Take $\gamma$ to be a curve with parametric representation $$\phi(t) = (e^t \sin(2t),e^t \cos(t)), t \in [0,2\pi].$$
I would like to find the length of $\gamma$. I believe that the length of $\gamma$ is $$L(\gamma) = \int_0^{2\pi} \| \phi'(t)\| dt = \int_0^{2\pi} \|(2e^t \cos(2t) + e^t \sin(2t),-e^t\sin(t) + e^t\cos(t))\| dt$$ $$=\int_0^{2\pi} \sqrt{(2e^t \cos(2t) + e^t \sin(2t))^2 + (-e^t\sin(t) + e^t\cos(t))^2} dt,$$
Althought this is beginning to become unweildy and I am not entirely sure how to continue with this calculation. Wolfram also seems to be having difficulty with figuring out this integral. Any recommendations on how to find $L(\gamma)$?