[tex]\bf \displaystyle \int~(x^2+1)cot(x^3+3x)\cdot dx\\\\
-------------------------------\\\\
\stackrel{substitution}{u=x^3+3x}\implies \cfrac{du}{dx}=3x^2+3\implies \cfrac{du}{dx}=3(x^2+1)
\\\\\\
\cfrac{du}{3(x^2+1)}=dx\\\\
-------------------------------\\\\
\displaystyle \int~\underline{(x^2+1)} cot(u)\cdot \cfrac{du}{3\underline{(x^2+1)}}\implies \cfrac{1}{3}\int~cot(u)\cdot du
\\\\\\
\cfrac{1}{3}( ln|sin(u)|)+C\implies \cfrac{ln|sin(x^3+3x)|}{3}+C[/tex]
