You can use SingularIntegralEquations.jl for Hilbert and log integrals:
julia> using ApproxFun, SingularIntegralEquations
julia> x = Fun()
Fun(Chebyshev(),[0.0, 1.0])
julia> hilbert(exp(x), 0.1) # 1/π * ∫ exp(t)/(t-x) dt
0.6363129376196711
julia> logkernel(exp(x), 0.1) # 1/π * ∫ exp(t)*log(abs(t-x)) dt
-0.7386375437288631
julia> hilbert(exp(x)/sqrt(1-x^2), 0.1) # works with Chebyshev singularities
1.1404096104609651