Mark Sheridan Waden, 50, has faced two weeks on trial in Brisbane Supreme Court accused of killing Priscilla Brooten, a 46-year-old US citizen.
Her body has never been found.
Defence barrister James Godbolt during his closing statement on Friday told the jury police searches for Ms Brooten had "a not high level of reliability".
"The investigation is inadequate in the extreme," Mr Godbolt said.
"She was in Australia illegally for many years and she lived her life in a way to stay under the radar."
Police efforts to find proof of life had relied on other organisations to run searches for Ms Brooten's seven known aliases, the jury heard.
Mr Godbolt said Ms Brooten did not have a bank account, Centrelink account, Medicare card or mobile phones in her own name.
"She is someone who doesn't want to be found," Mr Godbolt said.
The prosecution alleged Waden killed Ms Brooten and buried her in a trench at his home in Brisbane's northern suburbs on July 5, 2018 before dumping her body at a Brisbane rubbish tip almost a year later.
During his closing statement crown prosecutor Andrew Walklate told the jury Waden did murder Ms Brooten when "everything came to a head" in their troubled relationship, and arguments over money and infidelity.
Waden admitted to police he had lied to Ms Brooten's friends about her being raided by Border Force and deported when she did missed gym classes.
"He impersonated the deceased. There is a sustained deception both to friends and authorities … he dug a trench," Mr Walklate said.
"That is not the conduct of a man dealing with the aftermath of a momentary unlawful assault ... it was the intention to cause at least serious bodily harm."
Waden had removed hundreds of kilograms of soil from his yard, driven them to a rubbish tip during the night and pressure-washed his trailer afterwards, Mr Walklate said.
Mr Godbolt told the jury his client was working late shifts as an Uber driver at the time and could only run errands at night.
Jurors heard claims that decaying human remains had been buried at Waden's home were not supported by witness testimony or forensic evidence.
"There was never a body there," Mr Godbolt said.
Ms Brooten had consulted a doctor and a psychologist over her acts of serious self-harm and symptoms of major depression, the jury was told.
"Unfortunately it does loom in this case the possibility of suicide," Mr Godbolt said.
Jurors are due to start deliberations on Monday after receiving directions from Justice Peter Callaghan.
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