TITLE: How find this integral $\int\frac{\sin{x}}{\sqrt{2}+\sin{x}+\cos{x}}dx$
QUESTION [9 upvotes]: Find the  integral 
$$\int\dfrac{\sin x}{\sqrt{2}+\sin x+\cos x} \, dx$$
My idea: since
$$\sin x+\cos x=\sqrt{2}\sin(x+\dfrac{\pi}{4})$$
so
$$\int\dfrac{\sin x}{\sqrt{2}+\sin x +\cos x} \, dx=\int\dfrac{\sin x}{\sqrt{2}(1+\sin (x+\dfrac{\pi}{4})} \, dx$$
But then I don't know how to continue. Thank you

REPLY [4 votes]: In general, Weierstrass is probably a good idea for such trigonometric integrals.  However, your progress left the denominator much more manageable.  I would start as in David H's answer up until the step
$$\frac12\int\frac{\sin\theta-\cos\theta}{1+\sin\theta}d\theta$$
Instead of Weierstrass from here, simply multiply by $\frac{1-\sin\theta}{1-\sin\theta}$
$$\frac12\int\frac{(\sin\theta-\cos\theta)(1-\sin\theta)}{1-\sin^2\theta}d\theta=\frac12\int\frac{\sin\theta-\cos\theta-\sin^2\theta+\sin\theta\cos\theta}{\cos^2\theta}d\theta=$$
$$\frac12\int\sec\theta\tan\theta d\theta-\frac12\int\sec\theta d\theta-\frac12\int(\sec^2\theta-1)d\theta+\frac12\int\tan\theta d\theta$$
You should have no trouble with these remaining integrals.