Returns the reporting period month

Scenario: I need a formula that returns the reporting period month. Any date after the said date of the month, it will return a reporting period of the following month. 

If(

(Month([Date])="January" and Day([Date])>21)

or

(Month([Date])="March" and Day([Date])>25)

or

((Month([Date])="April" or Month([Date])="July" or Month([Date])="October" or Month([Date])="December") and Day([Date])>22)

or

(Month([Date])="May" and Day([Date])>20)

or

(Month([Date])="June" and Day([Date])>24)

or

((Month([Billing Month])="August" or Month([Date])="November") and Day([Date])>19)

or

(Month([Date])="September" and Day([Date]>30),

Month(AdjustMonth([Date], 1)), Month([Date])

)

This is not working :(