Maximising the Length of a Projectile Trajectory

Under uniform gravity and no drag, the arc length of a unit-speed projectile launched at angle $\theta$ is $L(\theta) = \tfrac{1}{2g}\left[2\sin\theta + \cos^2\theta \log\frac{1+\sin\theta}{1-\sin\theta}\right]$. Differentiating and setting to zero gives $2\csc\theta = \log\frac{1+\sin\theta}{1-\sin\theta}$, numerically $\theta \approx 56.465^\circ$. Full setup in the PDF.