Integrals yielding $e^{\pi}$ or $e^{-\pi}$

 Lately, I've been playing a lot with integrals, and coincidentally (with a bit of algebraic manipulation), I've come across these two beauties:

$$\int_{0}^{1} \left(\frac{5}{2} \left((x - \sqrt{x^2 - 1})^{2i} + x^4\right) - 1\right) \, dx = e^{\pi},\tag{1}$$

$$\int_{0}^{1} \left( -\frac{5}{2\left( x - \sqrt{x^2 - 1}\right)^{2i}} - \frac{5x^4}{2} + 1 \right) \, dx = e^{-\pi}.\tag{2}$$

$e^{\pi}$ is known as Gelfond's constant.

I have provided these integrals as a response to a question on MathSE. The proofs are left as exercises for the reader.