What's the best way to Integrate a composition of multiple functions The functions are totally different.
For example;
Integrate: $f(x)=\arctan(\ln(x^\frac{1}{(2x+e)})$
Uhmm.. Yeah kind of something like that.
How the hell do I do it
 A: As others already pointed out, it's not always possibile. This is because integration is, as I said, a form of art.
What you need to solve an integral, you may not need in solving another one.
Be it an integral of composite functions, more composite functions, nested composite functions et cetera. The fact is that when you write a composition of functions, the resulting function can be either simple or hard.
Trivial Example
Take $f(x) = 1/x$ and $g(x) = \ln(x)$.
$$f \circ g = \frac{1}{\ln(x)}$$
$$g \circ f = \ln\left(\frac{1}{x}\right)$$
The second one has a trivial integral.
The first one does not, and indeed is classified as a Special Function, the Logarithm Integral. Values are tabulated but you won't find an analytical primitive.
In your example, after some easy manipulation, you end up with the fancy but painful integral
$$\frac{1}{2}\int \arctan\left(z \frac{\ln(z) - \ln(2)}{z^2 + \ln(2)\ln(z)}\right)\ \text{d}x$$
Which has no close analytical form.
Unless of course you want an approximation for example in the case of very small $z$ or whatever.
General methods are rare, and they apply even more rarely.
