Prove the trigonometric identity $\frac{1-\cos\left(x\right)}{\sin\left(x\right)}=\frac{\sin\left(x\right)}{1+\cos\left(x\right)}$

Find the integral

$\int \frac { 32 x - 20 } { ( x - 1 ) ( 5 x - 3 ) } d x$

Find the integral

$\int\frac{2x+2}{\left(x^2+1\right)\left(x-1\right)^3}dx$

Find the integral

$\int\frac{1}{\left(x-1\right)^2\left(x+4\right)^2}dx$

Find the integral

$\int\frac{x-2}{\left(2x+1\right)\left(x+3\right)}dx$

Find the integral

$\int\frac{2x}{x^2-4}dx$

Find the integral

$\int\frac{\left(3x-5\right)}{\left(x-1\right)\left(x^2-1\right)}dx$

Find the integral

$\int\frac{\left(4x^2-8\right)}{x^3+2x^2}dx$