A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

A definite integral $\int_0^\infty\frac{2-\cos
x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

I need to find a value of this definite integral:
$$\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos
x\right)}dx.$$ Its numeric value is approximately $0.7875720991394284$,
and lookups in Inverse Symbolic Calculator Plus and WolframAlpha did not
return a plausible closed-form candidate.
Do you have any ideas how I can approach this problem?