Integral of sqrt(1+x^2)/x

# Integral of sqrt(1+x^2)/x

Indefinite integral
integral sqrt(1 + x^2)/x dx = sqrt(x^2 + 1) – log(sqrt(x^2 + 1) + 1) + log(x) + constant
Series expansion of the integral at x=0
(log(x/2) + 1) + x^2/4 + O(x^4)
(Puiseux series)
Series expansion of the integral at x=-i
(sqrt(2) sqrt(1 – i x) – log(1 + sqrt(2) sqrt(1 – i x)) – (i pi)/2) + ((x + 2 i sqrt(2) sqrt(1 – i x) + 3 i) (x + i))/(2 (1 + sqrt(2) sqrt(1 – i x))) + ((-i sqrt(2) sqrt(1 – i x) x – 16 i x + 17 sqrt(2) sqrt(1 – i x) + 24) (x + i)^2)/(16 (1 + sqrt(2) sqrt(1 – i x))^2) + ((6 i x^2 – 129 sqrt(2) sqrt(1 – i x) x – 396 x – 321 i sqrt(2) sqrt(1 – i x) – 454 i) (x + i)^3)/(192 (1 + sqrt(2) sqrt(1 – i x))^3) + O((x + i)^4)
(generalized Puiseux series)
Series expansion of the integral at x=i
(sqrt(2) sqrt(1 + i x) – log(1 + sqrt(2) sqrt(1 + i x)) + (i pi)/2) + ((x – 2 i sqrt(2) sqrt(1 + i x) – 3 i) (x – i))/(2 (1 + sqrt(2) sqrt(1 + i x))) + ((i sqrt(2) sqrt(1 + i x) x + 16 i x + 17 sqrt(2) sqrt(1 + i x) + 24) (x – i)^2)/(16 (1 + sqrt(2) sqrt(1 + i x))^2) + ((-6 i x^2 – 129 sqrt(2) sqrt(1 + i x) x – 396 x + 321 i sqrt(2) sqrt(1 + i x) + 454 i) (x – i)^3)/(192 (1 + sqrt(2) sqrt(1 + i x))^3) + O((x – i)^4)
(generalized Puiseux series)
Series expansion of the integral at x=∞
x – 1/(2 x) + 1/(24 x^3) + O((1/x)^4)
(Laurent series)