Integration using inverse trig and hyperbolic functions
Given that \(y = \operatorname { artanh } ( \cos x )\)
show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x$$
Hence find the exact value of
$$\int _ { 0 } ^ { \frac { \pi } { 6 } } \cos x \operatorname { artanh } ( \cos x ) d x$$
giving your answer in the form \(a \ln ( b + c \sqrt { 3 } ) + d \pi\), where \(a , b , c\) and \(d\) are rational numbers to be found.