The integrals below involve `x^2+a^2`
1) `int 1/(x^2+a^2) dx = 1/a tan^-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 tan^-1 (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/(x^2+a^2))`
6) `int 1/(x^2(x^2+a^2)) dx = -1/(a^2x)-1/a^3 tan^-1 (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) `int1/(x^2+a^2)^2 dx = x/(2a^2(x^2+a^2))+1/(2a^3) tan^-1 (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/(2a) tan^-1 (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/(2a^5) tan^-1 (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(x^2+a^2)^(n-1)) dx -1/a^2 int 1/(x^(m-2)(x^2+a^2)^n) dx`