In[]:=
Join[​​{{{Subscript[η,1],Subscript[η,2],Subscript[η,3]},"starting integral","double integral","integral post-residue-theorem","\"simplified\" y-integral"},​​{" "," "," "," "," "}},​​Table[​​{a,b,c}=RandomReal[{0,5},3];​​{​​    {a,b,c},​​    NIntegrate[ArcTan[x+b]/((x+a)^2+c^2),{x,0,∞}],​​    NIntegrate[(x+b)/((x+a)^2+c^2)1/(1+(x+b)^2y^2),{x,0,∞},{y,0,1}],​​    1/2NIntegrate[((a-b-Ic)(π+Arg[a-Ic]-ILog[Abs[a-Ic]]))/(c(1+((a-b)^2-c^2-2I(a-b)c)y^2))-((a-b+Ic)(π+Arg[a+Ic]-ILog[Abs[a+Ic]]))/(c(1+((a-b)^2-c^2+2I(a-b)c)y^2))+(Iπ+IArg[b-I/y]+Log[Abs[b-I/y]])/(1-2I(a-b)y-((a-b)^2+c^2)y^2)+(Iπ+IArg[b+I/y]+Log[Abs[I+by]/y])/(1+2I(a-b)y-((a-b)^2+c^2)y^2),{y,0,1}]//Chop,​​NIntegrate[((b-a)ArcTan[c/a])/c(1+((a-b)^2+c^2)y^2)/(1+2((a-b)^2-c^2)y^2+((a-b)^2+c^2)^2y^4)+2(a-b)(yArcTan[1/(by)])/(1+2((a-b)^2-c^2)y^2+((a-b)^2+c^2)^2y^4)+(1-((a-b)^2+c^2)y^2)/(1+2((a-b)^2-c^2)y^2+((a-b)^2+c^2)^2y^4)Log[Sqrt[(b^2+1/y^2)/(a^2+c^2)]],{y,0,1}]​​},​​     4]​​     ]//Transpose//Grid
Out[]=
{
η
1
,
η
2
,
η
3
}
{1.40194,0.599675,4.22374}
{0.401037,4.31027,3.21493}
{3.12963,4.98912,0.918632}
{4.13039,1.53107,3.15956}
starting integral
0.382093
0.649834
0.452913
0.285435
double integral
0.382093
0.649834
0.452913
0.285435
integral post-residue-theorem
0.382093
0.649834
0.452913
0.285435
"simplified" y-integral
0.382093
0.649834
0.452913
0.285435