using the reduction formula of secx^n
intsecx^ndx=intsecx^(n-2)*secx^2dx
integrate intsecx^(n-2)*secx^2dx by parts (uv- intvdu)
where u=secx^(n-2) and dv=secx^2
therefore intsecx^(n-2)*secx^2dx
=tanx*secx^(n-2)-(n-2)inttan^2*secx^(n-2) dx
=tanx*secx^(n-2)-(n-2)int(secx^2-1)*secx^(n-2)dx
=tanx*secx^(n-2)-(n-2)intsecx^n*secx^(n-2)dx
if I_"n"=intsecx^ndx
therefore
I_"n"=tanx*secx^(n-2)-(n-2)intsecx^n-secx^(n-2)dx
I_"n"=tanx*secx^(n-2)-(n-2)I_n+(n-2)I_"n-2"
(n-1)I_"n"=tanx*secx^(n-2)+(n-2)I_"n-2"
I_"n"=1/(n-1)*(tanx*secx^(n-2))+(n-2)/(n-1)I_"n-2"
Where I_"n"=intsecx^ndx and I_"n-2"=intsecx^(n-2)dx
Substitute every n with 4 in I_"n"=1/(n-1)*(tanx*secx^(n-2))+(n-2)/(n-1)I_"n-2"
Therefore
intsecx^4dx=1/3 secx^2*tanx-1/2intsecx^2dx
=1/3 secx^2*tanx-1/2tanx
This method can be used to integrate sec x to the power of anything as long as the power is bigger than 1