Some integrals with problematic evaluation

[parisse]

I wonder who asked you to post screenshots, certainly never me! You can't do anything with screenshots except looking at them...
Don't forget that there is no table lookup inside Giac. If the system finds a way to compute the antiderivative, it will use it even if there might be a another way which does not introduce floor and so on. For example take
int(1/(2+3*cos(x)^2))
If I returned atan(2*tan(x)/(sqrt(10)))/(sqrt(10)) without the floor term, the answer would not be continuous at x=pi/2+k*pi. Your antiderivative does not require a floor term, but I don't know how you get it, it does not help me much to see it. The general algorithm I'm using for fractions of trigonometric expressions is rewrite with tangents and make the change of variable with the tangent.
I have nothing against improving the integration algorithm, but if you want to really help me do that, you should
1/ enter the commandline
2/ enter the nice looking answer you expect
3/ explain how it is obtained