Integral sqrt(1+x^2)

# Integral sqrt(1+x^2)

Indefinite integral
integral sqrt(1 + x^2) dx = 1/2 (sqrt(x^2 + 1) x + sinh^(-1)(x)) + constant
Alternate form of the integral
1/2 sqrt(x^2 + 1) x + 1/2 log(sqrt(x^2 + 1) + x) + constant
Expanded form of the integral
1/2 sqrt(x^2 + 1) x + 1/2 sinh^(-1)(x) + constant
Series expansion of the integral at x=0
x + x^3/6 – x^5/40 + x^7/112 + O(x^9)
(Taylor series)
Series expansion of the integral at x=-i
(-1)^(floor(((3 pi)/2 – arg(x + i))/(2 pi))) (((-1)^(1/4) sqrt(x + i))/sqrt(2) – ((-1)^(3/4) (x + i)^(3/2))/(12 sqrt(2)) – (3 (-1)^(1/4) (x + i)^(5/2))/(160 sqrt(2)) + O((x + i)^(7/2))) + (-1/4 i (2 sqrt(2) sqrt(1 – i x) + pi) + (5 sqrt(1 – i x) (x + i))/(4 sqrt(2)) + (7 i sqrt(1 – i x) (x + i)^2)/(32 sqrt(2)) + (3 sqrt(1 – i x) (x + i)^3)/(128 sqrt(2)) + O((x + i)^4))
Series expansion of the integral at x=i
(-1)^(floor((pi – 2 arg(x – i))/(4 pi))) ((1/2 – i/2) sqrt(x – i) + (1/24 + i/24) (x – i)^(3/2) – (3/320 – (3 i)/320) (x – i)^(5/2) + O((x – i)^(7/2))) + (1/4 i (2 sqrt(2 i x + 2) + pi) + (5 sqrt(i x + 1) (x – i))/(4 sqrt(2)) – (7 i sqrt(i x + 1) (x – i)^2)/(32 sqrt(2)) + (3 sqrt(i x + 1) (x – i)^3)/(128 sqrt(2)) + O((x – i)^4))