Calculating $\int_{0}^{\pi}\text{sinh}\left(\sin\left(x\right)\right)\text{d}x$ I was wondering if it exists a beautiful exact value to
$$ \int_{0}^{\pi}\text{sinh}\left(\sin\left(x\right)\right)\text{d}x$$
which has a nice graph over $\left[0,\pi\right]$, but i can't get to compute it.

If you know the Beta and Gamma functions well, you can see the sequence $$a_n:=\int_0^\pi\sin^{2n+1}x\mathrm{d}x=2\int_0^{\pi/2}\sin^{2n+1}x\mathrm{d}x=\operatorname{B}\left(\frac{1}{2},\,n+1\right)=\frac{n!\sqrt{\pi}}{\Gamma\left(n+\frac{3}{2}\right)}$$satisfies$$a_0=2,\,\frac{a_{n+1}}{a_n}=\frac{2(n+1)}{2n+3}$$so by induction$$a_n=2^{n+1}\prod_{i=1}^{n-1}\frac{i}{2i-1}=\frac{2^{2n+1}n!^2}{(2n+1)!}.$$So your integral is $$2\sum_{n\ge 0}\frac{n!^2}{(2n+1)!^2}4^n=\pi L_0(1),$$with $L_n(x)$ the modified Struve function. (I'm not an expert on this function myself, but Wolfram Alpha gives the result, and I suspect it can be derived from the previous link's Eq. (1) using Legendre's duplication formula.)
