Problem

(a) Differentiate $x\,\operatorname{arcosh}(5x)$ with respect to $x$

(b) Hence, or otherwise, show that

$$\int_{1/4}^{3/5}\operatorname{arcosh}(5x)\,dx =\frac{3}{20}-\frac{2\sqrt2}{5}+\ln(p+q\sqrt2)^{\,k}-\frac14\ln r$$

where $p$, $q$, $r$ and $k$ are rational numbers to be determined.