How to find the integration of the given expression

Question

How to solve Integration of

$$\int \dfrac{\sqrt{x^2 +1}}{x+1}$$

I tried putting $x=2\tan t$. But not getting the final solution

Answer 1

Substitute $x=\tan t$ to integrate

\begin{align} & \int \frac{\sqrt{x^2 +1}}{x+1}dx\\ =&\int \frac{\sec^3t}{\tan t+1}dt =\int \frac{(\tan t -1)(\tan t+1)+2}{\tan t+1}\sec tdt \\ = &\int \left( \tan t\sec t -\sec t + \sqrt2 \sec(t-\frac\pi4) \right)dt\\ =&\sec t -\ln(\sec t+\tan t) + \sqrt2\ln(\sec(t-\frac\pi4) +\tan(t-\frac\pi4) )+C \end{align}