The integrals below involve `x^2-a^2` where `x^2>a^2`
1) `int 1/(x^2-a^2) dx = 1/(2a) ln ((x-a)/(x+a))`
OR `= -1/a coth^-1(x/a)`
2) `int x/(x^2-a^2) dx = 1/2 ln (x^2-a^2)`
3) `int x^2/(x^2-a^2) dx = x+a/2 ln ((x-a)/(x+a))`
4) `int x^3/(x^2-a^2) dx = x^2/2+a^2/2 ln (x^2-a^2)`
5) `int 1/(x(x^2-a^2)) dx = 1/(2a^2) ln ((x^2-a^2)/x^2)`
6) `int 1/(x^2(x^2-a^2)) dx = 1/(a^2x)+1/(2a^3) ln ((x-a)/(x+a))`
7) `int 1/(x^3(x^2-a^2)) dx = 1/(2a^2x^2)-1/(2a^4) ln (x^2/(x^2-a^2))`
8) `int 1/(x^2-a^2)^2 dx = (-x)/(2a^2(x^2-a^2))-1/(4a^3) ln ((x-a)/(x+a))`
9) `int x/(x^2-a^2)^2 dx = (-1)/(2(x^2-a^2))`
10) `int x^2/(x^2-a^2)^2 dx = (-x)/(2(x^2-a^2))+1/(4a) ln ((x-a)/(x+a))`
11) `int x^3/(x^2-a^2)^2 dx = (-a^2)/(2(x^2-a^2))+1/2 ln (x^2-a^2)`
12) `int 1/(x(x^2-a^2)^2) dx = (-1)/(2a^2(x^2-a^2))+1/(2a^4) ln (x^2/ (x^2-a^2))`
13) `int 1/(x^2(x^2-a^2)^2) dx = -1/(a^4x)-x/(2a^4(x^2-a^2))-3/(4a^5) ln ((x-a)/(x+a))`
14) `int 1/(x^3(x^2-a^2)^2) dx = -1/(2a^4x^2)-1/(2a^4(x^2-a^2))+1/a^6 ln (x^2/(x^2-a^2))`
15) `int 1/(x^2-a^2)^n dx = (-x)/(2(n-1)a^2(x^2-a^2)^(n-1))-(2n-3)/((2n-2)a^2) int 1/(x^2-a^2)^(n-1) dx`
16) `int x/(x^2-a^2)^n dx = (-1)/(2(n-1)(x^2-a^2)^(n-1))`
17) `int 1/(x(x^2-a^2)^n) dx = (-1)/(2(n-1)a^2(x^2-a^2)^(n-1))-1/a^2 int 1/(x(x^2-a^2)^(n-1)) dx`
18) `int x^m/(x^2-a^2)^n dx = int x^(m-2)/(x^2-a^2)^(n-1) dx +a^2 int x^(m-2)/(x^2-a^2)^n dx`
19) `int 1/(x^m(x^2-a^2)^n) dx = 1/a^2 int 1/(x^(m-2)(x^2-a^2)^n) dx -1/a^2 int 1/(x^m(x^2-a^2)^(n-1)) dx`