q''[t] = Inverse[a(q[t],q'[t])].q'[t]

n = 3 (* Sets exponent for the "a" matrix, fast if n=1, slow for n >=3*)

q = {{q1[t]}, {q2[t]}, {q3[t]}, {q4[t]}} (* configuration variables*)
dq = D[q, t] (* velocity variables*)
ddq = D[dq, t] (* acceleration variables*)

(* Setting up initial conditions*)
q0 = Thread[Flatten@q == 0.1] /. t -> 0
dq0 = Thread[Flatten@dq == -0.1] /. t -> 0

(* Large matrix that needs to be inverted A = Inverse[a[q,dq]]*)
A[q_, dq_, n_] := Inverse[
{{2*Cos[dq[[1]]], 2*Sin[q[[4]]], q[[3]] + Sin[q[[4]]], Cos[dq[[1]]] + Sin[dq[[4]]]},
 {2*Sin[q[[4]]], 2*q[[1]], Cos[dq[[1]]] + Sin[q[[3]]], Cos[dq[[1]]] + dq[[1]]},
 {q[[3]] + Sin[q[[4]]], Cos[dq[[1]]] + Sin[q[[3]]], q[[4]] + Sin[q[[4]]], Cos[dq[[1]]] + q[[2]]},
 {Cos[dq[[1]]] + dq[[4]], Cos[dq[[1]]] + dq[[1]], Cos[dq[[1]]] + q[[2]], dq[[4]] + Tan[dq[[4]]]}}^n]

{tvar, RHS} = Timing[Flatten[A[Flatten@q, Flatten@dq, n].dq]];
Print[tvar] (* print execution time for evaluating A matrix*)

ddqEqs = Thread[Flatten@ddq == RHS]; (* Setting up equations*)
NDSolve[Join[ddqEqs, q0, dq0], Flatten@q, {t, 0, 1}] (* Run numerical integration*)