Re: [Axiom-developer] Hard integrals

[Waldek Hebisch]

Alasdair McAndrew wrote:
> 
> Is there a list somewhere of integrals which can be solved in closed form
> by Axiom/FriCAS, but cannot be solved by the most recent versions of
> Maple/Mathematica?  I would be very interested in seeing such a list - but
> I don't myself have access at the moment to either Maple or Mathematica.

Personaly I have access to Maple 15 and Mathematica 8, which are
not the newest versions.  At

http://12000.org/my_notes/CAS_integration_tests/index.htm

Nasser M. Abbasi posted results of running Maple 2015.1 and
Mathematica 10.1 trough Rubi testsuite.  Below are some
examples from this testsuite, which Nasser reports as
unevaluated by Mathematica and Maple.  Note: Rubi testsuite
strongly favours pattern matchers and does not contain
many difficulties which complete algorithm is supposed to
handle.  Still, for some classes of integrals which FriCAS
can do in systematic way you can find there examples
which are beyond Mathematica and Maple.  BTW:
I can provide several examples which are beyond Rubi,
Maple 15 and Mathematica 8, but can not check them
with newer versions.

integrate(f^(c*(b*x+a)^3)*x^2,x)

   (1)
           3   3       2   2     2         3          +------------+2
         (b c x  + 3a b c x  + 3a b c x + a c)log(f) 3|   3
       %e                                            \|- b c log(f)
     + 
       -
             2                1     3   3       2   2     2         3
            a b c log(f)Gamma(-,(- b c x  - 3a b c x  - 3a b c x - a c)log(f))
                              3
         *
             +------------+
            3|   3
            \|- b c log(f)
     + 
           2              2     3   3       2   2     2         3
       2a b c log(f)Gamma(-,(- b c x  - 3a b c x  - 3a b c x - a c)log(f))
                          3
  /
                 +------------+2
       3        3|   3
     3b c log(f)\|- b c log(f)
                                         Type: Union(Expression(Integer),...)


integrate(exp(b^3*x^3+3*a*b^2*x^2+3*a^2*b*x+a^3)*x^4,x)

   (2)
                                3 3       2 2     2       3  +----+2
          2 2              2   b x  + 3a b x  + 3a b x + a  3|   3
       (3b x  - 6a b x + 9a )%e                             \|- b
     + 
                                                             +----+
            4              1    3 3       2 2     2       3 3|   3
       (- 3a  - 4a)b Gamma(-,- b x  - 3a b x  - 3a b x - a )\|- b
                           3
     + 
           3      2      2    3 3       2 2     2       3
       (12a  + 2)b Gamma(-,- b x  - 3a b x  - 3a b x - a )
                         3
  /
          +----+2
       5 3|   3
     9b  \|- b
                                         Type: Union(Expression(Integer),...)


integrate(exp(e*(d*x+c)^3)*(b*x+a)^3,x)                

   (3)
                                  3   3       2   2     2         3   +-----+2
          3          2      3    d e x  + 3c d e x  + 3c d e x + c e 3|   3
       (3b d x + 9a b d - 6b c)%e                                    \|- d e
     + 
               3 4     2     3       2 2 2     3 3       3
         ((- 3a d  + 9a b c d  - 9a b c d  + 3b c d)e + b d)
      *
                                                        +-----+
               1    3   3       2   2     2         3  3|   3
         Gamma(-,- d e x  - 3c d e x  - 3c d e x - c e)\|- d e
               3
     + 
              2   4        2   3     3 2 2
         (- 9a b d  + 18a b c d  - 9b c d )e
      *
               2    3   3       2   2     2         3
         Gamma(-,- d e x  - 3c d e x  - 3c d e x - c e)
               3
  /
           +-----+2
       4  3|   3
     9d e \|- d e
                                         Type: Union(Expression(Integer),...)
-- 
                              Waldek Hebisch