Two problems in this question, the first I was able to do but was getting an incorrect answer and the second I'm just plain stumped on.

FIRST Q:

$\displaystyle \int\cot^3{x}\csc^3{x}dx$

Here is what I've done so far... I'm coming close but my numbers aren't quite right.

$\displaystyle u=\csc^2{x}$

$\displaystyle du=-\cot{x}dx$

$\displaystyle \csc{x}=\sqrt{u}$

$\displaystyle \int (\csc^2{x}-1)\cot{x}\csc^3{x}dx$

$\displaystyle -\int (u-1)u\sqrt{u}du$

$\displaystyle -\int (u^{2}-u)u^{1/2}du$

$\displaystyle -\int u^{5/2}du + \int u^{3/2}du$

$\displaystyle \tfrac{2}{5}u^{5/2}-\tfrac{2}{7}u^{7/2}+C$

...and subbing in just leaves me with the wrong answer. Where did I go wrong?

SECOND Q:

$\displaystyle \int\csc{x}dx$

Thought to use $\displaystyle \csc{x}=\sqrt{1+\cot^2{x}}$ or $\displaystyle \frac{1}{\sin{x}}$, but neither really panned out

Thanks in advance.