The video contained what Shekau claimed was footage of Boko Haram gunmen opening fire on the military in the town of Bama, using heavy weapons mounted on flat-bed trucks.
Shekau also referred to fighting in the towns Baga and Gamboru Ngala near the border with Cameroon.
The Boko Haram leader has been declared a global terrorist by the United States, which in March put a $7 million (5.3 million euros) bounty on his head.
“I’m challenging Obama,” Shekau said in the video. He voiced similar challenges to French President Francois Hollande and Israeli Prime Minister Benjamin Netanyahu.
“They are no match for me,” he proclaimed.
Boko Haram has claimed to be fighting for the creation of an Islamic state in Africa’s most populous nation and largest oil producer, though it is believed to have various factions with differing aims.
Nigeria’s 160 million population is roughly divided between a mainly Muslim north and predominately Christian south.