Evaluation of $\int \frac{x\sin(\sin x)}{x+5} \ dx$

How do we find $\displaystyle\int\frac{x\sin(\sin x)}{x+5}dx$ ?

We don't. See Liouville's theorem and the Risch algorithm for more details. More to the point,

even the more humble looking $\displaystyle\int_0^\tfrac\pi2\sin(\sin x)~dx=\frac\pi2H_0(1)$ requires the presence of Struve

functions in its expression.