# How to control Boundary conditions after integrating over piecewise function?

The following code

Assuming[-1 < t < 1 && Element[a, Reals],
FullSimplify[
Integrate[
Piecewise[{{(1 + x)/(1 + t), -1 < x < t},
{(1 - x)/(1 - t),  t < x < 1},
{0,                True}}],
{x, -1, a}]]]


will result in $$\begin{cases} 1 & a>1 \\ \frac{(a+1)^2}{2 (t+1)} & a>-1\land a<t \\ \frac{t+1}{2} & a=t \\ \frac{(a-1)^2}{2 (t-1)}+1 & a>t\land a\leq 1 \end{cases}$$

By the assumptions above, that's the same as:

$$\begin{cases} 1 & a\ge 1 \\ \frac{(a+1)^2}{2 (t+1)} & a>-1\land a\leq t \\ \frac{(a-1)^2}{2 (t-1)}+1 & a>t\land a< 1 \end{cases}$$

which is the form I'd prefer.
Is there a way to automatically tell mathematica to do that?

The reason I want the second form is, that I'd need to integrate the output once again and in the first form, it would result in weird behavior, suddenly unnecessarily special-casing for $\frac{1}{\sqrt{2}}$ or things like that.

(Note that there is no case for $a\leq -1$. That case gets auto-handled by Mathematica as $\{0,True\}$ and doesn't show up when you use the \TeXFormat command)

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How general a solution do you need? In the example, for instance, just eliminating the equality will do the trick, as in % /. {y_, x_Equal} -> {y, False}, but in more general situations this might be incorrect. –  whuber Jan 30 at 19:27
@whuber What I really want is a solution that will work for the outer-most cases all the time (which will always be something<x<-1 or somethinng>x>1) which I want to turn to x<=-1 or x>=1 respectively, and additionally, I want to get rid of any single point definitions: What I'm doing is guaranteed to be continuous and won't need such things. Beyond that I might need some more but I'll have to see if anything weird happens, in which case I'll come back if it's not clear to me how to extend a solution that may be given here. –  kram1032 Jan 30 at 22:54
@whuber sometimes the formating of the conditions will also be messed up, having something depending on more variables in the middle, rather than just plain x, like something<2x-t<something or stuff like that. So the simplest form of pattern-matching probably won't cut it if I can't also automate a cleanup of all cases so they become something < x < something or similar... –  kram1032 Jan 30 at 23:02