As part of my Calculus II final we had a bonus question that had $$\int \sin(x)\sqrt{1+\cos^2(x)} \, dx \tag{*} $$ This set-up integral was not given and I know that (*) is easy to solve with U-Substitution.
As part of my incorrect set-up of the problem I wrote the integral as $$\int \sqrt{1+\cos^2(x)} \, dx \tag{**} $$
Try as I might with 1/2 Angles and Identities I was at a loss as how to approach an integral like (**). I ended up sitting till the end of the period without thinking of a method to approach the problem with.
So, how do you solve an integral like (**) that is resistant to the application of 1/2 Angle and Identity relationships? and what will put me on track to complete this integration?
This is a partial answer to the question in your comment, "how do you get to that new function?"
The function in this case is an elliptic integral of the second kind, $$E(\phi,k)=\int_0^\phi \sqrt{1-k^2\sin^2\theta}\,d\theta\ .$$ To evaluate your integral in these terms, $$\eqalign{\int \sqrt{1+\cos^2x}\,dx &=\int_0^x\sqrt{1+\cos^2\theta}\,d\theta+C\cr &=\int_0^x\sqrt{2-\sin^2\theta}\,d\theta+C\cr &=\sqrt2\int_0^x\sqrt{1-{\textstyle\frac{1}{2}}\sin^2\theta}\,d\theta+C\cr &=\sqrt2 E(x,{\textstyle\frac{1}{\sqrt2}})+C\ .\cr}$$ The Wolfram Alpha link given in one of the comments looks different from this, though if you carefully read the documentation you will see that they are just using a different notation.
As to how you evaluate this function, well really just the same as any other function - by finding out lots of facts about it. For example, it is obvious from the definition that $$E(\phi,-k)=E(\phi,k)$$ and that $$\frac{d}{d\phi}E(\phi,k)=\sqrt{1-k^2\sin^2\phi}\ .$$ Also, by substituting $t=\sin\theta$ or $z=\tan(\theta/2)$ you can derive the alternative formulae $$E(\phi,k)=\int_0^{\sin\phi} \sqrt{\frac{1-k^2t^2}{1-t^2}}\,dt =\int_0^{\tan(\phi/2)}\!\!\sqrt{1+(2-4k^2)z^2+z^4}\frac{dz}{(1+z^2)^2}\ .$$ And so on. . .
To actually get a numerical evaluation for a given $x$ you could use standard integral estimation methods such as Simpson's Rule, or you could calculate a Taylor series. As I am not an expert in numerical methods, there could very well be techniques which are superior to either of these. If you Google something like "elliptic integral numeric evaluation" I would think that you should find something.