calculus and analysis - Integral of trigonometric function gives different answer

I am working on an integral on the following trigonometric functions

$$\int_{-\pi}^\pi \frac{\cos[(4m+2)x] \cos[(4m+1)x]}{\cos x}dx$$

where $m$ is positive integer. I am running the following code in mathematica

Assuming[Element[m, Integers] && (m > 0),
 Integrate[
  Cos[(4 m + 2) x] Cos[(4 m + 1) x]*1/Cos[x], {x, -π, π}]]

It gives me result of $\pi$. Since $m$ could be any positive number, I expect the integral should be $\pi$ if I replace $m$ with an integer before the integral. However, I got zero if I replace $m$ with number first.

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