Learn Hardcore Danish: Jeg vil tage en anden bog.

Jeg vil tage en anden bog. ≈ “I want to take another book.” / “I intend to take another book.”
If you just want to state a neutral future (no special intention), Danish often uses the present tense instead:
I’ll take another book (later). → Jeg tager en anden bog (senere).

So vil focuses on desire/intention, not on future time alone.

Why is it tage and not tager?

Because vil is a modal verb, and in Danish, modal verbs are followed by the bare infinitive (the dictionary form) without -r and usually without at.

Infinitive: (at) tage = to take
Present tense: tager = take(s)

Examples:

Jeg tager en bog. = I take a book.
Jeg vil tage en bog. = I want to take a book.

So after vil, you must use tage, not tager.

Why don’t we say “Jeg vil at tage en anden bog” (with at)?

With vil, you do not use at before the following verb.

Correct: Jeg vil tage en anden bog.
Incorrect: Jeg vil at tage en anden bog.

Some verbs in Danish take at before another verb:

Jeg begynder at læse. = I’m starting to read.
Jeg prøver at forstå. = I’m trying to understand.

But the modal verbs (vil, kan, skal, må, bør) are followed directly by the infinitive without at:

Jeg vil tage…
Jeg kan læse…
Jeg skal købe…

Is vil tage a future tense, like English “will take”?

Not exactly. Danish doesn’t have a special “future tense” form like English will. Vil can sometimes refer to the future, but its main meaning is volition: wanting or intending.

Jeg vil tage en anden bog.
→ Focus on “I want / I intend to take another book.”
Simple future is often just the present in Danish:
I’ll take another book tomorrow. → Jeg tager en anden bog i morgen.

So yes, vil tage can be about the future, but what it really expresses is intention, not just future time.

What is the difference between tage and have here? Could I say “Jeg vil have en anden bog”?

Both are possible, but they mean slightly different things:

Jeg vil tage en anden bog.
→ Emphasizes the action of taking another book (picking it up, choosing another one).
Jeg vil have en anden bog.
→ Emphasizes wanting to have/receive/own another book; more about possession or desire.

In a library or bookstore context:

If you’re physically switching books, tage feels more natural.
If you’re telling someone what you want (e.g. ordering a different book), have is very common.

What exactly does anden mean here? Is it “another” or “second” or “different”?

Anden can cover several English ideas, depending on context:

Another one (a different one):
Jeg vil tage en anden bog. = I want to take another book / a different book.
The second in a sequence:
den anden bog = the second book.

In your sentence, with no clear sequence mentioned, en anden bog is usually understood as “a different book / another book (instead of this one).”

How is en anden bog different from en bog mere?

En anden bog = another / a different book (not the same one).
→ You’re changing which book it is.
En bog mere = one more book (an additional book, on top of the current one).
→ You’re increasing the number of books.

Examples:

If you don’t like the book you’re holding:
Jeg vil tage en anden bog. = I want to swap it for a different one.
If you already have one book and want an extra:
Jeg vil tage en bog mere. = I want to take one more book.

Why is it en bog and not et bog?

Because bog is a common gender noun in Danish, and common gender nouns use the article en.

en bog = a book
bogen = the book

If a noun is neuter, it takes et:

et hus = a house
huset = the house

There’s no reliable rule from English; you mostly have to learn the gender with each noun:

en bil (a car), en stol (a chair), et bord (a table), et æble (an apple), etc.

Can the word order be changed, like “Jeg vil en anden bog tage”?

No, not in a normal, neutral statement. The standard word order here is:

Subject – modal – infinitive – object
→ Jeg vil tage en anden bog.

Putting the object between vil and tage (Jeg vil en anden bog tage) is ungrammatical in modern Danish.

If you add more elements (like an adverb), vil + tage still stay together:

Jeg vil ikke tage en anden bog. = I don’t want to take another book.

How do I pronounce jeg vil tage en anden bog?

Very roughly (in a common standard accent):

jeg ≈ like English “yigh” (often reduced to something like /jɑ/ or /jæ/)
vil ≈ vil with a short i, a bit like English “vill”
tage ≈ roughly “TA-eh” or “TÆ-eh”; the -ge is very soft, often barely pronounced
en ≈ short, like English “en” in “en-list”
anden ≈ AN-den, both n sounds clear
bog ≈ close to “boh” with a long o (not like English “bog”). Many pronounce it like /boː/ or /bɔː/, often with a glottal stop (stød).

Spoken fast, the sentence can sound something like: “jæ vil TÆ-eh en AN-den boː”.

How would I say “other books” in plural, and does anden change?

Yes, anden changes in the plural:

Singular: en anden bog = another/other book
Plural: andre bøger = other books

So:

Jeg vil tage andre bøger. = I want to take other books.
andre is the plural form of anden/andet.

Is this sentence informal or formal? Would I use it in polite situations?

The sentence Jeg vil tage en anden bog. is neutral and perfectly fine in both informal and polite contexts.

If you want to sound a bit softer or more polite (especially when addressing a person), you could sometimes rephrase, for example:

Jeg tror, jeg vil tage en anden bog.
(“I think I’ll take another book.” – sounds a bit less direct.)

But grammatically and stylistically, your original sentence is completely acceptable in normal polite conversation.