calculus - Evaluation of $int_0^{pi/2} frac{1}{left( acos^2(x)+bsin^2(x) right)^n} , dx$

I would like to evaluate (using elementary methods if possible) : (for $a>0,\ b>0$)

$$
I_n=\int_0^{\pi/2} \frac{1}{( a\cos^2(x)+b\sin^2(x))^n} \, dx,\quad \ n=1,2,3,\ldots

$$
I thought about using $u=\tan(x)$ or $u=\frac{\pi}{2}-x$ but did not work. wolfram alpha evaluates the indefinite integral but not definite integral???