What's wrong with this syntax for a computed column?

There are several bugs, mostly caused by unbalanced parentheses and missing end

for cases

. Also, you don't need to specify the type.

So, I think this will work:

ALTER TABLE table_1
   ADD [calc_column] AS (CASE WHEN [payment_1] IS NULL AND  
                                   [payment_2] IS NULL AND
                                   [payment_3] IS NULL AND
                                   [payment_4] IS NULL AND
                                   [payment_5] IS NULL AND
                                   [payment_6] IS NULL
                              THEN NULL 
                              ELSE COALESCE([payment_1], 0) + COALESCE([payment_2], 0) +
                                    (CASE WHEN COALESCE([payment_3], 0) + COALESCE([payment_7], 0) + COALESCE([payment_4], 0) > COALESCE([payment_5], 0)
                                          THEN COALESCE([payment_1], 0) + COALESCE([payment_4], 0) + COALESCE([payment_3], 0)
                                          ELSE COALESCE([payment_8], 0) 
                                     END)
                         END);