Question

$\int \frac{\csc x}{\log \tan \frac{x}{2}}dx$

Answer 1

$\int \frac{\csc x}{\log \left(\tan x/2\right)}dx=\frac{1}{\sin x\left(\log \tan x/2\right)}$

$=\frac{1}{2}\sin x/2\cos x/2\times \log \left(\tan x/2\right)$

$=\frac{1}{2\tan x/2{\cos }^{2}x/2}\times \frac{}{\log \left(\tan x/2\right)}$

Now assume

$\log \left(\tan x/2\right)=t$

$\frac{{\sec }^2\left(x/2\right)}{2\tan x/2}dx=dt$

$\frac{dx}{2{\cos }^2\left(x/2\right)\tan x/2}=dt$

$\int \frac{1}{t}dt=\ln \left(t\right)+C$

$=\ln \left(\log \right(\tan x/2\left)\right)$