You can use $$ \frac{d}{dr} \cos^{-1} \left(f\left(r\right)\right) = -\left[1-\left(f\left(r\right)\right)^2\right]^{-1/2} \frac{df}{dr} $$ with $$ f\left(r\right) = \frac{M/r - m \alpha/M}{\sqrt{2 m E + m^2 \alpha^2 / M^2}} $$ to show that $$ \int dr \frac{M}{r^2} \left(2 m E + \frac{2 m \alpha}{r} - \frac{M^2}{r^2}\right)^{-1/2} = \cos^{-1} \left(\frac{M/r - m \alpha/M}{\sqrt{2 m E + m^2 \alpha^2 / M^2}}\right) + C $$

You can use $$ \frac{d}{dr} \cos^{-1} \left(f\left(r\right)\right) = -\left[1-\left(f\left(r\right)\right)^2\right]^{-1/2} \frac{df}{dr} $$ with $$ f\left(r\right) = \frac{M/r - m \alpha/M}{\sqrt{2 m E + m^2 \alpha^2 / M^2}} $$ to show that $$ \int dr \frac{M / r^2}{\sqrt{2 m E + 2 m \alpha / r - M^2 / r^2}} = \cos^{-1} \left(\frac{M/r - m \alpha/M}{\sqrt{2 m E + m^2 \alpha^2 / M^2}}\right) + C $$

I think you can use $$ \frac{d}{dr} \cos^{-1} \left(f\left(r\right)\right) = -\left[1-\left(f\left(r\right)\right)^2\right]^{-1/2} \frac{df}{dr} $$ with $$ f\left(r\right) = \frac{M/r - m \alpha/M}{2 m E + m^2 \alpha^2 / M^2} $$ to show that $$ \int dr \frac{M}{r^2} \left(2 m E + \frac{2 m \alpha}{r} - \frac{M^2}{r^2}\right)^{-1/2} = \cos^{-1} \left(\frac{M/r - m \alpha/M}{2 m E + m^2 \alpha^2 / M^2}\right) + C $$

You can use $$ \frac{d}{dr} \cos^{-1} \left(f\left(r\right)\right) = -\left[1-\left(f\left(r\right)\right)^2\right]^{-1/2} \frac{df}{dr} $$ with $$ f\left(r\right) = \frac{M/r - m \alpha/M}{\sqrt{2 m E + m^2 \alpha^2 / M^2}} $$ to show that $$ \int dr \frac{M}{r^2} \left(2 m E + \frac{2 m \alpha}{r} - \frac{M^2}{r^2}\right)^{-1/2} = \cos^{-1} \left(\frac{M/r - m \alpha/M}{\sqrt{2 m E + m^2 \alpha^2 / M^2}}\right) + C $$