Chapter 8 - Integration Techniques, L'Hopital's Rule, and Improper Integrals - 8.3 Exercises - Page 530: 54

$$\int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x =\ln |\sec x+\tan x| -2\sin x+c$$

$$ \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x $$ Since \begin{align*} \int \frac{\sin ^{2} x-\cos ^{2} x}{\cos x} d x&= \int \frac{1-2\cos ^{2} x}{\cos x} d x\\ &=\int (\sec x -2\cos x)dx\\ &=\ln |\sec x+\tan x| -2\sin x+c \end{align*}