This time XCAS and hp prime are really wrong! ! !

[parisse]

You should have noticed in the Terminal (on the Prime, or in green in Xcas) that "the discontinuities at 0 of sin(x) were not checked". Therefore it is your responsability to check the answer.
Running int(abs(sin(x)/e^x), x) returns exp(-x)*(-cos(x)*sign(sin(x))/2-sin(x)*sign(sin(x))/2), and you can check that the antiderivative is not continuous if sin(x)=0 because the cos does not vanish.
If you run int(abs(sin(x)/e^x), x, 0,10*pi) you will get 1/2+ a sum of exponentials. It's easy to guess that the answer as 1/2+sum(exp(-k*pi),k,0,inf)=1/2+1/(exp(pi)-1)
It's not too hard to implement automatic control if the integral has finite boundaries but much harder if one boundary is infinite if you want to have a reasonable responding time for common inputs.

About int(sin(x)*exp(-x^2)), you can get it as the imaginary part of int(exp(i*x)*exp(-x^2)) but unfortunately I don't have a simple form for im(erf(x-i/2)) and then it would be difficult to handle in a limit.

By the way, it is probably not intented, but I would appreciate if you could comment in a more friendly way. I do my best to improve Xcas, but I do not have all the ressources of Wolfram and I do not necessarily have the same priorities, for example I think that Xcas multivariate polynomial computing kernel is way more efficient than Mathematica.