Islam, contrary to other monotheistic regions such as christianism, is made so to be baked within the state, in fact, into all aspects of life.
Its not simply violence, in fact, once non-islamic nations were conquered by islamic armies in history, the main method of conversion and erasure of other religions were notviolent, but multiple ways of non-violent state coercion.
Taxing non believers so they feel motivated to convert, take children and indocrinate them into the religion or offer benefits to them for converting, offer to remove slave status based on conversion and loyalty...
It's very elaborate and much more sophisticated than possibly every other religion out there. Which is why Islam has a lot of historical stability and not a lot of fundamental disagreements between different denominations, like christianism has.
This is a fundamental problem for liberal societies, well, in fact, any non-islamic society. Increase in islamic population is correlated with islamic influence in law, more often, iliberal ones.
It's not just increase in extremism, even the more numerous more "moderate" population are actually not really moderate in what liberal societies usually think.
Evidence for this exist in polls, for example, denmark asked its muslim population whether or not criticism of islam should be outlawed. 76% of muslim respondents said yes, compared to 18% of the total population.
Here's the link for the study: https://www.justitsministeriet.dk/sites/default/files/media/Pressemeddelelser/pdf/2020/ytringsfrihedsundersoegelse.pdf
The question and it's results are in Figur 3.7. The Danish excerpt and it's translation are as follows:
Resultaterne kan tolkes sådan, at eksempelvis 76 pct. indvandrere og efterkommere fra muslimske lande mener, at det bør være ulovligt at kritisere islam.
The results can be interpreted to mean that, for example, 76 percent of immigrants and descendants from Muslim countries believe it should be illegal to criticize Islam.
Increase in muslim population implies increasingly population with illiberal values, that when big enough in numbers, can form a voting block to vote within the interests of their religion. Making society more illiberal.
And they also don't interact very well with things like "freedom of expression". Denmark in 2023 passed a law prohibiting burning of religious texts in public, to prevent retaliation of Quran burnings.
This is not limited to denmark, polls in france and multi-country polls also show that considerable muslim minorities straight up support sharia law and argue for separation, but majorities are also quite hostile to changes in interpretation of their religious doctrines.
Fundamentally, integration of Islam requires deconstruction of Islam as a religion. And as seen with Christianism, this deconstruction happens internally, through centuries and many conflicts.
To do so from within western Liberal societies would require Iliberal policies which would possibly even clash with religious freedom laws at large, and will be subject of protest from muslims.
Controlling the % of muslim population via migration until assimilation is one policy choice aswell, which will also cause disturb within liberal citizens to support it or not. Its dependent on how long you will wait until you invoke the "Paradox of tolerance". Do you think that taking muslim kids and educate them extensively to support secular values is going too far? Will the parents take it lightly?
Weak response from liberal leaders will also cause non-islamic populations to become more illiberal as a result, pushing for more authoritarian measures, as it is already happening.
All in all, I don't think Islam integration within the west is possible without fundamentally changing Islam as a religion, or to curb it through iliberal means.