Calculate the following integral by hands: $$\int_{0}^{\frac{\pi}{2}}\frac{\sin^{3}(t)}{\sin^{3}(t)+\cos^{3}(t)}dt$$
It seems apparently that integration by part does not work. I think integration by substitution could work but I can not figure out how to substitute.
I have calculated it by Wolfram and the result is $\frac{\pi}{4}$.