`int cot\ 4x\ csc^4 4x\ dx`

We write the expression under the integral sign as follows:

`cot\ 4x\ csc^4 4x\ dx` `=(csc^3 4x)\ cot\ 4x\ csc\ 4x`

Then, let `u = csc\ 4x` and so we have `du = -4\ csc\ 4x\ cot\ 4x\ dx`

That is, `-(du)/4 = csc\ 4x\ cot\ 4x\ dx`

Now we can perform the integral:

`int cot\ 4x\ csc^4 4x\ dx =int(csc^3 4x)\ cot\ 4x\ csc\ 4x\ dx`

`=-1/4intu^3du`

`=-u^4/16+K`

`=-(csc^4 4x)/16+K`