How to calculate $\int_0^\pi \sqrt{1+\cos^2(t)} dt$?

Question

How to calculate $\int_0^\pi \sqrt{1+\cos^2(t)} dt$ ? Seemly, this is not a elementary integrals ?

Answer 1

This is $$2\int_0^{\pi/2}\sqrt{2-\sin^2t}\,dt=2\sqrt2 E(1/\sqrt2)$$ where $$E(k)=\int_0^{\pi/2}\sqrt{1-k^2\sin^2t}\,dt$$ is a complete elliptic interval of the second kind.

In their book Pi and the AGM, Borwein and Borwein give the formula $$E(1/\sqrt2)=\frac{4\Gamma(3/4)^2+\Gamma(1/4)^2}{8\sqrt\pi}.$$