I want to evaluate this indefinite integral
\begin{gather*}
\int \ln^2(x+\sqrt{x^2-a^2})d x,\qquad \text{ where $a>0.$}
\end{gather*}
I know it is not so hard to evaluate this integral by hand, via using integration by parts, but I want to test if Maple can do this easily. So I asked Maple. Below is the result:
As you can see, if I enter just the command line >int(ln(x+sqrt(x^2-a^2))^2,x), Maple can not return the desired result, but just the command line itself. But if I use IntTutor, then it works well. My question is, why Maple can not give the result directly?
Maple does not have a full implementation of the Risch algorithm (especially in the mixed transcendental-algebraic case, which is the most difficult), and therefore does not produce an elementary antiderivative in all cases where such antiderivative exists. It does try the Risch algorithm, but does not succeed.