Need help in solving this integral. Could it be solved by substitution?

Question

The integral is $$\int{\frac{\cos(x) +x \sin(x)}{x(x + \cos(x))}}dx$$

Answer 1

$$\frac{\cos x +x \sin x}{x(x + \cos x)}=\frac1x-\frac{1-\sin x}{x + \cos x},$$ so $$\int \frac{\cos x +x \sin x}{x(x + \cos x)} \,dx=\ln\left|\frac{x}{x + \cos x}\right|+C.$$

Answer 2

Write it as $$\int \left(\frac{1}{x}-\frac{1-\sin x}{x+\cos x}\right)\,dx$$ and get $$\log x -\log(x+\cos x)+C$$