Is it possible to find the expression of the antiderivative $\int \frac{dx}{\cosh(x)+\sqrt{\cosh(2x)}}$

Question

I have been asked to express the integral $$\int \frac{dx}{\cosh(x)+\sqrt{\cosh(2x)}}$$

I thought about the substitution $$t=e^x$$

but it gave me a more complicate function. So, any idea will be appreciated.

Answer 1

Using $\cosh(x)=t$ we end with $$I=\int \frac{dt}{\sqrt{t^2-1} \left(t+\sqrt{2 t^2-1}\right)}$$ for which a CAS gives $$I=\frac{-2 t^3+\sqrt{2 t^2-1}+\sqrt{2-4 t^2} \sqrt{1-t^2} \left(F\left(\sin ^{-1}\left(\sqrt{2} t\right)|\frac{1}{2}\right)-E\left(\sin ^{-1}\left(\sqrt{2} t\right)|\frac{1}{2}\right)\right)+t}{ \sqrt{(t^2-1)(2 t^2-1)}}$$ Back to $x$ $$I=\text{csch}(x)-\sqrt{\cosh (2 x)} \coth (x)-i \left(F(i x|2)+E(i x|2)\right)$$