text: how do you integrate 1/(x^7+1)?   while not conceptually difficult, this is computationally problematic. the problem is that you have to factor x^7 +1. essentially, we are finding the 7th roots of -1, so the roots are -1, cos(pi/7)+-i sin(pi/7) cos(3pi/7)+-i sin(3pi/7) cos(5pi/7)+-i sin(5pi/7). the corresponding factors of x^7 +1 are x+1 x^2 - 2cos(pi/7) x +1 x^2 -2cos(3pi/7) x +1 x^2 -2cos(5pi/7) x +1. each of these can be written in the form (x-cos(kpi/7) )^2 +sin^2 (kpi/7).  the next step is to find the partial fraction decomposition of 1/(x^7 +1) using these factors. i would recommend a form looking like a/(x+1) for the first term and [b(x-cos(kpi/7))+c]/ [(x-cos(kpi/7)^2 +sin^2(kpi/7)] for the other terms. this will allow easier integration once the coefficients are found.