my abilites in integration (applied) are very limited, so my question is: if i would like to find out how to approach a problem like finding the antiderivative of the function $f(x) = (1+\frac{1}{x})^a$ (edit: i had here a wrong integrand before), where $a$ is a real constant, where would i go? (which text book for example or which source in the www?). How would i find out if a function has no simple closed expression? And if this is the case, considering the analogous "definite integration" problem by asking for a value of $\int^d_c f(x) dx$ for an interval $(c,d)$, would it yield an easier answer?
My trivial (non-helpful) observations are: If $a$ were a natural number: integration of rational functions is quite possible (Partial fraction decomposition, substition with trigonometric functions). Because the power is an arbitrary real number, i guess things like the Gamma function come into play. But i really have no clue.
Thanks.
Take into account that:
$$f(x)=\left(\frac{1}{1+x}\right)^a=(1+x)^{-a}.$$
If you want, to see it clearer, make the substitution $s=1+x\Rightarrow ds = dx.$ Therefore: $$\int f(x) \,dx = \int s^{-a} \,ds.$$