>>843
∫ 1/(x+1) dx = log|x+1|.

∫ 1/(x^2 +1) dx = arctan(x).

∫ 1/(x^3 +1) dx = (1/2)log|x+1| -(1/6)log|x^3+1| + (1/√3)arctan((2x-1)/√3).

∫ 1/(x^4 +1) dx = (A/8){log|(x^2+Ax+1)/(x^2-Ax+1)| + 2arctan(Ax+1) + 2arctan(Ax-1)},　A=√2.

∫ 1/(x^5 +1) dx = (1/4)log|x+1| - (1/20)log|x^5 +1| + (1/4√5)log|[2x^2+(√5-1)x+2]/[2x^2-(√5+1)x+2]|
　　　　　　　+ (1/10)B(+)arctan([4x-1+√5]/B(+)) + (1/10)B(-)arctan([4x-1-√5]/B(-)),　B(±)=√(10±2√5).

∫ 1/(x^6 +1) dx = (C/12)log|(x^2+Cx+1)/(x^2-Cx+1)| + (1/6)arctan(x/[1-x^2]) + (1/3)arctan(x),　C(±)=√3.

∫ 1/(x^8 +1) dx = (1/16)D(+)log|(x^2+D(+)x+1)/(x^2-D(+)x+1)| + (1/16)D(-)log|(x^2+D(-)x+1)/(x^2-D(-)x+1)|
　　　　　　　+ (1/8)D(+)arctan(D(+)x/[1-x^2]) + (1/8)D(-)arctan(D(-)x/[1-x^2]),　D(±)=√(2±√2).