I want to solve the following integration $$I = \int_0^\pi\int_0^{2\pi}\exp{\bigg[x\cos(\phi)\sin(\theta)+y\sin(\phi)\sin(\theta)+z\cos(\theta))\bigg]}\sin(\theta)\,d\phi \,d\theta $$ My Attempt:
First solve the $\phi$ part $$I = \int_0^\pi\exp{[z\cos(\theta)}]\sin(\theta) \Bigg[\int_0^{2\pi} \exp\bigg[x\cos(\phi)\sin(\theta)+y\sin(\phi)\sin(\theta))\bigg] \, d\phi\Bigg] \, d\theta\\ I = \int_0^\pi\exp[z\cos(\theta)]\sin(\theta) \, d\theta I_2 $$ where $$I_2 = \int_0^{2\pi} \exp\bigg[x\cos(\phi)\sin(\theta)+y\sin(\phi)\sin(\theta))\bigg] \, d\phi$$ Nothing seems to work here. I have tried integration by parts and substitution method but both just keeps expanding the terms. How can I solve this. Please help.