Problem

$$I_n=\int \cosh^n(2x)\,dx \qquad n\ge 0$$

(a) Show that, for $n\ge 2$

$$I_n=\frac{\cosh^{n-1}(2x)\sinh(2x)}{2n}+\frac{n-1}{n}I_{n-2}$$

(b) Hence determine

$$\int (1+\cosh 2x)^3\,dx$$

collecting any like terms in your answer.