f(0) + x f'(0) + 1/2 x^2 (f''(0) + (-1 + 2 gamma + log(2 π)) f'(0)) + 1/24 x^3 ((-9 + 12 gamma ^2 - 2 π^2 + 3 log^2(2 π) + 12 gamma (log(2 π) - 2) - 6 log(2 π)) f'(0) + 4 (f^(3)(0) + 3 (-1 + 2 gamma + log(2 π)) f''(0))) + 1/144 x^4 (f'(0) (24 gamma ^3 - 6 gamma (3 + 2 π^2 - 3 log^2(2 π) + 12 log(2 π)) + π^2 (14 - 6 log(2 π)) + 36 gamma ^2 (log(2 π) - 3) + 3 (11 + log^3(2 π) - 3 log^2(2 π) - 9 log(2 π) - 8 polygamma(2, 1))) + 6 (f^(4)(0) + 6 f^(3)(0) (-1 + 2 gamma + log(2 π)) + 2 (-3 + 12 gamma ^2 - π^2 + 3 log^2(2 π) + 6 gamma (-3 + log(4) + 2 log(π)) - 6 log(2 π)) f''(0))) + O(x^5) (Taylor series)

f(exp(1/4 x (-3 x + 2 (x - 2) log(x) + 4 + 2 log(π) + log(4))) ((e^(1/12) x^(5/12))/(A sqrt(2 π)) - (e^(1/12) (1/x)^(7/12))/(12 (A sqrt(2 π))) - (e^(1/12) (1/x)^(19/12))/(1440 (A sqrt(2 π))) + (157 e^(1/12) (1/x)^(31/12))/(51840 A sqrt(2 π)) + (65911 e^(1/12) (1/x)^(43/12))/(87091200 A sqrt(2 π)) - (918227 e^(1/12) (1/x)^(55/12))/(1045094400 (A sqrt(2 π))) + O((1/x)^(61/12))))

d/dx(f(G(x))) = 1/2 G(x) (-2 x + 2 (x - 1) polygamma(0, x) + 1 + log(2 π)) f'(G(x))