How much do I have to write?

05AB1E, 3 bytes

Code:

ÝJg

Uses the CP-1252 encoding. Try it online!

Explanation:

Ý     # Range [0 .. input]
 J    # Join into one string
  g   # Get the length of the string

Python 2, 55 46 bytes

lambda n:len(`range(abs(n)+1)`)+2*~n+3*n*(n<0)

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Getting better.

Röda, 23 bytes

f x{[#([seq(0,x)]&"")]}

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Explained:

f x{[#([seq(0,x)]&"")]}
f x{                  } /* Defines function f with parameter x. */
        seq(0,x)        /* Creates a stream of numbers from 0 to x. */
       [        ]       /* Creates an array. */
                 &""    /* Joins with "". */
     #(             )   /* Calculates the length of the resulting string. */
    [                ]  /* Returns the value. */