Integration of 1/(sin^m x cos^n x)

For the integral $$\int \frac{dx}{\sin^m x\,\cos^n x},$$ use the following cases:

• If $$m$$ is odd, write
  $$\sin^m x=\sin^{m-1}x\sin x=(1-\cos^2x)^{\frac{m-1}{2}}\sin x$$
  and substitute
  $$u=\cos x,\quad du=-\sin x\,dx\,. $$
• If $$n$$ is odd, write
  $$\cos^n x=\cos^{n-1}x\cos x=(1-\sin^2x)^{\frac{n-1}{2}}\cos x$$
  and substitute
  $$u=\sin x,\quad du=\cos x\,dx\,. $$
• If both $$m,n$$ are even, apply power-reduction:
  $$\sin^2x=\frac{1-\cos2x}{2},\quad\cos^2x=\frac{1+\cos2x}{2}\,. $$
• Alternatively, in general divide by $$\cos^{m+n}x$$ and set
  $$t=\tan x,\quad dx=\frac{dt}{1+t^2}\,, $$
  so that
  $$\sin^m x\cos^n x=\frac{t^m}{(1+t^2)^{\frac{m+n}{2}}}$$
  and the integral becomes a rational function in $$t$$.