$\int {\csc }^3\left(x\right) \mathit{\ⅆ}x$

$\= -\csc \left(x\right)\cot \left(x\right)-\int \cot \left(x\right)\cdot \left(\csc \left(x\right)\cot \left(x\right)\right) \mathrm{dx}$

$\= -\csc \left(x\right)\cot \left(x\right)-\int {\cot }^2\left(x\right)\csc \left(x\right) \mathrm{dx}$

$\= -\csc \left(x\right)\cot \left(x\right)-\int \left({\csc }^2\left(x\right)-1\right)\csc \left(x\right) \mathrm{dx}$

$\= -\csc \left(x\right)\cot \left(x\right)-\int {\csc }^3\left(x\right) \mathrm{dx}\+\int \csc \left(x\right) \mathrm{dx}$

$2\int {\csc }^3\left(x\right) \mathrm{dx}$

$\= -\csc \left(x\right)\cot \left(x\right)\+\int \csc \left(x\right) \mathrm{dx}$

$\= -\csc \left(x\right)\cot \left(x\right)\+\ln \(\left|\csc \left(x\right)-\cot \left(x\right)\right|\)$

$\int {\csc }^3\left(x\right) \mathrm{dx}$

$\=-\frac{1}{2}\csc \left(x\right)\cot \left(x\right)\+\frac{1}{2}\ln \(\left|\csc \left(x\right)-\cot \left(x\right)\right|\)$

Table 6.2.10(a)   evaluation of $\int {\csc }^3\left(x\right) \mathit{\ⅆ}x$ 

Evaluation in Maple

$\int {\csc }^3\left(x\right) \mathit{\ⅆ}x$ = $-\frac{1}{2}\frac{\cos \left(x\right)}{{\sin \left(x\right)}^2}\+\frac{1}{2}\ln \left(\csc \left(x\right)-\cot \left(x\right)\right)$$$

$\begin{array}{c}\int {\csc \left(x\right)}^3\ⅆx\hfill \\ & \text{=}& -\csc \left(x\right)\cot \left(x\right)-\int \csc \left(x\right){\cot \left(x\right)}^2\ⅆx\hfill & \left[\mathrm{parts}\,\csc \left(x\right)\,-\cot \left(x\right)\right]\\ & \text{=}& -\csc \left(x\right)\cot \left(x\right)-\int \csc \left(x\right)\left({\csc \left(x\right)}^2-1\right)\ⅆx\hfill & \left[\mathrm{rewrite}\,{\cot \left(x\right)}^2={\csc \left(x\right)}^2-1\right]\\ & \text{=}& -\csc \left(x\right)\cot \left(x\right)-\int {\csc \left(x\right)}^3\ⅆx-\ln \left(\csc \left(x\right)+\cot \left(x\right)\right)\hfill & \left[\mathrm{rewrite}\,\csc \left(x\right)\left({\csc \left(x\right)}^2-1\right)={\csc \left(x\right)}^3-\csc \left(x\right)\right]\\ & \text{=}& -\frac{\csc \left(x\right)\cot \left(x\right)}{2}-\frac{\ln \left(\csc \left(x\right)+\cot \left(x\right)\right)}{2}\hfill & \[\mathrm{so}\mathrm{lve}\end{array}$

Table 6.2.10(b)   Annotated stepwise evaluation of $\int {\csc }^3\left(x\right) \mathit{\ⅆ}x$ via the Integration Methods tutor