proc dosql { {s} } {
  global db
   if {[catch {pg_exec $db $s} reply]} {
      puts "sql error : $reply,[pg_result $reply -error]"
      return "Error"
   }
   if {[pg_result $reply -status] == "PGRES_COMMAND_OK"} {
      catch {pg_exec $db "insert into history values (DEFAULT, [pg_quote $s] , 'now' , NULL)"}
      return {}
   }
   if {[pg_result $reply -status] != "PGRES_TUPLES_OK"} {
      puts "sql error: [pg_result $reply -error]"
      return "Error"
   }
   set res [pg_result $reply -llist]
   catch {pg_exec $db "insert into history values (DEFAULT, [pg_quote $s] , 'now' , NULL)"}
   return $res
 # we only put non-errors in the history
 #   pg_result $reply -clear
   return
 } 

 # once only:
 dosql "create table history (i serial, l text, date timestamp , comment text)"

Now we can for instance

 dosql "select date,l from history"

----
!!!!!!
%| enter categories here |%
!!!!!!