General Form

Integrate[
Cos[((m π (x+a/2))/a)] Sin[((n π (y+c/2))/c)]Cos[((p π (x+a/2))/a)] Sin[((q π (y+b/2))/b)],
 {x, -a/2, a/2},{y, -b/2, b/2},Assumptions->{{m,n,p,q}∈Integers&& m≥0 && n≥0&&p≥0&&q≥0&&m+n≠0&&p+q≠0&&a>0&&b>0&&c>0} ]

Simplify[%, Assumptions->{m,p}∈Integers]

cq == bn && q == 0: always 0 since sin’s in the integrand are 0

m != p && cq == bn : always 0

Integrate[
Cos[((m π (x+a/2))/a)] Sin[((q π (y+c/2))/b)]Cos[((p π (x+a/2))/a)] Sin[((q π (y+b/2))/b)],
 {x, -a/2, a/2},{y, -b/2, b/2},Assumptions->{{m,n,p,q}∈Integers&& m≥0 && n≥0&&p≥0&&q≥0&&m+n≠0&&p+q≠0&&a>0&&b>0&&c>0} ]

Simplify[%, Assumptions->{m,p}∈Integers]

Integrate[
Cos[((m π (x+a/2))/a)] Sin[((q π (y+c/2))/b)]Cos[((p π (x+a/2))/a)] Sin[((q π (y+b/2))/b)]/.q->0,
 {x, -a/2, a/2},{y, -b/2, b/2},Assumptions->{{m,n,p,q}∈Integers&& m≥0 && n≥0&&p≥0&&q≥0&&m+n≠0&&p+q≠0&&a>0&&b>0&&c>0} ]

m == p != 0 && cq != bn

Integrate[
Cos[((m π (x+a/2))/a)] Sin[((n π (y+c/2))/c)]Cos[((p π (x+a/2))/a)] Sin[((q π (y+b/2))/b)]/.m->p,
 {x, -a/2, a/2},{y, -b/2, b/2},Assumptions->{{m,n,p,q}∈Integers&& m≥0 && n≥0&&p≥0&&q≥0&&m+n≠0&&p+q≠0&&a>0&&b>0&&c>0} ]

Simplify[%, Assumptions->{n,p}∈Integers]

m == p == 0 && cq != bn

Warning: different than the last case, factor 2

Integrate[
Cos[((m π (x+a/2))/a)] Sin[((n π (y+c/2))/c)]Cos[((p π (x+a/2))/a)] Sin[((q π (y+b/2))/b)]/.{m->0,p->0},
 {x, -a/2, a/2},{y, -b/2, b/2},Assumptions->{{m,n,p,q}∈Integers&& m≥0 && n≥0&&p≥0&&q≥0&&m+n≠0&&p+q≠0&&a>0&&b>0&&c>0} ]

Simplify[%, Assumptions->{n,q}∈Integers]

m == p != 0 && cq == bn (&& q != 0)

Integrate[
Cos[((m π (x+a/2))/a)] Sin[((q π (y+c/2))/b)]Cos[((p π (x+a/2))/a)] Sin[((q π (y+b/2))/b)]/.m->p,
 {x, -a/2, a/2},{y, -b/2, b/2},Assumptions->{{m,n,p,q}∈Integers&& m≥0 && n≥0&&p≥0&&q≥0&&m+n≠0&&p+q≠0&&a>0&&b>0&&c>0} ]

Simplify[%, Assumptions->{p,q}∈Integers]

m == p == 0 && cq == bn (&& q != 0)

Integrate[
Cos[((m π (x+a/2))/a)] Sin[((q π (y+c/2))/b)]Cos[((p π (x+a/2))/a)] Sin[((q π (y+b/2))/b)]/.{m->0,p->0},
 {x, -a/2, a/2},{y, -b/2, b/2},Assumptions->{{n,p,q}∈Integers&& n≥0&&p≥0&&q≥0&&m+n≠0&&p+q≠0&&a>0&&b>0&&c>0} ]

Simplify[%, Assumptions->q∈Integers]