The value of the definite integral \( \int_{1}^{\infty}\left(e^{x+1}+e^{3-x}\right)^{-1} d x \),....

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The value of the definite integral \( \int_{1}^{\infty}\left(e^{x+1}+e^{3-x}\right)^{-1} d x \), is
\( \mathrm{P} \)
(1) \( \frac{\pi}{4 e^{2}} \)
(2) \( \frac{\pi}{4 e} \)
(3) \( \frac{1}{e^{2}}\left(\frac{\pi}{2}-\tan ^{-1} \frac{1}{e}\right) \)
(4) \( \frac{\pi}{2 e^{2}} \)


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