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I have to compute the following integral, $$J(x;A,B)=\int_{0}^{x}dt\exp\left(-At-B\cos t\right),$$ where $B>0$ and $A>0$. Especially I am interested in $x=2\pi$ case. It seems that at least in this case this integral is something related to the Bessel function. However my attempts to find an appropriate integral representation were not successful. Can anyone help me with this integral?

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