Primitive of Square of Cosecant of a x over Cotangent of a x

Theorem

$\ds \int \frac {\csc^2 a x \rd x} {\cot a x} = \frac {-\ln \size {\cot a x} } a + C$

$\ds \frac \d {\d x} \cot x$

$=$

$\ds -\csc^2 x$

$\ds \leadsto \ \ $

$\ds \int \frac {\csc^2 x \rd x} {\cot x}$

$=$

$\ds -\ln \size {\cot a x} + C$

$\ds \leadsto \ \ $

$\ds \int \frac {\csc^2 a x \rd x} {\cot a x}$

$=$

$\ds \frac 1 a \paren {-\ln \size {\cot a x} } + C$

$\ds $

$=$

$\ds \frac {-\ln \size {\cot a x} } a + C$

simplifying

$\blacksquare$