The integrals below involve `x^n+-a^n`
1) `int 1/(x(x^n+a^n)) dx = 1/(na^n) ln (x^n/(x^n+a^n))`
2) `int x^(n-1)/(x^n+a^n) dx = 1/n ln (x^n+a^n)`
3) `int x^m/(x^n+a^n)^r dx = int x^(m-n)/(x^n+a^n)^(r-1) dx-a^n intx^(m-n)/(x^n+a^n)^r dx`
4) `int 1/(x^m(x^n+a^n)^r ) dx = 1/a^n int1/(x^m(x^n+a^n)^(r-1)) dx-1/a^n int 1/(x^(m-n)(x^n+a^n)^r) dx`
5) `int 1/(xsqrt(x^n+a^n)) dx = 1/(nsqrt(a^n)) ln ((sqrt(x^n+a^n)-sqrt(a^n))/(sqrt(x^n+a^n)+sqrt(a^n)))`
6) `int 1/(x(x^n-a^n)) dx = 1/(na^n) ln ((x^n-a^n)/x^n)`
7) `int x^(n-1)/(x^n-a^n) dx = 1/n ln (x^n-a^n)`
8) `int x^m/(x^n-a^n)^r dx = a^n intx^(m-n)/(x^n-a^n)^r dx+int x^(m-n)/(x^n-a^n)^(r-1) dx`
9) `int 1/(x^m(x^n-a^n)^r) dx = 1/a^nint 1/(x^(m-n)(x^n-a^n)^r) dx-1/a^nint 1/(x^m(x^n-a^n)^(r-1)) dx`
10) `int 1/(xsqrt(x^n-a^n)) dx = 2/(nsqrt(a^n)) cos^-1sqrt(a^n/x^n`
11) `int x^(p-1)/(x^(2m)+a^(2m)) dx = 1/(ma^(2m-p))sum_(k=1)^msin(((2k-1)ppi)/(2m)) tan^-1((x+acos[(2k-1)pi/(2m)])/(asin[(2k-1)pi/(2m]]))-1/(2ma^(2m-p)) sum_(k=1)^mcos(((2k-1)ppi)/(2m)) ln (x^2+2ax cos(((2k-1)pi)/(2m))+a^2)`
**where `0ltpleq2m`
12) `int x^(p-1)/(x^(2m)-a^(2m)) dx = 1/(2ma^(2m-p)) sum_(k=1)^(m-1) cos((kppi)/m) ln[x^2-2ax cos((kpi)/m)+a^2]-1/(ma^(2m-p))sum_(k=1)^(m-1) sin ((kppi)/m) tan^-1((x-a cos((kpi)/m))/(a sin((kpi)/m)))+1/(2ma^(2m-p))[ln(x-a)+(-1)^pln(x+a)]`
**where `0ltpleq2m`
13) `int x^(p-1)/(x^(2m+1)+a^(2m+1)) dx = (2(-1)^(p-1))/((2m+1)a^(2m-p+1))sum_(k=1)^msin((2kppi)/(2m+1))tan^-1((x+a cos[(2kpi)/(2m+1)])/(a sin[(2kpi)/(2m+1)]))-(-1)^(p-1)/((2m+1)a^(2m-p+1))sum_(k=1)^mcos((2kppi)/(2m+1))ln(x^2+2ax cos((2kpi)/(2m+1))+a^2)+((-1)^(p-1)ln(x+a))/((2m+1)a^(2m-p+1)`
**where `0ltpleq2m+1`
14) `int x^(p-1)/(x^(2m+1)-a^(2m+1)) dx = (-2)/((2m+1)a^(2m-p+1))sum_(k=1)^msin((2kppi)/(2m+1))tan^-1((x-a cos[(2kpi)/(2m+1)])/(a sin [(2kpi)/(2m+1)]))+1/((2m+1)a^(2m-p+1))sum_(k=1)^mcos((2kppi)/(2m+1))ln[x^2-2ax cos((2kpi)/(2m+1))+a^2]+ln(x-a)/((2m+1)a^(2m-p+1))`
**where `0ltpleq2m+1`