∫(tanx)^4 dx
=∫[(secx)^2 -1]^2 dx
=∫[(secx)^4-2(secx)^2+1] dx
=∫(secx)^2 [(secx)^2-2] dx + x
=∫[(secx)^2-2] d(tanx) + x
=∫[(tanx)^2-1] d(tanx) + x
=1/3 (tanx)^3 -tanx + x + c