integrate 1 divided by the product of (a²+x²) and(b²+x²)

Question

integrate

1 divided by

(a²+x²)(b²+x²)

Answer 1

Create partial fractions.

1/{(a^2 + x^2)(b^2 + x^2)} = (1/(a^2+b^2)){1/(b^2 + x^2) - 1/(a^2 + x^2)}

So, int( 1/{(a^2 + x^2)(b^2 + x^2)} ) dx = (1/(a^2+b^2)){(1/b).atan(x/b) - (1/a).atan(a/x))}