Let's consider the integral: $\int_0^\infty \frac{1-e^{-x}}{x} \cos{x} \ \mathrm{d}x$
If you use Mathematica 7, you will get the right answer $\frac{1}{2} \log{2}$. You can check convergence by Dirichlet's test: S. C. Malik, Savita Arora "Mathematical analysys" (second edition). 5.2 Tests for Convergence, page 391. "Dirichlet's test."
But the last version returns "integral does not converge". On the other hand, the last version calculates:
$\int_0^1 \frac{1-e^{-x}}{x} \cos{x} \ \mathrm{d}x + \int_1^\infty \frac{1-e^{-x}}{x} \cos{x} \ \mathrm{d}x$
and $\int_0^{-\infty} \frac{1-e^{x}}{x} \cos{x} \ \mathrm{d}x$.
The current version also knows answer to NIntegrate
of the function.
Hence, we see the BUG (obviously, the result is not correct). But Wolfram|Alpha support team answered, "After review, our internal development group believes the output given is correct." ([W|A #487459])
Does Wolfram Mathematica use general mathematics rules or special branch only, e.g. "Wolfram branch of maths"?