L = 20; b = 1; d = 1/2;
sys = {u[x]*D[u[x], x] + D[u[x], {x, 2}] + D[u[x], {x, 4}] +
 a/(2*L)*NIntegrate[
   D[u[xp], {xp, 3}]*Cot[[Pi]*(x - xp)/(2*L)], {xp, -L, x, L},
   Method -> {"PrincipalValue"}] - b*D[u[x]*D[u[x], x], x] +
 d*D[u[x], {x, 3}] == c*D[u[x], x], u[-L] == u[L]};

ParametricNDSolve[sys, u, {x, -L, L}, {a, c}]