To complete the answer: If you call F(x) the integral computed above, then CDF(x) = F(x) - F(a). Actually if you do the calculation, F(a) = - 0.5.

@ChristianRemling I found the way to apply your solution to the more general case. See the edit on my original post.

@ChristianRemling Do you thing it is possible to find a solution for the more general case proposed in the title? I'm trying to solve the case $\log({a_0}^2 + {a_1}^2 + 2 a_0 a_1 \cos(\omega t) + {a_2}^2)$ and just adding this little ${a_2}^2$ prevent me to find a nice solution.

You are totally right! This is the good approach.

Just the answer I was looking for. Perfect! Good idea to use the complex notation.