We want to evaluate the definite integral
[math]I = \displaystyle \int_0^{\pi/2} \frac{1}{1 + \cos^4{x}} \, dx. \tag*{}[/math]
Rewriting the integrand in terms of secant and tangent after dividing the numerator and denominator by [math]\cos^4{x}[/math] yields
[math]\begin{align*} I &= \displaystyle \int_0^{\pi/2} \frac{\sec^4{x}}{\sec^4{x} + 1} \, dx\\ &= \int_0^{\pi/2} \frac{\sec^2{x}}{(1 + \tan^2{x})^2 + 1} \cdot \sec^2{x} \, dx\\ &= \int_0^{\pi/2} \frac{\tan^2{x} + 1}{\tan^4{x} + 2\tan^2{x} + 2} \cdot \sec^2{x} \, dx. \end{align*} \tag*{}[/math]
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