I have relatively simple integral but I could not figure out how to solve. It is $$ \int_{0}^{1}dz \frac{(1-z)(1-z^2)A^2}{B^2z+(1-z)^2A^2} $$ $$B>>A$$ EDIT: You should use an approximation to you consider B>>A (B is much bigger than A).
Solution should seem like $$ log(\frac{B^2}{A^2})-\frac{7}{6} $$