Often one way is to reduce to constant intervals.
In many cases order of integration does not matter (but you may check for your specific case), so interchange them and consider Int(Int(f(a,b),b=1-a..1+a),a=0..10). You may use Upper Case instead of lower case for integration untill the end.
The inner integral can be mapped to any interval, in the following I map it to -1 ... 1. Then your problem has vanished.
Int(f(a,b),b = 1-a .. 1+a);
IntegrationTools:-Change(%, b = a*t+1, t): combine(%);
| | f(a, a t + 1) a dt da
Int(Int(f(a,a*t+1)*a,t = -1 .. 1),a = 0 .. 10);
eval(%, f = '(a,b) -> exp(-a^3-b)');
value(%): subs(int=Int, %); # Maple does not know a symbolic solution
| | 3
| | exp(-a - a t - 1) a dt da
| 3 3
| exp(-a + a - 1) - exp(-a - a - 1) da
Edited to add:
For that example good luck reduced the task even to dim = 1.
You can rewrite that final integrand as exp(-a^3)*exp(-1)*sinh(a)*2
to reduce (additive) cancellation errors if you want high precision
(though Maple should care for that on its own).