Find the sum of all integers that satisfy these conditions: \[
|x|+1>7\text{ and }|x+1|\le7.
\]
First, let's deal with $|x| + 1 > 7$.  Subtracting 1 from both sides gives $|x| > 6$, so the integers that satisfy $|x| + 1 > 7$ are those greater than 6 and those less than $-6$.  Since the inequality is strict ($>$, not $\ge$), $x$ cannot be 6 or $-6$.

Next, we consider $|x+1| \le 7$.  Writing this as $|x-(-1)| \le 7$, we see that $x$ must be within $7$ of $-1$ on the number line, which means it must be one of the integers from $-8$ to 6.  Since the inequality is nonstrict ($\le$, not $<$), $x$ can be $-8$ or 6.

The only integers that satisfy both inequalities are $-8$ and $-7$, and their sum is $\boxed{-15}$.