Find $\int \sin x \cos (\cos x) \cos(\sin x)+\cos x\sin(\cos x)\sin(\sin x)dx$

Question

How do I find: $$\int \sin x \cos (\cos x) \cos(\sin x)+\cos x\sin(\cos x)\sin(\sin x)dx$$? I made this one up by taking the derivative of $-\sin (\cos x)\cos (\sin x)$, and I wonder how someone would solve this monster.

Answer 1

First, split the integral:$$\int\sin x\cos (\sin x)\cos(\cos x)dx+\int\cos x\sin(\cos x)\sin(\sin x)dx$$Apply integration by parts to the first integral (where $\cos(\sin x)$ is $f$ and $\sin x\cos(\cos x)$ is $g'$): $$-\sin(\cos x)\cos (\sin x)-\int\cos x\sin(\cos x)\sin(\sin x)dx+\int\cos x\sin(\cos x)\sin(\sin x)dx$$Cancelling we get the answer. This was a very pretty solution.

Answer 2

$$ \begin{aligned} \int \sin x \cos (\cos x) \cos (\sin x)= & -\int \cos (\sin x) d(\sin (\cos x))\\=& -\cos (\sin x) \sin (\cos x)-\int \sin (\cos x) \sin (\sin x) \cos x d x \quad \textrm{ (By IBP)} \end{aligned} $$ Removing the last integral to the left yields $$\int \left(\sin x \cos (\cos x) \cos (\sin x)+\cos x \sin (\cos x) \sin (\sin x) \right)dx= -\cos(\sin x) \sin (\cos x)+C $$