Why is TI-Interactive so bad at integrating?

Hello!

Today i asked TI-interactive for doing this:

integral(abs(sin(x*2*pi)),x)

But the answer is exactly the same as I typed!?

After all TI was able to answer this one:

integral(sin(x*2*pi),x) = ............

From this answer I was able to integrate the first one myself and I ended up with:

integral(abs(sin(x*2*pi)),x) = (2*floor(2*x) - cos(pi*mod(2*x,1))) / (2*pi)

Now I discovered that the method that TI uses when it isn't able to integrate is very inexact.

Asking TI:

integral(abs(sin(x*2*pi)),x,5,63) = 36.923**8551611**

Making use of my integration:

integral(abs(sin(x*2*pi)),x,5,63) = 116/pi = 36.923**9467973**

Now I have two questions:

What is the name of the method which TI uses in a situation like this?

How comes it that TI isn't able to integrate a simple function, which I am able to integrate?