Question: I have been trying to find an integral that wolfapha would not compute an answer and I have finaly found out. 

**My problem** I don't know how to solve it. 

$$\int \frac{\mathrm{d}x}{x+\sqrt{-x^2}}$$

Some help would be greatly liked.

Answer: 
The most sensible interpretation of the problem I can find is to take $\sqrt {-x^2}$ as $\sqrt {(-x)^2}=|x|$ though I think the usual interpretation applies the $-$ after the square and would get $\sqrt{-(x^2)}$ and claim the square root is invalid. Accepting the first, you have $\int \frac {dx}{2x}$ which you can probably solve easily.
I would interpret the integral as either $\int \frac{1}{(1+i\, \text{sgn}\,x)} \frac{dx}{x}$, or invalid as in Ross' answer.