That pesky 1

Calculus Level 4

\[ \large \displaystyle \int_{0}^{\pi/2} \dfrac{\sin x + 1}{\sin x + \cos x} \, dx \]

If the above integral can be expressed in the form \[ \dfrac{\pi}{a} + \cos \left( \dfrac{ \pi }{b}\right) \ln \left( \dfrac{\sqrt{c }+ d}{\sqrt{c} - d} \right) ,\]

where \( a, b, c\) and \( d \) are positive integers, with \(c\) square-free, find \( a + b + c + d \).

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