Is there some kind of a variation of the Laplace's method or some other formula for the asymptotics of integrals of a type $$\int_a^bf(x)e^{mp(x)}\cos(mq(x)+x/2)dx, \ m\to\infty.$$ Here $f,p,q$ are continuous and differentiable on $[a,b]$ where $a$ and $b$ can be infinity.
I tried writing the integral as $$\int_a^bf(x)e^{mp(x)}\mathfrak{R}e^{ix/2}e^{imq(x)}dx, \ m\to\infty,$$ but with no ideas how to proceed. Any help is very welcome.
EDIT: I edited the expression in $\cos$ in the original integral. I'm interested in a particular case, when $a:=0$, $b=\pi/2$, $f(x):=(\sin(x))^r\sqrt{\cos(x)},\ r>0$, $p(x):=-1/2-\sin^2(x)$, $q(x):=1/2\sin(2x)+x$.