How to solve this improper integral?

[parisse]

I get errors with Xcas:
for 1/2 "Unable to handle singularities of 2*((sqrt(2)*ln(x-sqrt(2)*sqrt(x)+1))/8+(sqrt(2)*atan((sqrt(x)-(sqrt(2))/2)/sqrt(2)*2))/4-(sqrt(2)*ln(x+sqrt(2)*sqrt(x)+1))/8+(sqrt(2)*atan((sqrt(x)+(sqrt(2))/2)/sqrt(2)*2))/4) at [undef]"
I did not try on the emulator, but error handling is poor on the Prime, perhaps it is not returned to the user.
You can get the value of the integral by adding int(x^a/(1+x^2),x,-inf,0), replace -inf and inf by R large, add the upper half-circle integral of radius R and apply residue theorem, then take the limit as R->inf.
You can check for a=1/2 or a=1/3 like this
assume(t>0); simplify(eval(subst('int(x^(1/3)/(x^2+1),x,0,inf)',x=t^3)))