2 integrals to calculate:$\int_0 ^{2\pi} e^{\sin\phi}\sin(n\phi -\sin (\phi))d\phi$
$\int_0 ^{2\pi} e^{\cos\phi}\cos(n\phi -\cos (\phi))d\phi$
I tried to make a substitution:$\cos \phi=\frac{z+z^{-1}}{2}$ $\sin \phi=\frac{z-z^{-1}}{i2}$. And by the hint I will get to $\int_D e^zz^{-(n+1)}dz$, $D:=\{|z|<1\}$. But I am confused with this part: $\cos(n\phi -\cos (\phi))$. Thanks in advance.
Too long for a comment: Even if by absurd any of these integrals were to actually possess a
closed form, it will most likely be in terms of Bessel and Struve functions. I write this because
$\displaystyle\int_0^{2\pi}e^{a\sin x}~dx~=~\int_0^{2\pi}e^{a\cos x}~dx~=~2\pi~I_0(a),~$ while $\displaystyle\int_0^\pi\sin(nx-\sin x)~dx$ can be written
in terms of $\pi~H_{-|n|}(1),~$ and $\displaystyle\int_0^\pi\cos(nx-\cos x)~dx$ can be expressed in terms of $\pi~J_n(1)$ for
even values of $n=2m,~$ and $\displaystyle\int_0^\tfrac\pi2\cos(nx-\cos x)~dx$ can be parsed in terms of $\dfrac\pi2~H_{-|n|}(1)$
for odd values of $n=2m+1.~$ More information on this topic can be found here.
