∫（tan³t+1)/(sec²t) dt
＝ ∫sin³t / cost dt +∫cos²tdt

＝∫(1 - cos²t) sint / cost dt+1/2 ∫2cos²t-1+1dt

＝ - ∫(1 - cos²t) / cost dcost +sin2t/4+t/2
＝ - ∫1/cost dcost  + ∫cost dcost +sin2t/4+t/2

= - ln|cost| + cos²t/2 +sin2t/4+t/2+C