Solve the trigonometric integral $\int\frac{1}{1+\cos\left(x\right)^2}dx$

 Function Plot

Plotting: $\frac{\sqrt{2}}{2}\arctan\left(1.4142012\tan\left(\frac{x}{2}\right)-0.9999912\right)-5.6\times 10^{-17}\ln\left(\left(\tan\left(\frac{x}{2}\right)-\frac{\sqrt{2}}{2}\right)^2+\frac{1}{2}\right)+\frac{\sqrt{2}}{2}\arctan\left(1.4142012\tan\left(\frac{x}{2}\right)+0.9999912\right)+5.6\times 10^{-17}\ln\left(\left(\tan\left(\frac{x}{2}\right)+\frac{\sqrt{2}}{2}\right)^2+\frac{1}{2}\right)+C_0$