He divorced his wife thrice and he wants to take her back

Firstly: 

If a man says to his wife, “You are divorced, you are divorced, you are divorced,” it counts as three divorces according to the majority of scholars. But if he intended the second and third times as emphasis for the first, then it counts as one divorce. 

Some of the scholars regarded his saying “You are divorced, you are divorced, you are divorced” as being like saying “You are thrice divorced” and it only counts as one divorce. 

Your saying to her “You are divorced” after she gave birth counts as one divorce. So this was the second divorce. 

But if she was still in nifaas (post-partum bleeding) at the time of this divorce, then it is a haraam, innovated divorce, and the scholars differed as to whether it counts as such. The view favoured by the scholars of the Standing Committee for Issuing Fatwas is that it does not count as such. 

It says in Fataawa al-Lajnah al-Daa’imah (20/58): Innovated divorces are of several kinds, including: when a man divorces his wife at the time of menses, nifaas or in a period of purity (i.e., not menstruating) when he has had intercourse with her. The correct view is that this does not count as divorce. End quote. 

Shaykh Ibn Baaz (may Allaah have mercy on him) said: Allaah has prescribed that women are to be divorced when they are pure and free of nifaas or menses, and when the husband has not had intercourse with them. This is the shar’i divorce. If he divorces her during menses or nifaas or during a period of purity when he has had intercourse with her, then this is an innovated divorce, and it does not count as such according to the correct scholarly opinion, because Allaah says (interpretation of the meaning):

“O Prophet (صلى الله عليه وسلم)! When you divorce women, divorce them at their ‘Iddah (prescribed periods) and count (accurately) their ‘Iddah (periods)”

[al-Talaaq 65:1]. 

What is meant is when they are in a state of purity and the husband had not had intercourse with them. This is what the scholars said concerning divorcing them at their ‘iddah (prescribed periods): they should be pure and not have had intercourse or be pregnant. This is divorcing them at their ‘iddah (prescribed periods).

End quote from Fataawa al-Talaaq (p. 44). 

If you did not ask a scholar about the ruling on this second divorce then it does not count as such. But if you did ask one who is qualified to issue fatwas, then you must act according to his fatwa.  

Your saying on the third occasion “I divorce you” also counts as a divorce.  

If a man divorces his wife a third time, then she becomes irrevocably divorced from him and she is not permissible for him until she has married another man, because Allaah says (interpretation of the meaning):

“And if he has divorced her (the third time), then she is not lawful unto him thereafter until she has married another husband. Then, if the other husband divorces her, it is no sin on both of them that they reunite, provided they feel that they can keep the limits ordained by Allaah. These are the limits of Allaah, which He makes plain for the people who have knowledge”

[al-Baqarah 2:230]. 

It should be noted that the nikaah al-tahleel which some people do, which is a marriage to a second husband, intended to make a thrice-divorced woman permissible for her first husband by means of the second husband divorcing her, is haraam and the one who does that is cursed; it is also an invalid marriage and it does not make the woman permissible for her first husband. 

See also the answer to question no. 109245. 

And Allaah knows best.