Indefinite integral of dx/(sin(x)+tan(x)

I replaced tan(x) with sin(x)/cos(x) so then I flipped the sinx/cosx term and wrote the integral as:

cosx/(sin(x) + sin(x) dx

becomes

cosx/(2 sinx) dx

Then I took the 1/2 out of the integral

1/2 times integral cos x / sin x dx

Let u = sin x du = cos x dx

1/2 times integral du / u

integrate

1/2 * ln u + C

replace sin x

1/2 * ln sin x + C

Is that correct?

And how do you do this with replacing sin x with 2u/ 1 + u^2 in the original problem and so on?

kind of hung up that way with the tangent function?

Thanks