`sec^(3)x. `

Explanation

Let `I=int sec^(3)xdx=intsecx.sec^(2)dx`
`=secx int sec^(2)xdx-int [(d)/(dx)(sec x)int sec^(2)x dx]dx`
`=sec xtan x- intsec x tan x.tan dx`
`=sec xtan x- intsec x tan^(2)xdx`
`=sec xtan x- int sec x(sec^(2)x-1)dx`
`=sec xtan x- int (sec^(3)x-secx)dx`
`=sec xtan x- int sec^(3)xdx+f secx dx`
`therefore I=sec x tan x-l+log |sec x+tan x|+c_(1)`
`therefore 2I=sefc x tan x+log|sec x +tan x|+c_(1)`
`therefore I=(1)/(2)sec x tan x+(1)/(2)log |sec x+tan x|+c," where "c=(c_(1))/(2)`