Answer
$-\frac{1}{4}cotx(cscx^{3}x)+\frac{5}{8}cotx(cscx)-\frac{3}{8}\ln|cscx+cotx|+C$
Work Step by Step
$cot^{2}x$ = $csc^{2}x-1$
${\int}cot^{4}x(cscx)dx$ = $\int(csc^{2}x-1)^{2}cscxdx$ = ${\int}[csc^{5}x-2csc^{3}x+cscx]dx$
apply reduction formla
${\int}csc^{3}xdx$ = $-\frac{1}{2}cotx(cscx)+\frac{1}{2}\int(cscx)dx$ = $-\frac{1}{2}cotx(cscx)-\frac{1}{2}\ln|cscx+cotx|+C$
so that
${\int}csc^{5}xdx$ = $-\frac{1}{4}cotx(cscx^{3}x)+\frac{3}{4}{\int}cscx^{3}xdx$ = $-\frac{1}{4}cotx(cscx^{3}x)-\frac{3}{4}(\frac{1}{2}cotx(cscx)+\frac{1}{2}\ln|cscx+cotx|)+C$ = $-\frac{1}{4}cotx(cscx^{3}x)-\frac{3}{8}cotx(cscx)-\frac{3}{8}\ln|cscx+cotx|+C$
so
${\int}cot^{4}x(cscx)dx$ = ${\int}csc^{5}xdx-2{\int}csc^{3}xdx+{\int}cscxdx$ = $-\frac{1}{4}cotx(cscx^{3}x)-\frac{3}{8}cotx(cscx)-\frac{3}{8}\ln|cscx+cotx|+cotx(cscx)+ln|cscx+cotx|-ln|cscx+cotx|+C$ = $-\frac{1}{4}cotx(cscx^{3}x)+\frac{5}{8}cotx(cscx)-\frac{3}{8}\ln|cscx+cotx|+C$
