Evaluate $\int_{-\pi}^{\pi} \frac{x^2}{1+\sin{x}+\sqrt{1+\sin^2{x}}} \mathop{dx}$ https://ift.tt/eA8V8J

I came across this integral:$$\int_{-\pi}^{\pi} \frac{x^2}{1+\sin{x}+\sqrt{1+\sin^2{x}}} \mathop{dx}$$ I tried $u=x+\pi$ $$\int_{-\pi}^{\pi} \frac{(x+\pi)^2}{1-\sin{x}+\sqrt{1+\sin^2{x}}} \mathop{dx}$$ but had no success.

I also tried $u=-x$: $$\int_{-\pi}^{\pi} \frac{x^2}{1-\sin{x}+\sqrt{1+\sin^2{x}}} \mathop{dx}$$ Does this help? Any suggestions.

Answer is $\dfrac{\pi^3}{3}$ by the way.

from Hot Weekly Questions - Mathematics Stack Exchange
Saelee