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?

You can do integration by inverting the matrix representation of the differentiation operator with respect to a clever choice of a basis and then apply the inverse of the operator to function you wish to integrate.
For example, consider the basis $\mathcal{B} = \{e^{ax}\cos bx, e^{ax}\sin bx \}$. Differentiating with respect to $x$ gives 
\begin{align*}
\frac{d}{dx}e^{ax} \cos bx &= ae^{ax} \cos bx - be^{ax} \sin bx\\
\frac{d}{dx} e^{ax} \sin bx &= ae^{ax} \sin bx + be^{ax} \cos bx
\end{align*}
and the matrix representation of the linear operator is
$$T = \begin{bmatrix}
a & b\\
-b & a
\end{bmatrix}$$
To then solve something like $\int e^{ax}\cos bx\operatorname{d}\!x$, this is equivalent to calculating 
$$T^{-1}\begin{bmatrix}
1\\
0
\end{bmatrix}_{\mathcal{B}} = \frac{1}{a^{2} + b^{2}}\begin{bmatrix}
a\\
b
\end{bmatrix}_{\mathcal{B}}.$$
That is, 
$$\int e^{ax}\cos bx\operatorname{d}\!x = \frac{a}{a^{2}+b^{2}}e^{ax}\cos bx + \frac{b}{a^{2} + b^{2}}e^{ax}\sin bx$$
