We can solve the integral $\int\frac{1}{\cos\left(x\right)-1}dx$ by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of $t$ by setting the substitution

Hence

Substituting in the original integral we get

The integral of a function times a constant ($2$) is equal to the constant times the integral of the function

Solve the product $-\left(1+t^{2}\right)$

Take the constant $\frac{1}{-2}$ out of the integral

Multiply the fraction and term in $2\cdot \left(\frac{1}{-2}\right)\int\frac{1}{t^{2}}dt$

Rewrite the exponent using the power rule $\frac{a^m}{a^n}=a^{m-n}$, where in this case $m=0$

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $-2$

Replace $t$ with the value that we assigned to it in the beginning: $\tan\left(\frac{x}{2}\right)$

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$