I am being stuck in caculating this integral: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$$ I tried to change to another variable: $x = - t$ then $dx = - dt$, also I got: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos(-t)}{\sqrt{1-t^2}(1+e^t)}dt=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos(-x)}{\sqrt{1-x^2}(1+e^x)}dx.$$ Therefore $$2J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\left[\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}+\dfrac{\arccos(-x)}{\sqrt{1-x^2}(1+e^x)}\right]dx=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{e^x\arccos x+\arccos(-x)}{\sqrt{1-x^2}(1+e^x)}dx$$The numerator looks very complex and I really do not know how to do next. Can you guys give me some ideas ?
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