I'm working with several definite integrals and I arises the following problem. I'm going to try explain in the most generic way possible:
Consider this definite integral
\begin{equation}\label{one}\int_a^b f(x)\,g(x)\,dx\hspace{6cm}(1)\end{equation}
where $(a,b)\in\mathbb{R^+}$ is bounded or not and the functions $f$ and $g$ are two consecutive functions of the list $$\ln(\ln\,x),\ln(x),x^\alpha,e^x,e^{e^x}$$
Then, performing a suitable change of variable, one can express integral $(1)$ in terms of Gamma or Incomplete Gamma functions.
Well, my question is what can I do when $f$ and $g$ are not consecutive functions in the above list, in fact, are alternate functions, that is $\ln(\ln\,x)$ and $x^\alpha$ or $\ln(x)$ and $e^x$ ...
On the one hand, I wonder if there is any special function which has an integral representation with this kind of functions in the integrand.
On the other hand, I also wonder if there is any other way to evaluate this integral.
Any help or suggestion is wellcomend.
For the case of $$I=\int \log(x)\,e^x\,dx$$ one integration by parts will give $$I=e^x \log (x)-\text{Ei}(x)$$ where appears the exponential integral function.
$$J=\int x^a \log (\log (x))\,dx=\frac{x^{a+1} \log (\log (x))-\text{Ei}((a+1) \log (x))}{a+1}$$
These special functions are precisely introduced for such cases.