There is an undocumented(!) function that essentially performs a Mellin transform through lookup. In your case, however, only one of your four examples actually has a sensible Mellin transform (and thus, a Meijer $G$ representation):
Integrate[Log[x], {x, 0, 1}]; (* force autoloading of the internal function *)
Integrate`ImproperDump`Mellin[Sin[4 x], x]
(* Sqrt[Pi] Integrate`ImproperDump`MeijerGfunction[{}, {}, {1/2}, {0}, 4 x^2] *)
Check:
Simplify[% /. Integrate`ImproperDump`MeijerGfunction[a_, b_, c_, d_, z_] :>
MeijerG[{a, b}, {c, d}, z], x > 0]
(* Sin[4 x] *)
Try a more elaborate example:
Integrate`ImproperDump`Mellin[Hypergeometric2F1[1/2, -1/3, 1, x], x]
(* Integrate`ImproperDump`MeijerGfunction[{4/3, 1/2}, {}, {0}, {0}, -x]/(Sqrt[Pi] Gamma[-1/3]) *)
As it is a lookup-based function, it might return unevaluated on functions that nevertheless have a $G$ function representation:
Integrate`ImproperDump`Mellin[Log[1 + x]/x, x]
(* Integrate`ImproperDump`Mellin[Log[1 + x]/x, x] *)
MeijerG[{{0, 0}, {}}, {{0}, {-1}}, x]
(* Log[1 + x]/x *)
MeijerG
? $\endgroup$ – David G. Stork Jan 10 '16 at 3:09