hard indefinite integral with unreal solution

Question

Can someone give me an idea on how I should solve this?

$$\int \frac{1}{2 \sin{\left (x \right )} + 5 \cos{\left (x \right )}}\, dx$$

I tried to enter a replacement, but I'm not sure how to continue. Here's the exercise.

result

Answer 1

Hint:

Use substitution: $\:t=\tan \dfrac x2,\quad \mathrm dt=\frac12(1+t^2)\,\mathrm dx$, and the trigonometric formulæ: $$\sin x = \frac{2t}{1+t^2},\qquad \cos x = \frac{1-t^2}{1+t^2}.$$

Answer 2

Hint: Take $\theta$ such that $\cos\theta=\frac2{\sqrt{29}}$ and that $\sin\theta=\frac5{\sqrt{29}}$. Then$$\frac1{2\sin(x)+5\cos(x)}=\frac1{\sqrt{29}\sin(x+\theta)}.$$

Answer 3

Hint:

$$2\sin x+5\cos x=\sqrt{2^2+5^2}\cos\left(x-\arccos\dfrac5{\sqrt29}\right)$$

Now use this