General antiderivative of a csc-cot and reciprocal power term

Split the antiderivative into two pieces using linearity: $\int 2\csc(x)\cot(x)\,dx$ and $\int -\dfrac{6}{x}\,dx$. For the first piece, recall that $\frac{d}{dx}[-\csc(x)] = \csc(x)\cot(x)$, so $\int \csc(x)\cot(x)\,dx = -\csc(x)$, giving $\int 2\csc(x)\cot(x)\,dx = -2\csc(x)$.

For the second piece, use $\int \frac{1}{x}\,dx = \ln|x|$, so $\int -\dfrac{6}{x}\,dx = -6\ln|x|$. Combining both results and adding the arbitrary constant gives the general antiderivative $G(x) = -2\csc(x) - 6\ln|x| + C$.