Difficult integration problem

(From Maple's help pages, example by Mark van Hoeij.)

Integrating algebraic functions.

Consider the algebraic function

Integrating this function is difficult because of the roots appearing in the defining expression.

Here is the function:

> f:=(x^2-1+3*(1+x^2)^(1/3))/(1+x^2)^(2/3)*x/(x^2+2)^2;

Let us try to integrate it:

> int(f,x);

Looks like Maple could not do it.

There is, however, a method to deal with non--trivial algebraic functions like the one above. That is to convert the explicit roots into RootOf constructions.

> f:=convert(f,RootOf);

Now this function can be integrated:

> Int(f,x) = int(f,x);

>

The answer produced by Maple is an elementary function.

Compare this result to the integration performed by the server