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I need to solve a definite integral of exponential function combined with a rational function, which is depicted as follow:

$$I = \int\limits_{0}^\infty {\frac{{\exp \left( -t-\frac{1}{t} \right)}}{(1+t)^k}dt}$$ where k is a positive integer ($k \ge 1$). Can the integral $I$ be expressed in terms of special functions? (e.g. Bessel function, incomplete Gamma function, etc.).

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