**Question**

Solve the integral,$$\int\_{-\frac{\pi}2}^{\frac{\pi}2} \frac{\sin^3x}{\tan^3x+\cot^3x} dx$$

**Attempt**

I converted the equation in terms of $\sin(x)$ and $\cos(x)$ using the definition of $\tan(x)$ and $\cot(x)$, and then applied the substitution $t=\sin(x)$, however, this has proved itself to be quite a difficult integral to resolve in and of itself.

I would be great full for any suggestions of a more compact method. Any hints would be also be greatly appreciated.