Really advanced techniques of integration (definite or indefinite) Okay, so everyone knows the usual methods of solving integrals, namely u-substitution, integration by parts, partial fractions, trig substitutions, and reduction formulas. But what else is there? Every time I search for "Advanced Techniques of Symbolic Integration" or "Super Advanced Integration Techniques", I get the same results which end up only talking about the methods mentioned above. Are there any super obscure and interesting techniques for solving integrals?
As an example of something that might be obscure, the formula for "general integration by parts " for $n$ functions $f_j, \ j = 1,\cdots,n$ is given by
$$
\int{f_1'(x)\prod_{j=2}^n{f_j(x)}dx} = \prod_{i=1}^n{f_i(x)} - \sum_{i=2}^n{\int{f_i'(x)\prod_{\substack{j=1 \\ j \neq i}}^n{f_j(x)}dx}}
$$
which is not necessarily useful nor difficult to derive, but is interesting nonetheless.
So out of curiosity, are there any crazy unknown symbolic integration techniques?

According to Wikipedia, the "tangent half-angle substitution" (a.k.a. the Weierstrauss substitution) is the "world's sneakiest substitution".  It consists of subbing $x=\arctan(2t)$, which allows you to write $\sin(x) = 2t/(1+t^2)$, $\cos(x) = (1-t^2)/(1+t^2)$, and $dt = 2/(1+t^2) dx$.  An example (poached from Wikipedia) is the following:
$$
\begin{eqnarray}
\int \csc(x) dx & = & \int \frac{dx}{\sin(x)} \\
& = & \int \left(\frac{2t}{1+t^2}\right)^{-1} \frac{2}{1+t^2} dt \\
& = & \int \frac{dt}{t} \\
& = & \ln(t) +C\\
& = & \ln(\tan(x/2)) + C
\end{eqnarray}
$$

Also, it's worth mentioning that for integrands which possess a symmetry, one can often evaluate the integral using ideas from representation theory.  A simple example of this is the fact that the integral of an odd function over a domain symmetric about the origin vanishes.  The symmetry here is parity.  
A much more sophisticated example is when the integrand consists of functions with symmetry under a representation of the rotation group--e.g. spherical harmonics.  The analysis of such integrals goes under the name "Wigner-Eckart Theorem", and is extremely important in physics.  
The most fruitful application of this technique is usually to show that an integral is in fact $0$.  In cases where an integral is non-zero, symmetry may tell you some information but not a complete answer.  E.g. for an integral like $\int_{-1}^1 f(x) dx$ where $f(x)$ is even, you can't say what the exact value of the integral is, but you can say that it equals $2\int_0^1 f(x) dx$.
