[Math] Evaluate $\int \cosh^3 (x) \sinh^2 (x )dx $

Question

Evaluate $$\int \cosh^3 (x) \sinh^2 (x )dx $$

So my original thought was to apply the identity that $\sinh^2(x)=\cosh^2(x)-1$. This means that my integral becomes
$$\int \cosh^5(x)-\cosh^3(x) dx$$
which is worse to integrate I think. What would be the best approach to tackle this or where can i go from here? Any help would be appreciated.

Answer 1

Let $u=\sinh(x)$. Then $du=\cosh(x)dx$. Hence, the integral is $$\int (u^2+1)u^2du.$$