I'll use parametric values to avoid handling large integers all the way through:
#int dx/(m^2-n^2x^2) = int dx/((m+nx)(m-nx))#
Dividing in partial fractions:
#1/((m+nx)(m-nx)) = A/(m+nx) +B/(m-nx) = (A(m-nx)+B(m+nx))/((m+nx)(m-nx)) = (m(A+B)-n(A-B)x)/((m+nx)(m-nx))#
Giving: #A=B=1/(2m)#, so:
#int dx/(m^2-n^2x^2) = 1/(2m) int dx/(m+nx) + 1/(2m) int dx/(m-nx)#
#int dx/(m^2-n^2x^2) = 1/(2mn) int (d(m+nx))/(m+nx) - 1/(2mn) int (d(m-nx))/(m-nx)#
#int dx/(m^2-n^2x^2) = 1/(2mn) ln (abs(m+nx) / abs (m-nx)) + C#
Substituting:
#m = 100#
#n = sqrt(1234)#
we have:
#int dx/(10000+1234x^2) = 1/(200sqrt(1234))ln ( abs(100+xsqrt(1234))/abs(100-xsqrt(1234)))+C#
