General Form

Integrate[
Sin[((m π (x+a/2))/a)] Cos[((n π (y+c/2))/c)]Sin[((p π (x+a/2))/a)] Cos[((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]

m == p == 0:  always 0, since the sin’s are 0

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

Integrate[
Sin[((m π (x+a/2))/a)] Cos[((n π (y+c/2))/c)]Sin[((p π (x+a/2))/a)] Cos[((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∈Integers]

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

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

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

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

Integrate[
Sin[((m π (x+a/2))/a)] Cos[((q π (y+c/2))/b)]Sin[((p π (x+a/2))/a)] Cos[((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&&p>0&&m≠p&&a>0&&b>0&&c>0} ]

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

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

Integrate[
Sin[((m π (x+a/2))/a)] Cos[((n π (y+c/2))/c)]Sin[((p π (x+a/2))/a)] Cos[((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&&c q == b n&&a>0&&b>0&&c>0} ]

Simplify[%, Assumptions->p∈Integers]

Simplify[%, Assumptions->q∈Integers]

m == p && cq == bn && p != 0 && q == 0 => n==0

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

Simplify[%, Assumptions->p∈Integers]