$$\int\frac{\sqrt{1+x^2}}{x}dx$$
I tried letting $x=\tan\theta\ $ where $\frac{-\pi}{2} < \theta < \frac{\pi}{2}$ so that $dx = \sec^2\theta\,d\theta$ and after making the substitution one gets to $$\int\frac{\sec^3\theta}{\tan\theta} d\theta$$ which is equivalent to $$\int\frac{1}{\cos^2\theta\sin\theta}d\theta$$
After this, I don't know how to proceed. I tried looking for the same integral elsewhere and I found a solution that involves a method called partial fraction decomposition, I believe. But, I have not been taught that method yet and this integral appears on the section of the book that I am currently working on.