Does immigration help to sustain welfare in rich European countries?
Here is a chart from the OECD, which shows the net fiscal impact of migrants on their recipient country (i.e. by how much to they contribute or withdraw from the welfare state, albeit excluding in-kind benefits such as healthcare). For most Western countries, Denmark included, migrants are modest net positive contributors ("they pay in more than they take out").

This is because migrants tend to be disproportionately working age, whereas the native population contains many retired or young people. Thus, migrants tend to pay a lot of income tax relative to the average native. Luxembourg and Switzerland stand out because both are small economies that attract many high-wage knowledge workers.

On the vertical axis of this chart is the net contribution to the fiscal budget. So a value of 1% means $(\$\text{tax paid}-\$\text{welfare received})/\$\text{GDP}=0.01$. Note that this is different to (and less than) migrants' overall contribution to GDP because, for example, migrants keep much of their contribution to GDP in the form of after-tax wages.



I took the chart from Oxford's migration observatory, which has more information specific to the case of the UK.