Prove that $\int_0^{\pi/2}\frac{2-(4+a)\sin^2(x) - 2 a \sin^4(x)-a^2\sin^6(x)}{(1+a\sin^2(x))^{5/2}}dx=0$

Question

Numerically, I found that the following holds (with errors in the order of $10^{-16}$): $$\int_0^{\pi/2}\frac{2-(4+a)\sin^2(x) - 2 a \sin^4(x)-a^2\sin^6(x)}{(1+a\sin^2(x))^{5/2}}dx=0 \quad \forall a>0$$

I tried to prove it but failed. The integrand doesn't seem to have any symmetric properties that make the integral trivial. Can anyone prove or disprove it? Thanks.

Answer 1

HINT: use that the indefinite integral is given by $$-\frac{\sin (2 x) (a \cos (2 x)-a-4)}{\sqrt{2} (a (-\cos (2 x))+a+2)^{3/2}}$$