Possible intermediate steps:
integral 1/(sqrt(1-x^2) (1+x^2)) dx
For the integrand,1/(sqrt(1-x^2) (x^2+1)) substitute x = sin(u) and dx = cos(u) du.Then sqrt(1-x^2) = sqrt(1-sin^2(u)) = cos(u) and u = sin^(-1)(x):
= integral 1/(sin^2(u)+1) du
For the integrand 1/(sin^2(u)+1),substitute s = tan(u/2) and ds = 1/2 sec^2(u/2) du.Then transform the integrand using the substitutions sin(u) = (2 s)/(s^2+1),cos(u) = (1-s^2)/(s^2+1) and du = (2 ds)/(s^2+1):
= integral 2/((s^2+1) ((4 s^2)/(s^2+1)^2+1)) ds
Simplify the integrand 2/((s^2+1) ((4 s^2)/(s^2+1)^2+1)) to get (2 (s^2+1))/(s^4+6 s^2+1):
= integral (2 (s^2+1))/(s^4+6 s^2+1) ds
Factor out constants:
= 2 integral (s^2+1)/(s^4+6 s^2+1) ds
For the integrand (s^2+1)/(s^4+6 s^2+1),factor the denominator into quadratic irreducible terms:
= 2 integral (s^2+1)/((s^2-2 sqrt(2)+3) (s^2+2 sqrt(2)+3)) ds
For the integrand (s^2+1)/((s^2-2 sqrt(2)+3) (s^2+2 sqrt(2)+3)),use partial fractions:
= 2 integral ((1+sqrt(2))/(2 sqrt(2) (s^2+2 sqrt(2)+3))+(1-sqrt(2))/(2 sqrt(2) (-s^2+2 sqrt(2)-3))) ds
Integrate the sum term by term and factor out constants:
= 1/sqrt(2)-1 integral 1/(-s^2+2 sqrt(2)-3) ds+1+1/sqrt(2) integral 1/(s^2+2 sqrt(2)+3) ds
The integral of 1/(-s^2+2 sqrt(2)-3) is -(tan^(-1)(s/sqrt(3-2 sqrt(2))))/sqrt(3-2 sqrt(2)):
= 1+1/sqrt(2) integral 1/(s^2+2 sqrt(2)+3) ds-(tan^(-1)(s/sqrt(3-2 sqrt(2))))/sqrt(2 (3-2 sqrt(2)))+(tan^(-1)(s/sqrt(3-2 sqrt(2))))/sqrt(3-2 sqrt(2))
The integral of 1/(s^2+2 sqrt(2)+3) is (tan^(-1)(s/sqrt(3+2 sqrt(2))))/sqrt(3+2 sqrt(2)):
= -(tan^(-1)(s/sqrt(3-2 sqrt(2))))/sqrt(2 (3-2 sqrt(2)))+(tan^(-1)(s/sqrt(3-2 sqrt(2))))/sqrt(3-2 sqrt(2))+(tan^(-1)(s/sqrt(3+2 sqrt(2))))/sqrt(2 (3+2 sqrt(2)))+(tan^(-1)(s/sqrt(3+2 sqrt(2))))/sqrt(3+2 sqrt(2))+constant
Substitute back for s = tan(u/2):
= (tan^(-1)((tan(u/2))/sqrt(3-2 sqrt(2))))/sqrt(3-2 sqrt(2))-(tan^(-1)((tan(u/2))/sqrt(3-2 sqrt(2))))/sqrt(6-4 sqrt(2))+(tan^(-1)((tan(u/2))/sqrt(3+2 sqrt(2))))/sqrt(6+4 sqrt(2))+(tan^(-1)((tan(u/2))/sqrt(3+2 sqrt(2))))/sqrt(3+2 sqrt(2))+constant
Substitute back for u = sin^(-1)(x):
= (tan^(-1)(x/(sqrt(3-2 sqrt(2)) (sqrt(1-x^2)+1))))/sqrt(3-2 sqrt(2))-(tan^(-1)(x/(sqrt(3-2 sqrt(2)) (sqrt(1-x^2)+1))))/sqrt(6-4 sqrt(2))+(tan^(-1)(x/(sqrt(3+2 sqrt(2)) (sqrt(1-x^2)+1))))/sqrt(6+4 sqrt(2))+(tan^(-1)(x/(sqrt(3+2 sqrt(2)) (sqrt(1-x^2)+1))))/sqrt(3+2 sqrt(2))+constant
Factor the answer a different way:
= 1/2 ((2+2 sqrt(2)-sqrt(6+4 sqrt(2))) tan^(-1)(x/(sqrt(3-2 sqrt(2)) (sqrt(1-x^2)+1)))+(-2+2 sqrt(2)+sqrt(6-4 sqrt(2))) tan^(-1)(x/(sqrt(3+2 sqrt(2)) (sqrt(1-x^2)+1))))+constant
Which is equivalent for restricted x values to:
= (tan^(-1)((sqrt(2) x)/sqrt(1-x^2)))/sqrt(2)+constant