I don't believe that there is any trigonometric substitution involved here actually.
The integrand #secx(secx+tanx)# can be expanded out to get #sec^2x +secxtanx# which are actually two simple functions to be antidifferentiated.
#int sec^2x dx = tanx + c# because #d/dx(tanx) = sec^2x#.
#int secxtanx dx = secx# which we will prove using substitution.
Expressing that in terms of sine and cosine, we get #secxtanx = sinx/cos^2x# so the integral becomes:
#int sinx/cos^2x dx#, to which we apply the substitution #u=cosx#.
#u=cosx#
#:. (du)/dx=-sinx#
#:. -du = sinxdx#
Now, using the change of variable rule, we get:
# - int 1/u^2 du#
#=1/u + c#
#= secx + c#
#:. int secx(secx+tanx) dx = secx + tanx + c#
And there you have it!
