even if my posting was more around to locate the error it is of interest to know the actual solution here is some related:

Moll et al "A formula for a quartic integral: a survey of old proofs and some new ones", front.math.ucdavis.edu/0707.2118

Thx, I was not able to prove that the singularities are liftable
(as the smooth plot suggests), sigh.

However modifying your approach (without further justification)
gives a result (where it is a matter of taste whether a series
over LerchPhi (=hypergeometrics) is really that better than some
integral). For this integrate the summands of your g2 formally
(the disturbing terms will not appear) and then apply the 
'Fundamental Theorem':

 'Sum(Int(-(2*k)!*4^(-k)/k!^2/(2*k+1)*x^(2*k+1)/(1+x^4),x ),
    k = 0 .. infinity)';
  value(%): 
  G:=subs(sum=Sum,%);

  'Int(1/2*Pi/(1+x^4),x = 0 .. 1)  +  eval(G,x=1) - eval(G,x=0)'; 
  
  value(%); evalf(%);


                                                            2  1/2
           1/2         1/2             1/2         1/2    Pi  2
  1/16 Pi 2    ln(2 + 2   ) - 1/16 Pi 2    ln(2 - 2   ) + -------- +
                                                             16

        /infinity                                                  \
        | -----   /   (-2 k - 2)                                  \|
        |  \      |  2           (2 k)! LerchPhi(-1, 1, k/2 + 1/2)||
        |   )     |- ---------------------------------------------||
        |  /      |                     2                         ||
        | -----   \                 (k!)  (2 k + 1)               /|
        \ k = 0                                                    /


                           0.92254817570232

you certainly can give us a link to the concurrent results, yes?

That might it be, GAMMA(0,z)/GAMMA(z); plot(%,z=0..6); shows it ...

I would perfer something towards that, but may be for complete 'routines' (and not for a ... hm ... cryptic replacement of lines as for the '4997' didits problem).

Obe reason may be that I do not want to wait and in bad cases it might be, that it is done only together with a new release (well, I do not buy them all ...).

Of course I am aware that would be on my own risk and nobody will guarantee, that no side effects will take place ...

converting Ei( 1 , z ) to Sum returns GAMMA( z ), not GAMMA( 0 , z )

BTW: strange that Maple knows about the (modified) exponential integral Ei( z ),  but I can
not find it through the help (it should stand for the  formal series in  Robert's  reply)

Edited for the "BTW": ok, should have looked closer at the help pages ... there it is, sorry

perhaps of this?
   
  GAMMA(a,z) = GAMMA(a) - z^a/a * `1F1`(a,1+a,-z);  # from the help page for GAMMA
  subs(a=0,%);

    Error, numeric exception: division by zero

  Ei(1,x): 
  %=convert(%,Sum);
                         Ei(1, x) = GAMMA(x)

I always ask me: would such a 'hotfix' be permanent or needed for each restart?

in this case it has an exact solution, so it is clear

The mentioned problems by jpmay or tobybailey are only caused by a somewhat careless use.

1. Using identify does not make too much sense for low precision like the standard 10 Digits,
at least I prefer to have 14 or more and one should be able (if possible) to re-enter with more
Digits. This would kick-out the zeta thing.




2. The other was using evalhf. Either you increase precison to the shown deciaml places (which
is a bit risky w.r.t. exactness). Or you take the better way: work in same reliable precison, that
is ah:= evalhf(1/3) and a:=evalf(a) - after that it makes sense to proceed (use Digits:=14 before).

Within Int( ln(2+2*cos(t)^2+cos(t)^4), t = 0 .. Pi) we have a (trigonometric)
polynomial of degree 4, which splits over the Reals:

  r1,r2:=(-1+I)^(1/2), -(-1-I)^(1/2);
  p1,p2:= 'eval((x-r)*(x-conjugate(r)), r= r1)', 'eval((x-r)*(x-conjugate(r)), r= r2)';
  'p1*p2': '%'=evalc(collect(expand(%),x));

                                   1/2           1/2
                 r1, r2 := (-1 + I)   , -(-1 - I)


                              _ |                     _ |
       p1, p2 := (x - r) (x - r)|      , (x - r) (x - r)|
                                |r = r1                 |r = r2


                                     4      2
                        p1 p2 = 2 + x  + 2 x


The parabolas p have real coefficients and no roots on the interval, so log
turns the product to a sum, thus we have 2 integrals of the following form:

  Int(ln((cos(t)-conjugate(r))*(cos(t)-r)),t=0..Pi);
  ``=PDEtools[dchange](t=arccos(x),%,[x]);
  ``=value(rhs(%)): # <------- in terms of hypergeometrics
  ``=simplify(rhs(%),size);
  
  J:=rhs(%):

                   Pi
                  /
                 |                 _
                 |    ln((cos(t) - r) (cos(t) - r)) dt
                 |
                /
                  0


                          1
                         /                  _
                        |   ln((x - r) (x - r))
                     =  |   ------------------- dx
                        |             2 1/2
                       /        (1 - x )
                         -1


         /                           /     1  \1/2
         |                           |1 - ----|
         |                           |      2 |
         |             _             \     r  /
    = Pi |ln(-r) + ln(-r) + ln(1/2 + -------------)
         |                                 2
         \

                                      \
                                      |
                                      |
                        /     1  \1/2 |
         + ln(1/2 + 1/2 |1 - ----|   )|
                        |     _2 |    |
                        \     r  /    /


Hence the value is given by

  'eval(J,r=r1) + eval(J,r=r2)';
  ``=evalc(%):
  combine(%,ln);
  evalf(%);

                         J|       + J|
                          |r = r1    |r = r2


                     1/2          1/2 1/2   1/2          1/2 1/2
              13   10      (3 + 10   )    10      (3 + 10   )
      = Pi ln(-- + ----- + -------------------- + --------------)
              16     4              8                   4


                             = 3.658972529


Numerical cross check:

  Int( ln(2+2*cos(t)^2+cos(t)^4), t = 0 .. Pi); evalf(%);

                    Pi
                   /
                  |                   2         4
                  |    ln(2 + 2 cos(t)  + cos(t) ) dt
                  |
                 /
                   0


                             3.658972529


I was using Maple 11.02 with Digits:=14

I think the eaxt value is

  Pi*ln(1/16*(4+2*10^(1/2))*(3+10^(1/2))^(1/2)+13/16+1/4*10^(1/2));
  evalf(%);


                          1/2         1/2 1/2          1/2
                 (4 + 2 10   ) (3 + 10   )      13   10
           Pi ln(---------------------------- + -- + -----)
                              16                16     4

                           3.6589725292852

and I should find the time to write it up (with continuous anti-derivatives).

(Alex: I used the online version and got nothing, but thx for the result!)

I have no reason to doubt Jacques' answer ... but played a bit more,
may be the following may help (but does not solve the problem).

Problems already occure for Int(1/(cos(t)-(-1-I)^(1/2))*t*sin(t),t = 0 .. Pi)
or -Int(cos(t)*t*sin(t)/(-cos(t)-(1+I)^(1/2)),t = 0 .. Pi).

Do not know too much about that (and the question is a bit unclear).

Besides the certainly existing standard literature 2 links to NAG articles

http://www.nag.co.uk/doc/TechRep/Pdf/tr2_00.pdf and
http://www.nag.co.uk/Numeric/CL/financial/GLFEN27.pdf (view at the problem as a multi asset option and use quasi-random numbers)

20 items is quite high, so one way would be to put them into clusters, may be a PCA (principal component analysis) helps.

The only direct and brute way that I am aware of is for 'margin calculations', but that's on daily runs only (i.e. over-night risk)

The practical problem may be deeper: the pdf for the items usually would not be the same over livetime (except working with historical data, while a better approach might be to work with derived data). The more serious part is: the correlations are not constant and from my vague knowledge one should search for "dispersion trades" (i.e. a major risk comes from there).

For Maple I have seen some 'classical' stuff, but can not remember and just do not have Maple here to look up the Toolbox