Maxima -- GPL CAS based on DOE-MACSYMA / Feature Requests / #190 integrate( x*exp(atan(x)) / (1+x^2)^(3/2),x ) not solved

Barton Willis - 2024-06-11

To find an antiderivative, Maxima's integration code tries various methods. When no method is successful, Maxima returns a nonform. Unfortunately, Maxima's methods are not exhaustive, so it's not guaranteed that if an antiderivative can be expressed in a particular form that Maxima will find it.

I cannot find any evidence that Maxima is unable to find this antiderivative due to a bug--I think it is due to an algorithmic weakness. Possibly this report should be filed as a feature request, not a bug.

We are, of course, interested in extending the usefulness of Maxima, so please consider filing a feature request.

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- Trevor Clavijo - 2024-06-11
  
  I agree with you so much.
  Sorry to bother you guys.
  
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Robert Dodier - 2024-06-11

Ticket moved from /p/maxima/bugs/4309/

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Just for fun, I tried to see if I could get maxima to do this. The substitution t=atan(x) works well:

(%i45) triginverses:all;
(%o45)                                all
(%i46) integrate(x*exp(atan(x))/(1+x^2)^(3/2),x);
                               ⌠     atan(x)
                               ⎮ x %e
(%o46)                         ⎮ ─────────── dx
                               ⎮   2     3/2
                               ⌡ (x  + 1)
(%i47) changevar(%o46, t=atan(x),t,x);
                            ⌠   t    2
                            ⎮ %e  sec (t) tan(t)
(%o47)                      ⎮ ────────────────── dt
                            ⎮      2        3/2
                            ⌡  (tan (t) + 1)
(%i49) trigsimp(%o47);
                          ⌠   t
                          ⎮ %e  sin(t) abs(cos(t))
(%o49)                    ⎮ ────────────────────── dt
                          ⎮         cos(t)
                          ⌡

If we assume cos(t)>=0, we're left with integrate(exp(t)*sin(t),t):

(%i50) integrate(exp(t)*sin(t),t);
                               t
                             %e  (sin(t) - cos(t))
(%o50)                       ─────────────────────
                                       2
(%i51) subst(t=atan(x),%);
                          x              1          atan(x)
                    (──────────── - ────────────) %e
                           2              2
                     sqrt(x  + 1)   sqrt(x  + 1)
(%o51)              ───────────────────────────────────────
                                       2
(%i55) factor(%o51);
                                         atan(x)
                               (x - 1) %e
(%o55)                         ─────────────────
                                        2
                                2 sqrt(x  + 1)

And differentiating this result produces the original integrand.

I don't know anything about the Risch algorithm, but isn't the integrand an elementary function? Or maybe exp(atan(x)) doesn't count?

FWIW, this is quite a bit different from what Wolfram Alpha returns for the integral.

integrate( x*exp(atan(x)) / (1+x^2)^(3/2),x ) not solved

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#190 integrate( x*exp(atan(x)) / (1+x^2)^(3/2),x ) not solved

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