[tex]\displaystyle\int\tan^4x\sec x\,\mathrm dx[/tex]
[tex]\displaystyle\int(\sec^2x-1)^2\sec x\,\mathrm dx[/tex]
[tex]\displaystyle\int(\sec^4x-2\sec^2x+1)\sec x\,\mathrm dx[/tex]
[tex]\displaystyle\int\sec^5x\,\mathrm dx-2\int\sec^3x\,\mathrm dx+\int\sec x\,\mathrm dx[/tex]
The power reduction formula for secant will be a great help here (unless you want to integrate by parts several times only to arrive at the same result); for [tex]n\neq1[/tex], we have
[tex]\displaystyle\int\sec^nx\,\mathrm dx=\dfrac1{n-1}\sec^{n-1}x\sin x+\frac{n-2}{n-1}\int\sec^{n-2}x\,\mathrm dx[/tex]
which gives
[tex]\displaystyle\frac14\sec^4x\sin x-\frac78\sec^2x\sin x+\frac18\int\sec x\,\mathrm dx[/tex]
[tex]=\dfrac14\sec^4x\sin x-\dfrac78\sec^2x\sin x+\dfrac18\ln|\sec x+\tan x|+C[/tex]
