integration - 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.

Answer

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 :)