Because I like a challenge:

  =SUMPRODUCT(
    (bracket_min<income)*
    (
      ((income<=bracket_max)*(income-bracket_min))
      +
      ((income>bracket_max)*(bracket_max-bracket_min))
    )
    *bracket_rate
  )

Where income is your income, bracket_min is the range of bracket minimums, bracket_max is the range of bracket maximums, and bracket_rate is the range of bracket tax rates.

Demo on Google Sheets:

https://docs.google.com/spreadsheets/d/1z0vx8TJeWr-hbJ3q6E7r...