It is not difficult to verify that $ \frac{\mathrm d}{\mathrm dx} \left[ \log\Big(x+\sqrt{x^2+1}\Big) \right] = \frac{1}{\sqrt{1+x^2}} $ for $x\geq 0$, say.
How would one calculate the indefinite integral $ \int \frac{1}{\sqrt{1+x^2}} \ \!\mathrm dx$ without knowing this? I have tried many of the usual tricks, without success.
The title of the question is chosen because Mathematica outputs $\text{Arcsinh}(x)+C$ as the answer.