Evaluate $\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$

Question

$$\int\frac{\cot x}{\cos^2 x-\cos x+1}\,\,dx$$ Please guide me by which term it should be substituted to get the result of this integration. I have tried it by using $\cos x =t$, but it went so long and more problematic.

Answer 1

Hint

Starting from $$I=\int \frac{\cot (x)}{\cos ^2(x)-\cos (x)+1}dx$$ and using $\cos(x)=t$, $x=\cos^{-1}(t)$, $dx=-\frac{dt}{\sqrt{1-t^2}}$, $\cot(x)=\frac{t}{\sqrt{1-t^2}}$, you should arrive to $$I=-\int\frac{t}{\left(1-t^2\right) \left(t^2-t+1\right)}dt$$ and partial fraction decomposition leads to $$\frac{-t}{\left(1-t^2\right) \left(t^2-t+1\right)}=\frac{1-2 t}{3 \left(t^2-t+1\right)}+\frac{1}{2 (t-1)}+\frac{1}{6 (t+1)}$$

I am sure that you can take from here.

Added later

You could also use the tangent half-angle substitution (Weierstrass substitution) and so, using $y=\tan(\frac x2)$, arrive to $$I=\int \frac{1-y^4}{3 y^5+y}dy=\int \Big(\frac{1}{y}-\frac{4 y^3}{3 y^4+1}\Big)dy$$ which is even simpler than the previous one.

Answer 2

Hint : \begin{align} \int\frac{\cot x}{\cos^2 x-\cos x+1}\;\mathrm dx&=\int\frac{\frac{\cos x}{\sin x}}{\cos^2 x-\cos x+1}\cdot\frac{\sin^2 x}{\sin^2 x}\;\mathrm dx\\[10pt] &=\int\frac{\cos x}{\cos^2 x-\cos x+1}\cdot\frac{\sin x}{1-\cos^2 x}\;\mathrm dx\\[10pt] &=\int\frac{y}{y^2 -y+1}\cdot\frac{\mathrm dy}{y^2 - 1}\qquad\Rightarrow\qquad\text{set}\;y=\cos x\\[10pt] &=-\frac{1}{3}\underbrace{\int\frac{2y-1}{y^2 -y+1}\mathrm dy}_{\large\color{red}{\text{set}\;t\,=\,y^2 -y+1}}+\frac{1}{2}\underbrace{\int\frac{\mathrm dy}{y-1}}_{\large\color{red}{\text{set}\;u\,=\,y-1}}+\frac{1}{6}\underbrace{\int\frac{\mathrm dy}{y+1}}_{\large\color{red}{\text{set}\;v\,=\,y+1}} \end{align}

Answer 3

\begin{align} &\int\frac{\cot x}{\cos^2 x-\cos x+1}\,dx\\ =& \int\frac{\cot x(\cos x+1)}{\cos^3 x+1}\,dx = \int\frac{\cot x(\cos x+1)\csc^3x}{\cot^3 x+\csc^3x}\,dx\\ =& \int\frac{\cot^2 x\csc^2x+\cot x\csc^3x}{\cot^3 x+\csc^3x}\,dx =-\frac13 \int\frac{d(\cot^3 x+\csc^3x) }{\cot^3 x+\csc^3x}\\ =&-\frac13\ln|\cot^3 x+\csc^3x|+C \end{align}