Numerical integration of $\int_0^{\infty} t^{a-1} \cos(k \ln(t)) \left\{\frac{1}{t}\right\} \, dt $

For $0<a<1$ and $k>0$, I am studying the following integral, $$h(a,k)=\int_0^{\infty} t^{a-1} \cos(k \ln(t)) \left\{\frac{1}{t}\right\} \, dt .$$

1. How to numerically integrate this integral in softwares like Mathematica or Python?

2. How to plot $h(a,k)$ as real valued function?

I have no background in numerical analysis but I can learn if the references are provided which could help me to study this integral.