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The function is even, so try

forget()

l,sigma=var("l,sigma")

assume(l<0)

assume(sigma>0)

2integrate((1+xx/sigma^2)**(l-1/2) , x, 0, oo)

The function is even, so try

forget()

l,sigma=var("l,sigma")

assume(l<0)

assume(sigma>0)

2integrate((1+x2 * integrate((1+x * x/sigma^2)**(l-1/2) , x, 0, oo)