(* Entry 4.251.1 *)

Integrate[x^(mu-1)*Log[x]/(b+x),{x,0,Infinity},Assumptions->{(0< Re[mu]<1)&&(b>0)}]

f1[b_,mu_]:=(b^(-1+mu)) π Csc[mu π] (-π Cot[mu π]+Log[b]);

(* Table gives *)

f2[b_,mu_]:=Pi*b^(mu-1)*(Log[b]-Pi*Cot[Pi*mu])/Sin[Pi*mu];

FullSimplify[f1[b,mu]/f2[b,mu]]

(* Numerics *)

g1[b_,mu_]:=NIntegrate[x^(mu-1)*Log[x]/(b+x),{x,0,Infinity}];

g2[b_,mu_]:=N[Pi*b^(mu-1)*(Log[b]-Pi*Cot[Pi*mu])/Sin[Pi*mu]];

quot[b_,mu_]:=N[g1[b,mu]/g2[b,mu]];

(* The function quot should be 1 *)

quot[2.34,0.23]