My eleventh integral problem

Calculus Level 5

\[ \large \displaystyle \int_{0}^{\pi/2} e^{x} \cos^{2} x \, dx \]

It is given that the above integral can be expressed in the form \[\large \dfrac{a e^{{b \pi}/{c}}-d}{f}\] where \( a, b, c, d, f \) are positive integers, \(\gcd(a,f) = \gcd(b,c) = 1\).