Generalising integration by parts for the product of more than two functions Just as the product rule can be generalised to the product of more than two functions, i.e. $$\frac{d}{dx} \left [ \prod_{i=1}^k f_i(x) \right ]
 = \sum_{i=1}^k \left(\frac{d}{dx} f_i(x) \prod_{j\ne i} f_j(x) \right)
= \left(  \prod_{i=1}^k f_i(x) \right) \left( \sum_{i=1}^k \frac{f'_i(x)}{f_i(x)} \right),$$ is there a way to generalise integration by parts to evaluate $$\int f_1(x) f_2(x) \cdots f_n(x) dx \qquad ?$$

For contextual purposes only, I'm trying to evaluate by hand 
$$\int x \cosh(x+1)e^x\sin(x) dx \quad 
.
$$
I know that I could let $u=x \cosh(x+1)$ and $v \ '=e^x\sin(x)$ but that would require integration by parts to be performed at least three times.
Is there a more-efficient way to evaluate this integral using the proposed 'generalisation'?
Thanks!

The Wikipedia article on integration by parts gives you the generalization you're looking for:
$$\Bigl[ \prod_{i=1}^n u_i(x) \Bigr]_a^b = \sum_{j=1}^n \int_a^b \prod_{i\neq j}^n u_i(x) \, du_j(x),$$
where $u_i(x)$ are your $n$ functions of $x$ that are terms of the product that comprise your integrand.
