What is the ruling of Islamic law regarding the days during which bleeding stopped due to taking birth control pills, and which were not followed by a complete period of purity, and is it obligatory to repeat the prayers during these days?

If the bleeding exceeds fifteen days, you are considered a woman with Istihada (non-menstrual vaginal bleeding), and you must refer to your previous habit, or act based on discernment (tamyiz), or observe six or seven days as is common for most women.

If the blood flows continuously so that you do not find enough time to perform prayer with a valid purification, you should take precautions, perform ablution (wudu') for each prayer after its time has entered, and pray the obligatory (fard) prayers and any supererogatory (nawafil) prayers you wish.

If the bleeding stops at a known time, you should wait for that time, then cleanse yourself (istinja'), perform ablution, and pray. After this, all rulings for pure women apply to you.

You must make up the prayers you missed, thinking they were due to menstruation, but which turned out to be Istihada.