I have a problem with calculating the following integral
Given is a function $T(E)$ with constants $E$, $a$ and $U_0$, defined as
$$\sqrt {2m} \int_{x_1}^{x_2}\left(E+\frac{U_0}{\cosh^2(ax)}\right)^{-\frac{1}{2} }dx$$
I know that $\cosh^2 x=1-\sinh^2 x$ and how to rewrite them as exponential functions, but the negative square root really gives me some trouble. Maybe anyone got some tips on how to start as even wolframalpha could not give me a solution to this. Kind Regards
Hint. One may write $$ \sqrt {2m} \int_{x_1}^{x_2}\left(E+\frac{U_0}{\cosh^2(ax)}\right)^{-\frac{1}{2} }dx=\sqrt {2m} \int_{x_1}^{x_2}\frac{\cosh(a x)}{\left(E(1+\sinh^2(ax))+U_0\right)^{1/2}}dx $$ then one may notice that $$ \left(\sinh (ax)\right)'=a\cosh (ax) $$ obtaining a standard integral of the form $$ \int_{\alpha_0}^{\alpha_1}\frac{du}{\sqrt{\alpha+u^2}}. $$