Maxima can solve some integrals involving absolute values,
including the example given (integrate(exp(-abs(x)-%i*x),x,minf,inf)),
via the package abs_integrate.

(%i2) load (abs_integrate);
(%o2)
     /usr/local/share/maxima/5.32.1/share/contrib/integration/abs_integrate.mac
(%i3) integrate(exp(-abs(x)-%i*x),x,minf,inf);
Principal Value
(%o3)                                  1

My recommendation is to close this report as "not a bug" or something
like that; no matter how extensive the integration routines are,
there will always be some integrals which it cannot solve,
and although it's regrettable, it's not a bug.


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