Indefinite Integrals with $\frac{1}{\sin x}$

 $\int \frac {dx }{\sin a x}dx = \frac {1}{a} \ln { \left( \frac {1} {\sin ax} - \cot ax \right ) } = \frac {1}{a} \ln { \tan \frac {ax}{2} } +C$ $\int \frac {dx}{ \sin^2 ax }= -\frac {\cot ax} { a} +C$ $\int \frac {1}{ \sin^3 ax }dx = -\frac {\cot ax}{2a \sin ax}+ \frac {1}{2a} \ln{ \tan \frac{ax}{2} }+C$ $\int \frac {\cot ax dx}{\sin^n ax}= -\frac{1}{na \sin^nax} +C$ $\int \sin ax dx = -\frac {\cos ax}{a} +C$ $\int \frac {xdx} {\sin ax} = \frac {1}{a^2} \left \{ ax + \frac {(ax)^3}{18}+ \frac{7(a x)^5}{1800} + \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n+2)!} + \cdot \cdot \cdot \right \} +C$ $\int \frac {dx}{x \sin ax } = -\frac{1}{ax} + \frac {ax} {6} + \frac{7(a x)^3}{1080}+ \cdot \cdot \cdot + \frac {2(2^{2n-1}-1)Bn(ax)^{2n+1}}{(2n-1)(2n)!} + \cdot \cdot \cdot +C$ $\int x \sin^2 ax dx = - \frac {x \cot ax}{a} + \frac {1}{a^2} \ln {\sin a x} - \frac {x^2}{2} +C$ $\int \frac {x dx}{ \cos^2 ax} = -\frac {x\cot ax}{a} + \frac {1}{a^2} \ln { \sin ax } +C$ $\int \frac {dx}{q+\frac {p}{\sin ax}}= \frac{x}{q}-\frac{p}{q} \int \frac{dx}{p+q\sin ax} +C$ $\int \sin^n ax dx = -\frac {\cot ax}{a(n-1)\sin^{n-2}ax} + \frac {n-2}{n-1} \int \frac {dx}{\sin^{n-2} a x } +C$