integrating a function within another function Is there a general solution for the following integration?

$$\int{f(g(x))}{dx}$$

Hypothesis (Probably wrong and you should ignore) from observations:

$$\int{f(g(x))}{dx}=\frac{F(g(x))}{g^{\prime}(x)}+c$$
Where $F(x)=\int{f(x)}{dx}$ and $c$ is a constant
No, there is no general solution for the primitive of $f\circ g$.

The derivative of your proposed solution is 

$$\frac{g'(x) ( g'(x) f(g(x)) ) - F(g(x))g''(x) }{(g'(x))^2} = f(g(x)) - \frac{F(g(x))g''(x) }{(g'(x))^2}$$

If there was a formula with just the primitive of $f$, it would be easy to integrate any function $v$ : choose $f(x) = \ln(x)$ and define $g(x) = e^{v(x)}$. then $f(g(x)) = v(x)$

Another remark : $\exp$ and $-x^2$ are very simple functions to integrate, but the primitive of $e^{-x^2}$ cannot be expressed with a finite number of simple functions
This is incorrect as if you take derivative from both sides of your hypothesis, you get
$$f(g(x))=\frac{f(g(x))\cdot (g'(x))^2-F(g(x))\cdot g''(x)}{(g'(x))^2},$$
Which obviously does not hold.