Answer
$$\int\csc^5xdx=-\frac{\csc^3x\cot x}{4}-\frac{3\csc x\cot x}{8}-\frac{3}{8}\ln|\csc x+\cot x|+C$$
Work Step by Step
$$A=\int\csc^5xdx$$
Use Reduction Formula 100, which states that
$$\int\csc^naxdx=-\frac{\csc^{n-2}ax\cot ax}{a(n-1)}+\frac{n-2}{n-1}\int\csc^{n-2}axdx$$
for $n=5$ and $a=1$
$$A=-\frac{\csc^3x\cot x}{4}+\frac{3}{4}\int\csc^3xdx$$
Apply Formula 100 one more time, for $n=3$ and $a=1$:
$$A=-\frac{\csc^3x\cot x}{4}+\frac{3}{4}\Big(-\frac{\csc x\cot x}{2}+\frac{1}{2}\int\csc xdx\Big)$$ $$A=-\frac{\csc^3x\cot x}{4}+\frac{3}{4}\Big(-\frac{\csc x\cot x}{2}-\frac{1}{2}\ln|\csc x+\cot x|\Big)+C$$ $$A=-\frac{\csc^3x\cot x}{4}-\frac{3\csc x\cot x}{8}-\frac{3}{8}\ln|\csc x+\cot x|+C$$
