Given to solve,

`int sec^6 (4x)tan(4x) dx`

let `u = 4x , du = 4dx`

so,

`int sec^6 (4x)tan(4x) dx`

=` int sec^6 (u)tan(u) (du)/4`

=` (1/4) int sec^6 (u)tan(u) du`

let `secu = v`

so, `dv = (sec u)(tan u) du`

so ,

`(1/4) int sec^6 (u)tan(u) du`

= `(1/4) int sec^5 (u) tan(u) *sec(u) du`

=`(1/4) int (v)^5 dv`

= `(1/4) v^6/6`

=` (v^6)/24`

= `((sec (u) )^6)/24`

=`((sec (4x) )^6)/24 +c`