The Tough Integral

In an earlier blog entry, I admired the following immediate solution of Mathematica.

\(\int\limits_0^1 \dfrac{1}{x} \cos \left( \dfrac{\log x}{x} \right) \, dx = 0.323367\)

I now succeeded to get this integral with numerical methods of Euler. Horst Vogel inspired me in a private communication to try this with some substitutions. He reduced the problem to the integral

\(I(T) = \int\limits_0^T y'(t) \cos(t) \,dt, \qquad t=y e^y\)

which he could solve with a solver for differential equations. I did not use this trick, but used a more elementary way to compute

\(\int\limits_0^\infty \cos(y e^y) \,dy\)

However, it is still not clear, how Mathematica does this.

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