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Solution

Using property $\int_0^{\pi/2} f(\tan x),dx = \int_0^{\pi/2} f(\cot x),dx$ Add both: $I=\int_0^{\pi/2} \frac{1}{1+\tan^3 x}dx$ $I=\int_0^{\pi/2} \frac{\tan^3 x}{1+\tan^3 x}dx$ $2I=\int_0^{\pi/2}1dx=\frac{\pi}{2}$ $I=\frac{\pi}{4}$