I don't see them losing influence at all, to the contrary; muslim countries are modernizing and losing their traditional thinking. Saudi Arabia, Sudan, Somalia ..etc.
Liberals are clearly losing influence with the rise of the east non western countries will dominate the world economy in the next 50 years you think China India or Russia will care if you implement sharia laws in your country they don't put human rights as a condition to access their markets also almost all GDP projection show a minimum of 4 out of the top 10 economies will be muslim countries before the end of the century.
Also don't forget the strong population growth in the European union sure some will apostate but some won't it can reach a point where muslims can affect the internal policies and demand that Europe respect muslim countries .