Answer
$-\dfrac{\csc^3x\cot x}{4}-\dfrac{3\csc x\cot x}{8}-\dfrac{3}{8}\ln|\csc x+\cot x|+C$
Work Step by Step
Use the Reduction Formula:
$\int\csc^naxdx=-\dfrac{\csc^{n-2} (ax) \cot (ax) }{a(n-1)}+\dfrac{n-2}{n-1}\int\csc^{(n-2)}(ax) dx $
Let$\space I=\int\csc^5(x) dx\\=-\dfrac{\csc^3x\cot x}{4}+\dfrac{3}{4}\int\csc^3x \space dx\\=-\dfrac{\csc^3x\cot x}{4}+\dfrac{3}{4}(-\dfrac{\csc x \cdot \cot x}{2}+(1/2)\int\csc xdx)\\=-\dfrac{\csc^3x\cot x}{4}-\dfrac{3\csc x\cot x}{8}-\dfrac{3}{8}\ln|\csc x+\cot x|+C$
