An integral involving the golden ratio

 Several interesting integrals involving the golden ratio can be found at MathSE. Here I present another one that I haven't seen anywhere else on the web. I'm referring to this one:

$$\int_\frac{\pi}{2}^{\pi} \frac{\sqrt{5}}{\sin{x}-\cot{x}}\,dx=4\ln\phi.\tag{1}$$

I discovered $(1)$ serendipitously while exploring the advantages of using the sine half-angle substitution. 

Proof. Let's start by evaluating the indefinite integral:

$$\int \frac{1}{\sin{x}-\cot{x}}\,dx=\int \frac{\sin{x}}{\sin^2{x}-\cos{x}}\,dx=\int -\frac{\sin{x}}{\cos^2{x}+\cos{x}-1}\,dx.$$

Substitute $u=\cos{x} \rightarrow du=-\sin{x}\,dx$.

$$\int \frac{1}{\sin{x}-\cot{x}}\,dx=\int \frac{1}{u^2+u-1}\,du.$$

Factoring the denominator and performing partial fraction decomposition,

$$\begin{aligned}\int \frac{1}{u^2+u-1}\,du&=\frac{2}{\sqrt{5}} \int \frac{1}{2u-\sqrt{5}+1}\,du- \frac{2}{\sqrt{5}} \int \frac{1}{2u+\sqrt{5}+1}\,du\\&= \frac{1}{\sqrt{5}}\left(\ln{(2u-\sqrt{5}+1)}-\ln{(2u+\sqrt{5}+1)}\right)+C.\end{aligned}$$

Undoing the substitution $u=\cos{x}$, taking into account that $2\phi=\sqrt{5}+1$ and $-\sqrt{5}+1=2-2\phi$, and applying properties of logarithms,

$$\sqrt{5}\int \frac{1}{\sin{x}-\cot{x}}\,dx=\ln{\frac{\left|\cos{x}+1-\phi\right|}{\cos{x}+\phi}}+C.$$

Now, evaluating the definite integral,

$$\begin{aligned}\sqrt{5}\int_\frac{\pi}{2}^{\pi} \frac{1}{\sin{x}-\cot{x}}\,dx&=\ln{\frac{\left|\cos{x}+1-\phi\right|}{\cos{x}+\phi}}\\&=\ln{\frac{\phi}{\left|1-\phi\right|}}-\ln{\frac{\left|1-\phi\right|}{\phi}}\\&=\ln{\left(\frac{\phi}{\phi-1}\right)}-\ln{\left(\frac{\phi-1}{\phi}\right)}\\&=2\ln{\left(\frac{\phi}{\phi-1}\right)}.\end{aligned}$$

But $\phi-1=\frac{1}{\phi}$, thus

$$\int_\frac{\pi}{2}^{\pi} \frac{\sqrt{5}}{\sin{x}-\cot{x}}\,dx=4\ln{\phi}.$$

Other examples:

$$\int_{0}^{\frac{\pi}{2}} \frac{\sqrt{5}}{\csc{x}-\tan{x}}\,dx=\ln{\phi}.$$

The golden ratio is a famous ratio, but have you heard of the silver ratio, $\phi_s$? Well, the following integral is related to it. I invite you to prove it for yourself.

$$\int_{0}^\frac{3\pi}{2} \frac{\sqrt{2}}{\sin{x}-\cos{x}}\,dx=2\ln{\phi_s}.$$