Question Number 86269 by M±th+et£s last updated on 27/Mar/20
$$\int\frac{{sec}^{\mathrm{2}} \left({x}\right)}{\left({tan}\left({x}\right)−\mathrm{1}\right)^{\mathrm{4}} \left({tan}\left({x}\right)−\mathrm{2}\right)}\:{dx} \\ $$
Commented by john santu last updated on 28/Mar/20
$${u}\:=\:\mathrm{tan}\:{x}−\mathrm{2}\: \\ $$$$\Rightarrow\:\int\:\frac{{du}}{\left({u}+\mathrm{1}\right)^{\mathrm{4}} \left({u}\right)}\: \\ $$$${decomposition} \\ $$$$\frac{\mathrm{1}}{\left({u}+\mathrm{1}\right)^{\mathrm{4}} \left({u}\right)}\:=\:\frac{{p}}{{u}+\mathrm{1}}+\frac{{q}}{\left({u}+\mathrm{1}\right)^{\mathrm{2}} }+\frac{{r}}{\left({u}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$+\:\frac{{s}}{\left({u}+\mathrm{1}\right)^{\mathrm{4}} }\:+\:\frac{{t}}{{u}} \\ $$$$ \\ $$