Line integral of $\sin x$. 
How to evaluate:
  $$\int \sqrt{1+\cos^2x} dx$$ 

Is a simple antiderivative known to exist?
 A: A simple antiderivative is given by 
$$\tag{1}
\int_0^x\sqrt{1+\cos^2t}\,dt.
$$
This is an example of what is called an elliptic integral and it cannot be expressed in terms of other well-known elementary functions (which is what you probably mean by "simple", and doesn't agree with what I would call "simple"). 
But let me stress my point: the expression in (1) can be used to approximate values of the function, and in that sense it is not necessarily worse than, say $\log x$. 
A: This question is much like the other one.
Using $\cos^2(x) = 1-\sin^2(x)$
$$
   \int \sqrt{1+\cos^2(x)} \mathrm{d} x = \int \sqrt{2-\sin^2(x)} \mathrm{d} x = \sqrt{2} \int \sqrt{1-\frac{1}{2} \sin^2(x)} \mathrm{d} x = \sqrt{2} \operatorname{E}\left(x, \frac{1}{2}\right) + C
$$
where $\mathrm{E}(x, m)$ denotes the incomplete elliptic integral of the second kind:
$$ 
   \operatorname{E}\left(\phi, m\right)  = \int_0^\phi \sqrt{1-m \sin^2(\varphi)} \mathrm{d} \varphi
$$
In particular, no elementary andi-derivative exists.
