Hard integral of root function and hyperbolic function

Question

I need to calculate this integral:

$$\int^A_B\frac{\sqrt{x-B}}{\cosh^2x}dx$$

Is there any way to do this?

Answer 1

Hint: For $B=0$ and $A\to\infty$, we have

$$\int_0^\infty\frac{x^n}{\cosh^2x}dx=\frac{2^{n-1}-1}{4^{n-1}}\cdot n!\cdot\zeta(n)~,$$

see OEIS A$052665$. For $n=\dfrac12~,$ this becomes $\bigg(\dfrac1{\sqrt2}-1\bigg)\cdot\sqrt\pi\cdot\zeta\bigg(\dfrac12\bigg).~$ Since $\zeta(n)=\text{Li}_n(1)$,

if a closed form for the indefinite integral were to exist, it could only be expressed in terms of such special functions like polylogarithms, or hypergeometric series, obtained by expanding $~\text{sech}^2x$ into its binomial series, and then switching the order of summation and integration.