I don't know about the Danish system as a whole, but the income tax system for individuals is much simpler then the US. For example, there are three different tax forms Americans file (1040EZ with about 15 lines, 1040A with fifths, and the full 1040 with 79 lines). Your income tax is calculated in stages.
First you add up all potential sources of income -- wage income, independent contractor income (which usually has to go on a Schedule C so your software calculates the Self-Employment tax correctly), interest from your bank account, capital gains (which could come from selling Bitcoin at a profit), etc. This is your Gross Income.
Then you adjust the Gross Income by taking out certain other expenses the government really approves of. This are called "above the line" deductions. This year I paid student-loan interest, so I subtracted that interest from my wages to get my Gross Income. This gave me an "Adjusted Gross Income." Many retirement accounts, health savings accounts, etc. are here which means the Feds are using the Income tax system top subsidize a) health care, b) higher education, and c) retirement. You will also note that since it's done through the income tax system, the tax-break used to pay for all these things is much more advantageous for people who have a lot of income to tax (like Mitt Romney) then the poor schmuck whose so proud that he's full-time at Walmart.
After that you take out your exemptions ($3,950 per person claimed on your return, and there are lengthy legal tomes devoted to the thorny subject of who gets to claim whom for what), your Deduction ($6,200 for single or married filing single, $9,100 for Head of Household, or $12,400 for Married Filing Joint; or some completely different number you calculate yourself based on a completely different list of expenses The Government Really Approves of them the aforementioned above-the-line deductions). This results in a Modified Adjusted Gross Income.
But now that we've grossed our income, adjusted it, and modified the adjustment, surely it will be a simple matter of math to figure out the taxes owed? Don't be stupid.America has multiple tax brackets, and if you told an American just on the cusp of going from the 10% bracket to the 15% that his tax bill would increase by 50% if he'd made just bit more he will scream to high heaven and start a rebellion. So what they did is they made a table. It's in $50 increments. If you're in the 10% bracket every increment is $5 of taxes until you get to the end of the bracket. Then it's 15%, so it alternates going up by $7 and $8 (including pennies so you could just do $750 would make sense, so we don't). Then you hit the 25% bracket and alternate between $12 and $13, the 28% bracket is $14, but once you're halfway through the 28% bracket they stop the table and you have to calculate by hand for the rest of the 28% bracket, and the entirety of the 33%, the 35% and there 39.6% brackets.
Congratulations! We're half done! You see now that we know what you should theoretically pay we can start using tax credits. Tax credits are better then deductions because credits reduce the full amount you pay. There're child-care credits, education credits, and retirement credits. We're now about 2/3 done. Maybe 3/4.
Now we have to factor in your payments. Your withholding is just a very small part of what you officially paid. The thing about the credits in the last paragraph is they can make you own $0 taxes, but the can't make the Federal government cut you a fucking check for $5,000 (note: I actually had someone at my tax desk today get a $5,000 check from the Feds). Some education payments double as tax payments, there's a credit for buying gasoline for your tractor, there's the infamous Earned Income Credit which is de facto the major element of the modern welfare state because it gives people who made very little money and are raising families multi-thousand$ checks.
So my friend, if Denmark has managed to make it's income tax system more complicated then the US I will be both extremely disappointed and impressed.