I wonder, is there a closed-form solution to integral of type $$\int \left(\frac{a^x+b^x}{2}\right)^{1/x}dx,$$ for $a\ne b$ (both positive real numbers). If not, what about a special case when $a=1$, i.e., $$\int \left(\frac{1+b^x}{2}\right)^{1/x}dx.$$
In general, having a definite integral of this type, are there some methods to compute such integrals?
If such a closed form were to exist, then, for $a=\dfrac1b=e$, we'd have $\displaystyle\int\sqrt[\Large x]{\cosh x}~dx=F(x)$. Now, are you aware of the former expression possessing any closed form anti-derivatives ? In fact, even simpler integrands, like $x^x$ or $\sqrt[\Large x]x$, do not have one. Even the simple case as $b=0$, for instance, requires the exponential integral. Of course, all of this is merely an informal explanation. An actual proof would require applying either Liouville's theorem, or the Risch algorithm. The latter consists basically in an entire book, and the former requires deep knowledge of notions of abstract algebra.