The optimization process and recommendations:

1. Avoid Calling Functions With Indexed Columns (query line: 16): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `attempt_date` is indexed, the index won’t be used as it’s wrapped with the function `DATE_SUB`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
2. Avoid Calling Functions With Indexed Columns (query line: 17): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `attempt_date` is indexed, the index won’t be used as it’s wrapped with the function `DATE_SUB`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
3. Avoid Calling Functions With Indexed Columns (query line: 18): When a function is used directly on an indexed column, the database's optimizer won’t be able to use the index. For example, if the column `attempt_date` is indexed, the index won’t be used as it’s wrapped with the function `DATE_SUB`. If you can’t find an alternative condition that won’t use a function call, a possible solution is to store the required value in a new indexed column.
4. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

ALTER TABLE `er_login_attempts` ADD INDEX `er_attempts_idx_attempt_date` (`attempt_date`);

The optimized query:

SELECT
        count(cinc.id) AS cinc_minuts,
        count(deu.id) AS deu_minuts,
        count(quinze.id) AS quinze_minuts 
    FROM
        er_login_attempts cinc,
        er_login_attempts deu,
        er_login_attempts quinze 
    WHERE
        '[email protected]' IN (
            cinc.target_mail, deu.target_mail, quinze.target_mail
        ) 
        AND '1.11.11.111' IN (
            cinc.ip_from, deu.ip_from, quinze.ip_from
        ) 
        AND cinc.attempt_date BETWEEN MAX(cinc.attempt_date) AND DATE_SUB(MAX(cinc.attempt_date), INTERVAL 5 minutes) 
        AND deu.attempt_date BETWEEN MAX(deu.attempt_date) AND DATE_SUB(MAX(deu.attempt_date), INTERVAL 10 minutes) 
        AND quinze.attempt_date BETWEEN MAX(quinze.attempt_date) AND DATE_SUB(MAX(quinze.attempt_date), INTERVAL 5 minutes)