#I=intcsc^2xsec^3xdx#
Use #csc^2x=1+cot^2x#:
#I=int(1+cot^2x)sec^3xdx=intsec^3xdx+intcsc^2xsecxdx#
Again use #csc^2x=cot^2x+1#:
#I=intsec^3xdx+int(cot^2x+1)secxdx#
#color(white)I=intsec^3xdx+intsecxdx+intcotxcscxdx#
The two rightmost integrals are standard:
#I=intsec^3xdx+lnabs(secx+tanx)-cscx#
Let #J=intsec^3xdx#. To solve this, begin with integration by parts, letting:
#u=secx" "=>" "du=secxtanxdx#
#dv=sec^2xdx" "=>" "v=tanx#
Then:
#J=secxtanx-intsecxtan^2xdx#
Using #tan^2x=sec^2x-1#:
#J=secxtanx-intsecx(sec^2x-1)dx#
#color(white)J=secxtanx-intsec^3xdx+intsecxdx#
The integral of #secx# is common. We can add #J# to both sides since its reappeared on the right-hand side:
#2J=secxtanx+lnabs(secx+tanx)#
#J=1/2secxtanx+1/2lnabs(secx+tanx)#
Then the original integral equals:
#I=(1/2secxtanx+1/2lnabs(secx+tanx))+lnabs(secx+tanx)-cscx#
#color(white)I=color(blue)(1/2secxtanx-cscx+3/2lnabs(secx+tanx)+C#
