For some reason, the following forgets about one summand:

(1) -> integrate(exp(-x^2)+exp(x)/x,x)

(1)

The problem is in expintegratepoly$INTTR. There you find the following definition:

  -- returns either
  --   (q in GP, a in F)  st p = q' + a, and a=0 or a has no integral in F
  -- or (q in GP, r in GP) st p = q' + r, and r has no integral elem/UP
      expintegratepoly(p, FRDE) ==
        coef0:F := 0
        notelm := answr := 0$GP
        while p ^= 0 repeat
          ans1 := FRDE(n := degree p, a := leadingCoefficient p)
          answr := answr + monomial(ans1.ans, n)
          if ~ans1.sol? then         -- Risch d.e. has no complete solution
               missing := a - ans1.right
               if zero? n then coef0 := missing
                          else notelm := notelm + monomial(missing, n)
          p   := reductum p
        zero? notelm => [answr, coef0]
      [answr, notelm]

In principle, this function takes a polynomial p and tries to integrate every coefficient. If it finds an answer, it adds it to answr, otherwise to notelm. Note however, that the constant term of the p will never get added to notelm, even if it was not possible to integrate it. So maybe the last line should read:

      [answr, notelm+monomial(coef0, 0)]

This seems to "work", i.e., the integrals are then returned unevaluated. However, I'd rather have axiom to use the linearity of the integral... Note that

integrate(exp(-x^2)+sin(x),x)

(2)

is an example for an integral where coef0 does not vanish but notelm does. So the more drastic change:

          if ~ans1.sol? then         -- Risch d.e. has no complete solution
               missing := a - ans1.right
               if zero? n then coef0 := missing
               notelm := notelm + monomial(missing, n)
          p   := reductum p
        zero? notelm => [answr, coef0]
      [answr, notelm]

is not necessary - but does not produce wrong results either.

By the way, here are some other - strange - manifestation of the same bug:

integrate(exp(-x^2)+1/x,x)

(3)

integrate(exp(x)/x+1/x,x)

(4)

Although is certainly elementary, and so is its integral, the bug manifests itself.

Martin

Status: open => fix proposed

Name: #199 integratee^(x^2) => #199 #199 integrate(exp(-x^2)+exp(x)/x,x)

Name: #199 #199 integrate(exp(-x^2)+exp(x)/x,x) => #199 integrate(exp(-x^2)+exp(x)/x,x)

Status: fix proposed => closed