WASHINGTON (DPA) - President Donald Trump celebrated the death of Abu Bakr al-Baghdadi as he has other major events of his presidency - with grievances, partisan jabs and confounding statements and tweets.

Even Democrats acknowledged that a president battling an impeachment probe and bipartisan criticism for pulling US troops out of northern Syria got a much-needed win that could help his 2020 re-election campaign.

"Unfortunately, it's very good for Trump because it kind of credentials him at a time when his profile on foreign policy and national security has taken a major hit," said Ms Celinda Lake, a Democratic pollster who is advising former vice-president Joe Biden's presidential campaign.

Yet a win for Mr Trump never comes without complications. Mr Trump's vivid description on Sunday (Oct 27) of Baghdadi "whimpering and crying and screaming" was undermined on Monday when General Mark Milley, chairman of the Joint Chiefs of Staff, told a Pentagon news conference that he could not confirm those details and did not "know the source" of Mr Trump's information.

Defence Secretary Mark Esper had already thrown cold water on Mr Trump's account on Sunday, declining to confirm it in a televised interview.

Mr Trump said he might release some video of the Special Forces raid that led the elusive leader of the Islamic State in Iraq and Syria (ISIS) to blow up a suicide vest in a tunnel rather than surrender.

The Pentagon had aerial surveillance of the site, but it's not known if US forces wore body cameras that offered additional video.

Mr Trump also tweeted a photo of the military K-9 dog that the White House said had chased Baghdadi into the tunnel. The Pentagon refused to release the dog's name, saying it was classified because the animal was still on duty in the war zone.

Mr Trump appeared undaunted by criticisms of his rhetoric - or of the loud boos and chants to "Lock him up!" that he received while attending the World Series game on Sunday night in Washington - boasting that Baghdadi was "dead as a doornail" and attacking his predecessors in the White House.

"He should have been killed years ago," Mr Trump told a conference of police chiefs in Chicago. "Another president should've gotten him."

Mr Trump's campaign sent text messages and e-mails to supporters that took a similar tone - mixing cheers for the accomplishment with longstanding complaints that Mr Trump is under unfair attack from the media.

Politicians often use current events to raise money or build supporter networks. But Mr Trump broke with tradition by quickly slamming his political opponents, a tactic more likely to please ardent supporters than win converts.

"This is something that a president can use to unify people and build support for his national security agenda, but it's generally done in somewhat of a subdued way," said Mr Ryan Williams, a Republican consultant.

Mr Williams, an aide on Mr Mitt Romney's 2012 presidential campaign, recalled that President Barack Obama delivered a solemn address in May 2011 after a CIA-led team covertly entered Pakistan to kill Osama bin Laden.

Though Mr Trump is eager to claim credit for the Baghdadi mission, he refused to give Mr Obama credit for taking out Osama bin Laden, the mastermind of the 9/11 attacks.

"Stop congratulating Obama for killing Bin Laden. The Navy Seals killed Bin Laden," he tweeted during a 2012 presidential debate after Mr Romney gave Mr Obama kudos.

In a statement on Monday, Mr Biden called the Baghdadi mission a "win for American national security" and he praised Mr Trump for giving it the go-ahead.

"But as more details of the raid emerge, it's clear that this victory was not due to Donald Trump's leadership," Mr Biden said. "It happened despite his ineptitude as commander-in-chief."

A senior State Department official said the Pentagon had accelerated the mission because of the "chaotic situation" that followed Mr Trump's abrupt decision this month to withdraw US troops from northern Syria, abandoning US-backed Kurds in the area.

The political impact of the Baghdadi raid could shift over time.

Democrats cited Osama bin Laden's death frequently during the 2012 campaign, claiming "Bin Laden is dead and General Motors is alive" to highlight Mr Obama's top foreign and domestic accomplishments, including rescuing the automobile industry after the 2008 recession.

Polls showed Mr Obama got a bounce in his approval ratings, but it disappeared in a few months.

Mr Charlie Cook, editor of the non-partisan Cook Political Report, predicted a similar outcome for Mr Trump, arguing that his core support seldom shifts, for good news or bad.

"If 4.2 per cent GDP growth in the 2nd quarter of 2017 and 50-year lows in unemployment don't help him much, this probably won't," he said in an e-mail.

Yet Ms Lake, the Biden pollster, argued that Mr Obama was helped in the long run - and Mr Trump may be as well - even if polls don't reflect a clear bump.

For Mr Obama, the Osama bin Laden raid helped assuage doubts among conservative and male voters who tend to view African American, female and Democratic candidates less favourably on foreign policy and defence, she said.

For Mr Trump, the Baghdadi mission could prevent defections from independent voters concerned about his lacklustre foreign policy record, including his battles with European allies, his failures to nail down accords with China and North Korea, and his abandonment of US military allies in Syria.

Mr Trump now has a response to those criticisms.

"To me, it's more of an influence and with the whole building block of things rather than a direct hit," Ms Lake said. "And those building blocks are rearranged here."