"You miss all the highs. I would miss playing for this team. This week can't be beaten. I can walk away with my head held high," he added.
The last day of the series saw a bit of drama as centuries by Lokesh Rahul and Rishabh Pant had given India a slim hope of clinching, or at least saving, the Test.
Rahul played a fighting knock of 149 runs before a spectacular delivery by England spinner Moeen Ali ended his resistance.
Pant, who scored 114, gave him excellent support from the other end as the duo built up a partnership of 204 runs to frustrate the England bowlers for a lengthy period.
However, once the duo were back in the pavilion, the hosts quickly wrapped up the Indian tailenders to complete a well deserved series victory.
It was a memorable day for James Anderson as well, as the England pacer's second innings figures of 4/63 saw him go past Australian legend Glenn McGrath to take the fourth spot on the list of all-time highest Test wicket takers.
Anderson, who now has 564 Test wickets to his name, is leading McGrath by one wicket. He is preceded on the list by Muttiah Muralitharan, Shane Warne and Anil Kumble.
"Today it showed how tough Test cricket can be. I leave with great memories. To see Jimmy go past Glenn McGrath was brilliant. Just has been a privilege to have played with England's greatest cricketer," Cook said.
"No disrespect to any other cricketer. He almost takes it for granted that he will hit the length from ball one. It is fitting that he got the last wicket," he added.
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