Q.$\displaystyle\int \dfrac{\sec^2x}{\tan^2x-1}\,dx$

Several rational and irrational integrands reduce to a small table of standard forms.

Type I — pure quadratic denominators. With constant $a>0$:

Type II — general quadratic denominators $\int\frac{dx}{ax^2+bx+c}$ and $\int\frac{dx}{\sqrt{ax^2+bx+c}}$: make the coefficient of $x^2$ unity, complete the square to write $ax^2+bx+c$ as a sum/difference of two squares, and reduce to Type I.

Type III — linear over quadratic $\int\frac{px+q}{ax^2+bx+c}\,dx$ (and the $\sqrt{\ }$ version): write $px+q=A\frac{d}{dx}(ax^2+bx+c)+B$, find $A,B$ by matching coefficients, then split into a log/derivative-over-function piece plus a Type-I/II piece. …