The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus: ∫ sec 3 ⁡ x d x = 1 2 sec ⁡ x tan ⁡ x + 1 2 ∫ sec ⁡ x d x + C = 1 2 + C = 1 2 + C, | x | < 1 2 π {\textstyle {\begin{aligned}\int \sec ^{3}x\,dx&={\tfrac {1}{2}}\sec x\tan x+{\tfrac {1}{2}}\int \sec x\,dx+C\\[6mu]&={\tfrac {1}{2}}+C\\[6mu]&={\tfrac {1}{2}}+C,\qquad |x|<{\tfrac {1}{2}}\pi \end{aligned}}} where gd − 1 {\textstyle \operatorname {gd} ^{-1}} is the inverse Gudermannian function, the integral of the secant function. Wikipedia