I need to evaluate a number of integrals over a unit circle, whereby the integrands are very long fourth order tensor components which are functions of phi but also of other tensor components, i.e. i have a tensor valued function of phi and another tensor:
B1111 (phi,C1312,etc.) = int_0^2*pi[((4*cos(phi)^2*((-1*C1312^2 + C1212*C1313)* cos(phi)^4 + A*(2*C1313*C2212 - 2*C1312*C2213 - 2*C1312*C2312 + 2*C1212*C2313)*cos(phi)^3*sin(phi) + A^3*(-2*C2213*C2223 - 2*C2223*C2312 + 2*C2222*C2313 + 2*C2212*C2323)*cos(phi)sin(phi)^3 + A^4(-1*C2223^2 + C2222*C2323)*sin(phi)^4))/(((C1313*cos(phi)^2 + A^2*C2323*sin(phi)^2)*((-1*C1112^2 + C1111*C1212)*cos(phi)^4 - (C1312*cos(phi)^2 + A^2*C2223*sin(phi)^2)*((-1*C1112*C1113 + C1111*C1312)* cos(phi)^4 + A*(C1113*(-1*C1122 - 1*C1212) + C1112*(-1*C1123 + C1312) + C1111*(C2213 + C2312))*cos(phi)^3*sin(phi) + A^4*(C1212*C2223 - 1*C2212*C2312)sin(phi)^4) - 1(C1113*cos(phi)^2 + A*(C1123 + C1312)*cos(phi)*sin(phi) + A^2*C2312*sin(phi)^2)*((C1113*C1212 - 1*C1112*C1312)cos(phi)^4 + A(C1123*C1212 - 1*C1122*C1312 + 2*C1113*C2212 - 1*C1112.*C2213 - 1*C1112*C2312)*cos(phi)^3*sin(phi) + A^3*((C1123 + C1312)*2 - 1*2*C2223 - 1*2*C2223 + C2212*(-1*C2213 + C2312))*cos(phi)*sin(phi)^3)))]dphi
Such integrals cannot be solved analytically, so I do it numerically. However, certain tensor arguments containing zero valued components lead to singularities in the integrals. I can get close to zero, but I'm interested in an integration solution at which the aforementioned components are exactly zero. An option would be to keep those components that are zero in symbolic form, integrate numerically and then see what the limit of the resulting sum is when the symbolic form components go to zero. There seems to be work on mixed numerical-symbolic integration methods, but I'm not inside those methods, and haven't thought yet of any other approach to solving this either. Can someone help?