Maybe you want:
j = 3; Integrate[Product[1/(1/c[i]^2 - (u[i] + x[i])^2), {i, 1, j}], x]
Or:
j = 3; Integrate[Product[1/(1/c[i]^2 - (u[i] + x[i])^2), {i, 1, j}], x[1], x[2], x[3]]
(*if j=3 integration a function with respect to three variables,then we need x[1] ,x[2] ,x[3] *)
j = 3; Integrate[Product[1/(1/c[i]^2 - (u[i] + x[i])^2), {i, 1, j}], Sequence @@ Table[x[i], {i, 1, j}]]
(*More automatic*)
?