How to evaluate the integral $\int_0^{2\pi} \theta\exp(x\cos(\theta) + y\sin(\theta))) d\theta$

Not hard to see that \begin{align*} \int_0^{2 \pi } \theta \cdot e^{x \cos \theta +y \sin \theta } \, \mathrm{d}\theta &=\int_{0}^{2\pi }\theta \sum_{k=0}^{\infty }\frac{\left ( x \cos \theta +y \sin \theta \right )^{k}}{k!}\, \mathrm{d}\theta \\ &=\int_{0}^{2\pi }\theta \sum_{k=0}^{\infty }\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}y^{j}\sin^{j}\theta x^{k-j}\cos^{k-j}\theta \, \mathrm{d}\theta \\ &=\sum_{k=0}^{\infty }\frac{1}{k!}\sum_{j=0}^{k}\binom{k}{j}y^{j}x^{k-j}\int_{0}^{2\pi }\theta \sin^{j}\theta \cos^{k-j}\theta \, \mathrm{d}\theta \end{align*} For the last integral, with the help of Mathematica we get the following complex result \begin{align*} \int_{0}^{2\pi }\theta \sin^{m}\theta \cos^{n}\theta \, \mathrm{d}\theta =&\frac{(-1)^{m+n} \pi ^2 \csc \left(\frac{n \pi }{2}\right) \csc \left(\frac{\pi m}{2}+\frac{n \pi }{2}\right) \Gamma \left(\frac{n}{2}+\frac{1}{2}\right)}{4 \Gamma \left(\frac{1}{2}-\frac{m}{2}\right) \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}+\frac{\pi ^2 \csc \left(\frac{n \pi }{2}\right) \csc \left(\frac{\pi m}{2}+\frac{n \pi }{2}\right) \Gamma \left(\frac{n}{2}+\frac{1}{2}\right)}{4 \Gamma \left(\frac{1}{2}-\frac{m}{2}\right) \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}\\ &+(-1)^{m+n} 2^{\frac{m}{2}-\frac{1}{2}} \cos \left(\frac{m \pi }{2}\right) \Gamma \left(\frac{m}{2}+\frac{1}{2}\right) \Gamma \left(-\frac{m}{2}-n-\frac{1}{2}\right) \Gamma (n+1) \, _2F_1\left(\frac{1-m}{2},\frac{m+1}{2};\frac{m}{2}+n+\frac{3}{2};\frac{1}{2}\right)\\ &+\frac{(-1)^{m+2 n} 2^{\frac{m}{2}-\frac{1}{2}} \pi \Gamma \left(\frac{m}{2}+\frac{1}{2}\right) \Gamma (n+1) \, _2F_1\left(\frac{1-m}{2},\frac{m+1}{2};\frac{m}{2}+n+\frac{3}{2};\frac{1}{2}\right)}{\Gamma \left(\frac{m}{2}+n+\frac{3}{2}\right)}\\ &+\frac{(-1)^{m+n} \Gamma \left(\frac{n}{2}\right) \Gamma \left(\frac{m}{2}+1\right) \, _3F_2\left(\frac{1}{2},1,\frac{m}{2}+1;\frac{3}{2},1-\frac{n}{2};1\right)}{2 \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}\\ &+\frac{\Gamma \left(\frac{n}{2}\right) \Gamma \left(\frac{m}{2}+1\right) \, _3F_2\left(\frac{1}{2},1,\frac{m}{2}+1;\frac{3}{2},1-\frac{n}{2};1\right)}{2 \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}\\ &+\frac{(-1)^n \Gamma \left(\frac{m}{2}\right) \Gamma \left(\frac{n}{2}+1\right) \, _3F_2\left(\frac{1}{2},1,\frac{n}{2}+1;\frac{3}{2},1-\frac{m}{2};1\right)}{2 \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}\\ &+\frac{(-1)^{m+2 n} \Gamma \left(\frac{m}{2}\right) \Gamma \left(\frac{n}{2}+1\right) \, _3F_2\left(\frac{1}{2},1,\frac{n}{2}+1;\frac{3}{2},1-\frac{m}{2};1\right)}{2 \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}\\ &+\frac{(-1)^n \pi \Gamma \left(\frac{m}{2}+\frac{1}{2}\right) \left(\pi \csc \left(\frac{m \pi }{2}\right) \csc \left(\frac{\pi m}{2}+\frac{n \pi }{2}\right)+\pi \sec \left(\frac{n \pi }{2}\right)\right)}{4 \Gamma \left(\frac{1}{2}-\frac{n}{2}\right) \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}\\ &+\frac{(-1)^{m+2 n} \pi \Gamma \left(\frac{m}{2}+\frac{1}{2}\right) \left(\pi \csc \left(\frac{m \pi }{2}\right) \csc \left(\frac{\pi m}{2}+\frac{n \pi }{2}\right)+\pi \sec \left(\frac{n \pi }{2}\right)\right)}{4 \Gamma \left(\frac{1}{2}-\frac{n}{2}\right) \Gamma \left(\frac{m}{2}+\frac{n}{2}+1\right)}\\ &+\frac{(-2)^{m+n} \pi ^{5/2} \Gamma \left(\frac{m}{2}+\frac{1}{2}\right) \sec \left(\frac{\pi m}{2}+n \pi \right)}{\Gamma \left(\frac{1}{2}-\frac{n}{2}\right) \Gamma \left(-\frac{m}{2}-\frac{n}{2}+\frac{1}{2}\right) \Gamma (m+n+1)} \end{align*} Looking at this horrible result, If I'm not doing the wrong way, I don't there will be a closed form for the integral, as least it won't be too short.