f(t)=cos(at)/t, f'(t)=-asin(at)/t-cos(at)/t2.
ℒ{f(t)}=∫[0-,∞]e-stf(t)dt=∫[0-,∞]e-st(cos(at)/t)dt.
Let u=f(t), du=f'(t)dt, dv=e-stdt, v=-e-st/s, then integrating by parts:
ℒ{f(t)}=-e-stf(t)/s+(1/s)∫e-stf'(t)dt=f(0-)/s+(1/s)ℒ{f'(t)} (integral evaluated for t∈[0-,∞]).
f(0-)=1/t→∞; similarly f'(0-)→∞. This precludes evaluation of the transform.