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The value of $\int_{0}^{\frac{\pi}{4}} e^{\sec x} \dfrac{\sin\Big( x + \dfrac{\pi}{4}\Big)}{(1 - \sin x) \cos x} dx$

can be expressed as $\dfrac{(a + \sqrt{b})e^{\sqrt{c}} - e}{\sqrt{d}}.$

Find $\displaystyle{a+b+c+d}.$

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