$$\int {\frac{1}{x^\frac13+x^\frac14}+\frac{log(1+x^\frac16)}{x^\frac13+x^\frac12}}dx$$ I have to sovle this function integral. Please help me out I have tried a lot but its approach is very long. I have also seen its solution bit still its quite long approach. Please suggest the shortest possible method.
Note: The solution should also be in elementary form.
For the first part, write $$ \frac{1}{x^{1/4}+x^{1/3}}=\frac{1}{(1+x^{1/12})x^{1/4}} $$ and then let $u=x^{1/12}$. Using a geometric sum, you end up with $$ 12\int \frac{u^8}{1+u}\,du=12\int -1+u-u^2+u^3-u^4+u^5-u^6+u^7+\frac{1}{1+u}\,du. $$ which is easy to integrate. I leave it to you.
For the second part, write $$ \frac{\log(1+x^{1/6})}{x^{1/3}+x^{1/2}}=\frac{\log(1+x^{1/6})}{(1+x^{1/6})x^{1/3}}, $$ and then let $u=x^{1/6}$. You end up with $$ 6\int\frac{u^3\log(1+u)}{1+u}\,du. $$ Let $s=\log(1+u)$, and you will find (again, modulo constant), $$ 6\int (e^s-1)^3s\,ds. $$ This is easily calculated, integrating by parts.