Integral of $$$\cos{\left(\ln\left(e^{\operatorname{atan}{\left(\ln\left(\cos{\left(t \right)}\right) \right)}} + 1\right) \right)}$$$
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Find $$$\int \cos{\left(\ln\left(e^{\operatorname{atan}{\left(\ln\left(\cos{\left(t \right)}\right) \right)}} + 1\right) \right)}\, dt$$$.