Network Rail engineers worked through the night on Thursday to rerail and remove the train involved and repair damaged track, points and cabling.
These works were delivered ahead of schedule allowing the introduction of a normal service on Friday, rather than the revised service previously advertised.
The RAIB is continuing its investigation into the cause of the derailment.
David Simpson, Network Rail route managing director Scotland, said: “Every effort has been made to resolve this incident as quickly as possible and restore a full service to passengers.
“Our engineers worked quickly and efficiently through the night to recover the railway in time for start of service on Friday, delivering all four lines back into operation 24-hours earlier than originally planned.
“We are continuing to work with the RAIB to identify the cause of the incident.”
Also in Europe today, City.Mobi is celebrating the success of its new guide to London.
City.Mobi offers the most comprehensive mobile travel guides available, with over 800 cities in 200 countries listed. Each is developed by the City.Mobi team to combine into a single global travel directory.
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City.Mobi guides include user reviews and traveller utilities such as a translation guide, currency converter, news and local weather guide.