How do you know if an integral is impossible? Are there tricks one can use while looking at integral expressions to determine if one is easily calculatable or not? For example:
$$
A = \int \frac{dx}{\ln(1+x^2)} \quad B = \int \frac{dx}{x\ln(x^2)}
$$
Wolfram finds an answer for $B$, but not one for $A$, even though $A$ looks easier. I would like to know why and also how/if one can look at an integral and see if it is a solvable one or not. With "solvable", I mean for a university student and not using non-real numbers or very advanced maths. :) I know most integrals can't be solved analytically but I'm thinking of those that can appear on exams. 
 A: Integration is "difficult". There isn't one systematic approach that will work by hand for most integrals. When it comes to integrating in exams, your best bet is to just practice integrating lots of different things until you start to recognise patterns.
Many integrable functions are of the form:
$$\int f'(x)g'(f(x))\mathrm{d}x = g(f(x))+c$$
Your second example can be seen to be of this form:
$$\frac{1}{\ln(x^2)} = \frac{1}{2\ln(x)}$$
$$\frac{1}{2x\ln(x)} = \frac{1}{2}\frac{1}{x}\frac{1}{\ln(x)} = \frac{1}{2}\ln'(x)\ln'(\ln(x))$$
$$\int \frac{1}{x\ln(x^2)} = \int \frac{1}{2}\ln'(x)\ln'(\ln(x)) = \frac{1}{2}\ln(\ln(x))+c$$
The key is to be able to notice these patterns, and I think the only way you can do that is through practice.
As for integrability "in theory", this is entirely different to whether or not you can integrate by hand in an exam, and the comments below your question are quite good :)
A: Risch’s algorithm will help as it turns a problem involving an integral into an algebraic problem.
It is based on the form of the function being integrated and on methods for integrating rational functions, radicals, trigonometrical, logarithmic, exponential and “special” functions, e.g. Riemann’s Zeta function.
Risch himself called it a decision procedure, because it is a method for deciding whether a function has an elementary function as an indefinite integral, and if it does, for determining that indefinite integral.
BE WARNED: The complete description of the Risch algorithm takes over 100 pages.
Integration is much more difficult than differentiation. Most fuctions, if you were to randomly combine them, cannot be integrated in terms of elementary functions e.g. f(x) = x^x is not integrable, however Risch’s algorithm gives its value at the point x=0, simplied(!!) by leaving out an infinite number of very small terms.
Most people will be surprised at it’s apparent complexity, but it’s value at x=0 is shown here:
https://i.stack.imgur.com/0DFNi.gif
Thank the good Lord for computer algorithm apps.
