A friend and I have been swapping difficult integrals for the holidays to stump each other and he recently sent me this one that I haven't been able to figure out (mission accomplished, I guess 🙂 ).
I've tried a few substitutions of the form
$$x-1 = f(t)$$
but if they cancel out one side, they won't simplify on the other because of the presence of both the exponential and the log. At best I could simplify the problem to
$$2 + \int_0^1 e^{1-\frac{1}{x^2}} + \frac{1}{\sqrt{1-\log x}}\:dx$$
by shifting the integral over to the interval $[0,1]$ to see if I could spot any patterns. The integral on the right evaluates to $1$, which is a surprisingly clean answer.
Wolfram gives a complicated looking antiderivative but one of the rules of our little game was that we would invoke no special functions beyond the standard transcendentals and hyperbolics/trig. Even if this was the intended solution, I'm not sure how to simplify the bound at $1$ with the $\operatorname{erf}$s
I suspect he had some clean trick in mind since that was the theme of the game, but I'm stumped.
Best Answer
Hint: Observe you have \begin{align} I=\int^2_1 e^{1-\frac{1}{(x-1)^2}}+1 +1+\frac{1}{\sqrt{1-\log(x-1)}}\ dx = \int^2_1 [f(x)+f^{-1}(x)]\ dx = 3. \end{align} Draw a picture (for any $f$)!