Integral with simple fractions: $\int_\ \frac{\cos x }{\sin x \sqrt{1+\cos^2x}} \, dx$

Question

I have a problem with this integral

$$\int_\ \frac{\cos x }{\sin x \sqrt{1+\cos^2x}} \, dx$$

Using substitution $u = \sin x $ we get

$$\int_\ \frac{1 }{\ u \sqrt{2-u^2}} \, du$$

I think the next step is to use simple fractions, but I don't know how to do it.

Answer 1

Hint

Multiply both numerator and denominator by $u$ and then use the substitution $2-u^2=t^2.$ To get $$\int \frac{u}{u^2\sqrt{2-u^2}} \, du=\int \frac{1}{t^2-2} \, dt.$$ Now you can use partial fractions.