How can I multi dimensional integration quickly?

Questions
1

I tired to Integrate with several dimensions but it calculate so slow. I wait for 20 minutes but calculate so much.
Therefore I want to know how can I integrate several times in a quick methods.
in below I tried to integrate

for 3 dimension

Integrate[Integrate[Integrate[(1 + Subscript[c, 1]*Subscript[x, 1] + 
    Subscript[c, 2]*Subscript[x, 2] + 
    Subscript[c, 3]*Subscript[x, 3])^(-4), {Subscript[x, 1], 0, 1}],{Subscript[x,2],0,1}],{Subscript[x,3],0,1]

for 4 dimension

Integrate[Integrate[Integrate[Integrate[
 exp (-Sum[
     Subscript[c, i]^2*(Subscript[x, i] - Subscript[u, i])^2, {i, 1, 
      4}]), {Subscript[x, 1], 0, 1}],{Subscript[x,2],0,1}],{Subscript[x,3],0,1}],{Subscript[x,4],0,1}]]

For 5 dimension

Integrate[Integrate[Integrate[Integrate[Integrate[
 exp (-Sum[
     Subscript[c, i]*Abs[Subscript[x, i] - Subscript[u, i]], {i, 0, 
      5}]), {Subscript[x, 1], 0, 1}], {Subscript[x, 2], 0, 1}], {Subscript[x, 3], 0, 1}]], {Subscript[x, 4], 0, 1}]], {Subscript[x, 5], 0, 1}]]
2

It takes a long time because it tries to figure out the complicated integrability conditions in terms of the coefficient. You can skip those steps with GenerateConditions->False in the three integrals.