Jeg vil sammenligne den her bog med den anden.

Because vil is a modal verb, and after modal verbs Danish normally uses the bare infinitive.

• Jeg vil sammenligne = I want to / will compare
• Jeg kan læse = I can read
• Jeg skal gå = I have to go

So at is not used after vil. A form like jeg vil at sammenligne would be wrong in normal Danish.

It can suggest either idea, depending on context.

In everyday Danish, jeg vil + infinitive often sounds like:

• I want to ...
• I intend to ...
• sometimes I will ...

So Jeg vil sammenligne ... can mean either I want to compare ... or I will compare .... Danish often leaves that distinction to context.

Because after vil, the main verb stays in the infinitive form.

• vil sammenligne = want to compare / will compare
• sammenligner = compares / am comparing

Compare:

• Jeg vil sammenligne den her bog ... = I want to compare this book ...
• Jeg sammenligner den her bog ... = I am comparing / compare this book ...

So sammenligne is required here because of vil.

Sammenligne means to compare. The usual pattern is:

sammenligne A med B

So in this sentence:

• den her bog = the first thing being compared
• den anden = the second thing being compared

This is a very normal and common structure in Danish.

Because the standard Danish construction is sammenligne ... med ....

So:

• sammenligne den her bog med den anden

In English, you may hear both compare with and compare to, but in Danish med is the normal choice here.

Both are possible.

• den her bog = very common, especially in everyday spoken Danish
• denne bog = also correct, but often sounds a bit more formal or written

So this sentence uses the more conversational form.

You could also say:

Jeg vil sammenligne denne bog med den anden.

That would still be correct.

Because Danish often forms this/that + noun with this pattern:

• den + her + noun for common gender singular
• det + her + noun for neuter singular
• de + her + noun for plural

Examples:

• den her bog = this book
• det her hus = this house
• de her bøger = these books

So den is not extra — it is part of the normal demonstrative pattern.

Because after den her, the noun stays in the indefinite form.

So Danish says:

• den her bog
• not den her bogen

This is important, because English speakers often expect something like the this book, but Danish does not work that way.

The same is true with denne:

• denne bog
• not denne bogen

Yes. Bogen her can also mean this book.

So both of these are possible:

• den her bog
• bogen her

A rough difference is:

• den her bog is a very common neutral way to say this book
• bogen her can sound a bit more like this book here, often with a pointing or contrastive feeling

In many situations, both are fine.

Usually not if you mean this book.

Den bog by itself often means something more like:

• that book
• or the book in a specific context

So if you want this book, use:

• den her bog
• or denne bog

Without her, the meaning usually changes.

Because the noun bog is understood and left out.

So:

• den anden = the other one / the second one
• den anden bog = the other book / the second book

Danish often leaves out a noun when it is already clear from context.

Here, den anden really means the other book.

It can mean both, depending on context.

• If there are two things, den anden often means the other one
• In an ordered sequence, it can mean the second

In this sentence, most learners would understand it as the other one, because one book is being compared with another book.

They change according to gender and number.

• anden = singular, common gender
• andet = singular, neuter
• andre = plural

Examples:

• den anden bog = the other book
• det andet hus = the other house
• de andre bøger = the other books

Since bog is a common gender noun, the sentence uses den anden.