An strange integral: $\int\frac{\mathrm dx}{\sqrt{1+\sin^2x}}$

Question

I find this integral on my mathematical analysis exercise book: $$\int\frac{\mathrm dx}{\sqrt{1+\sin^2x}}$$ I tried to rewrite $1+\sin^2x$ as $2-\cos^2x$, expecting trigonometric substitution will be useful and obviously I failed. One of my classmates said it may be ellipse integral, but as far as I know, elliptic integrals are of the following form: $$\int\frac{\mathrm dx}{\sqrt{1-k^2\sin^2x}}$$ By the way, the integral calculator on MathDF failed to solve it. So maybe it is true that this integral cannot be expressed in elementary form, then I will tell the author that they mistyped the formula AGAIN.