How do you integrate $\csc^3(x)$? How do you integrate
$
\int \csc^{3}x\,dx
$ ?
I know that you have to change the integral into $\int \csc x (1 + \cot^2 x)\,dx$
But what do you do next?
 A: how about $ \int \csc ^3 x \, dx = \int \frac 1{\sin ^3 x} \, dx = \int \frac{\sin x \, dx}{\sin ^4 x} = - \int \frac{du}{(1-u^2)^2}$ where $u = \cos x.$
we can now use partial fraction to decompose $$ \frac 1{(u^2 -1)^2} = \frac{1}{(u-1)^2(u+1)^2}=\frac{A}{u-1}+\frac{B}{(u-1)^2}+\frac{C}{u+1}+\frac{D}{(u+1)^2}\tag 1$$
you find that $B = D = \frac14.$  find the constants $A, C.$  now you can integrate every term in $(1).$
A: HINT: $$\sin(x)^3=\sin(x)^2\sin(x)=(1-\cos(x)^2)\sin(x)$$
setting $t=\sin(x)$ we get $$dx=-\frac{dt}{\sin(x)}$$ and our integral will be 
$$-\int \frac{dt}{(1-t^2)\sin(x)^2}=-\int \frac{dt}{(1-t^2)^2}$$
A: $$\int\csc^3(x)\space\text{d}x=$$

Use the reduction formula, where $m=3$:
$$\int \csc^m(x)\space\text{d}x=-\frac{\cos(x)\csc^{m-1}(x)}{m-1}+\frac{m-2}{m-1}\int\csc^{m-2}(x)\space\text{d}x$$

$$-\frac{\cot(x)\csc(x)}{2}+\frac{1}{2}\int\csc(x)\space\text{d}x=$$

Multiply numerator and denominator of $\csc(x)$ by $\cot(x)+\csc(x)$:

$$-\frac{\cot(x)\csc(x)}{2}+\frac{1}{2}\int-\frac{-\cot(x)\csc(x)-\csc^2(x)}{\cot(x)+\csc(x)}\space\text{d}x=$$

Substitute $u=\cot(x)+\csc(x)$ and $\text{d}u=(-\csc^2(x)-\cot(x)\csc(x))\space\text{d}x$:

$$-\frac{\cot(x)\csc(x)}{2}-\frac{1}{2}\int\frac{1}{u}\space\text{d}u=$$
$$-\frac{\cot(x)\csc(x)}{2}-\frac{\ln\left|u\right|}{2}+\text{C}=$$
$$-\frac{\cot(x)\csc(x)}{2}-\frac{\ln\left|\cot(x)+\csc(x)\right|}{2}+\text{C}=$$
$$-\frac{\cot(x)\csc(x)-\ln\left|\cot(x)+\csc(x)\right|}{2}+\text{C}$$
