How to control Boundary conditions after integrating over piecewise function?

Question

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)