Primitive of Power of Sine of a x over Power of Cosine of a x

Theorem

Reduction of Power of Sine

$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {-\sin^{m - 1} a x} {a \paren {m - n} \cos^{n - 1} a x} + \frac {m - 1} {m - n} \int \frac {\sin^{m - 2} a x} {\cos^n a x} \rd x + C$

Reduction of Power of Cosine

$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m + 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \rd x + C$

Reduction of Both Powers

$\ds \int \frac {\sin^m a x} {\cos^n a x} \rd x = \frac {\sin^{m - 1} a x} {a \paren {n - 1} \cos^{n - 1} a x} - \frac {m - 1} {n - 1} \int \frac {\sin^{m - 2} a x} {\cos^{n - 2} a x} \rd x + C$