$u\={\sin }^{n-1}\left(x\right)$

$\mathrm{dv}\=\sin \left(x\right){\cos }^m\left(x\right)\mathrm{dx}$

$\mathrm{du}\=\left(n-1\right){\sin }^{n-2}\left(x\right)\cos \left(x\right)\mathrm{dx}$

$v\=-\frac{{\cos }^{m\+1}\left(x\right)}{m\+1}$

$\int {\cos }^m\left(x\right){\sin }^n\left(x\right) \mathit{\ⅆ}x$

$\=-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+1}-\left(-\frac{n-1}{m\+1}\right)$ $\int {\cos }^{m\+2}\left(x\right){\sin }^{n-2}\left(x\right) \mathrm{dx}$

$\=-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+1}\+\frac{n-1}{m\+1}$ $\int {\cos }^m\left(x\right){\sin }^{n-2}\left(x\right)\left(1-{\sin }^2\left(x\right)\right) \mathrm{dx}$

$\=-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+1}\+\frac{n-1}{m\+1}$ $\left(\int {\cos }^m\left(x\right){\sin }^{n-2}\left(x\right) \mathrm{dx}-\int {\cos }^m\left(x\right){\sin }^n\left(x\right) \mathrm{dx}\right)$

$\=-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+1}\+\frac{n-1}{m\+1}$ $\int {\cos }^m\left(x\right){\sin }^{n-2}\left(x\right) \mathrm{dx}-\frac{n-1}{m\+1}$ $\int {\cos }^m\left(x\right){\sin }^n\left(x\right) \mathrm{dx}$

$\left(1\+\frac{n-1}{m\+1}\right)\int {\cos }^m\left(x\right){\sin }^n\left(x\right) \mathit{\ⅆ}x$

$\=-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+1}\+\frac{n-1}{m\+1}$ $\int {\cos }^m\left(x\right){\sin }^{n-2}\left(x\right) \mathrm{dx}$$$

$\int {\cos }^m\left(x\right){\sin }^n\left(x\right) \mathit{\ⅆ}x$

$\=\frac{m\+1}{m\+n}\left(-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+1}\+\frac{n-1}{m\+1}\int {\cos }^m\left(x\right){\sin }^{n-2}\left(x\right) \mathrm{dx}\right)$

$\=-\frac{{\cos }^{m\+1}\left(x\right){\sin }^{n-1}\left(x\right)}{m\+n}\+\frac{n-1}{m\+n}\int {\cos }^m\left(x\right){\sin }^{n-2}\left(x\right) \mathrm{dx}$

Table 6.2.8(a)   Derivation of the second reduction formula in Table 6.2.1

Initialize

Integrate by parts

$q_1≔\mathrm{Parts}\left(Q\,{\sin }^{n-1}\left(x\right)\right)$

$-\frac{{\cos \left(x\right)}^{m\+1}{\sin \left(x\right)}^{n-1}}{m\+1}-\left(\∫\left(-\frac{{\cos \left(x\right)}^{m\+1}{\sin \left(x\right)}^{n-1}\left(n-1\right)\cos \left(x\right)}{\left(m\+1\right)\sin \left(x\right)}\right)\ⅆx\right)$

Massage to the form in line 1 of Table 6.2.8(a)

$q_2≔\mathrm{collect}\left(\mathrm{simplify}\left(q_1\right)\,\mathrm{int}\right)$

$-\frac{\left(-n\+1\right)\left(\∫{\cos \left(x\right)}^{m\+2}{\sin \left(x\right)}^{n-2}\ⅆx\right)}{m\+1}-\frac{{\cos \left(x\right)}^{m\+1}{\sin \left(x\right)}^{n-1}}{m\+1}$

Replace ${\cos }^2\left(x\right)$ with $1-{\sin }^2\left(x\right)$ and massage to form in line 4 of Table 6.2.8(a)

$q_3≔\mathrm{collect}\left(\mathrm{simplify}\left(\mathrm{Expand}\left(\mathrm{eval}\left(q_2\,{\cos }^{m\+2}\left(x\right)\={\cos }^m\left(x\right)\cdot \left(1-{\sin }^2\left(x\right)\right)\right)\right)\right)\,\mathrm{int}\right)$

$\frac{\left(-n\+1\right)\left(\∫{\cos \left(x\right)}^m{\sin \left(x\right)}^n\ⅆx\right)}{m\+1}\+\frac{\left(n-1\right)\left(\∫{\cos \left(x\right)}^m{\sin \left(x\right)}^{n-2}\ⅆx\right)}{m\+1}-\frac{{\cos \left(x\right)}^{m\+1}{\sin \left(x\right)}^{n-1}}{m\+1}$

Solve for the unknown integral

$\mathrm{collect}\left(\mathrm{normal}\left(\mathrm{isolate}\left(Q\=q_3\,Q\right)\right)\,\mathrm{int}\right)$

$\∫{\cos \left(x\right)}^m{\sin \left(x\right)}^n\ⅆx\=\frac{\left(n-1\right)\left(\∫{\cos \left(x\right)}^m{\sin \left(x\right)}^{n-2}\ⅆx\right)}{m\+n}-\frac{{\sin \left(x\right)}^{n-1}{\cos \left(x\right)}^{m\+1}}{m\+n}$