Is it prescribed for the worshipper to secretly say "Astaghfirullah Al-Atheem (I ask Allah, the Almighty, for forgiveness)" when rising from bowing and saying "Allahumma Rabbana Lakal Hamd (O Allah, our Lord, to You be all praise)"?

Rising from bowing (Ruku') is not a place for supplication (Du'a) in prayer, except in times of calamities. What is to be said in this position is what has been authentically reported from the Prophet, peace and blessings be upon him. It is recommended to make abundant supplication during prostration (Sujud), for the Prophet, peace and blessings be upon him, said: "As for bowing, glorify the Lord therein. And as for prostration, strive in supplication, for it is most likely that you will be answered." And whoever's saying of "Allahumma Rabbana wa laka al-hamd" (O Allah, our Lord, and to You be all praise) coincides with the saying of the angels, his past sins will be forgiven.