I know that $$\frac{d\bigg(\det|C_1(x)\;\;C_2(x)\;\;C_3(x)|\bigg)}{dx}$$
$$=\bigg(\det|C_1'(x)\;\;C_2(x)\;\;C_3(x)|\bigg)+\bigg(\det|C_1(x)\;\;C_2'(x)\;\;C_3(x)|\bigg)+\bigg(\det|C_1(x)\;\;C_2(x)\;\;C_3'(x)|\bigg)$$
Is there also a formula for the integration of a determinant?
$$\int{\bigg(\det|C_1(x)\;\;C_2(x)\;\;C_3(x)|\bigg)}dx$$