$$\int_0^{2\pi} \sqrt{a^2 +b^2+2ab \cos\varphi}\,\mathrm{d}\varphi$$
Where $a$ and $b$ are constants. I had to find the distance travelled by a point at a distance of $b$ from the centre of a rolling disc with radius $a$ in one full rotation. I got the final answer as this integral, but I was not able to solve this integral.
First consider the integrand \begin{align} \sqrt{a^{2} + b^{2} + 2 a b \cos(\phi)} \end{align} which can be seen as \begin{align} \sqrt{a^{2} + b^{2} + 2 a b \cos(\phi)} &= \sqrt{(a+b)^{2} - 4ab \sin^{2}\left(\frac{\phi}{2}\right) } \\ &= (a+b) \sqrt{ 1 - \left( \frac{4ab}{(a+b)^{2}} \right) \, \sin^{2}\left( \frac{\phi}{2}\right) }. \end{align} Now, \begin{align} I &= \int_{0}^{2\pi} \sqrt{a^{2} + b^{2} + 2 a b \cos(\phi)} \, d\phi \\ &= (a+b) \, \int_{0}^{2\pi} \sqrt{ 1 - \left( \frac{4ab}{(a+b)^{2}} \right) \, \sin^{2}\left( \frac{\phi}{2}\right) } \, d\phi \\ &= 2(a+b) \int_{0}^{\pi} \sqrt{1- k^{2} \sin^{2}(x) } \, dx \end{align} where $(a+b)^{2} k^{2} = 4ab$. The integral remaining if the complete elliptical integral of the second kind. The "solution" of the integral is \begin{align} I = 4(a+b) E\left(\frac{4ab}{(a+b)^{2}}\right). \end{align}
Hence, \begin{align} \int_{0}^{2\pi} \sqrt{a^{2} + b^{2} + 2 a b \cos(\phi)} \, d\phi = 4(a+b) E\left(\frac{4ab}{(a+b)^{2}}\right). \end{align}