Det Con Stuart Galloway, from Kilmarnock, East Ayrshire, falsely told the women their alleged abuser had been arrested and was going on trial.
He was caught when one alleged victim contacted another officer. Galloway resigned from Police Scotland in 2014.
The 43-year-old admitted wilfully neglecting and violating his duty.
Jailing him at Glasgow Sheriff Court, Sheriff John McCormick told Galloway: "Public trust in police officers is essential. This is particularly so where allegations of sexual assault require investigation.
"Failure to investigate such allegations can have serious consequences for the alleged victims of crime as well as those accused of crime."
Sheriff McCormick added: "Here there is not only a failure to investigate but a concerted attempt to explain delays and conceal any failure."
The court previously heard that in November 2009 a 41-year-old woman - known as witness A - reported that her father had abused her physically, mentally and sexually from around 1979 to 1989.
The inquiry was passed to the family protection unit and an officer took statements from the witness.
The court heard that in her statement, witness A spoke about the male also raping and assaulting her mother, a woman aged 61, referred to as witness B.
On 17 March 2010, a statement was obtained from witness A and the file was passed to Galloway within the family protection unit in September 2010.
He met both witnesses but failed to detain, or arrest the alleged perpetrator or report the case to the Crown Office and Procurator Fiscal Service.
The officer also provided both witnesses with a series of false updates.
These included that the man had pleaded guilty to abuse and rape, that court dates had been arranged in 2014 and that these had subsequently been cancelled for reasons such as court papers not being submitted on time, the judge being unavailable and the man being beaten up in prison.
After the final date for sentencing was cancelled on 8 April 2014, witness A felt something was not right and later went to Pollok police office in Glasgow.
When the matter was looked into Galloway confessed "I messed up" and admitted what he had done.
A former police officer who failed to act on reports that a mother and daughter had been sexually abused has been jailed for eight months.