I'm currently trying to solve the integral: $$ I(\vec{a},\vec{b})=4\pi\int\limits_0^1\frac{\mathrm{d}u}{1-(\vec{a}u+\vec{b}(1-u))^2}, $$ but I can't seem to find a good starting point. I know that if $\vec{a}$ and $\vec{b}$ are scalars this integral would be easilly solved by a partial fraction decomposition (where the relation $x^2-y^2=(x-y)(x+y)$ is really helpful), yielding an answer in terms of the natural logarithm.
Right now however I am stuck with a vector relation, which doesn't make the calculations any easier. I tried the partial fraction decomposition, this doesn't make things easier. I also tried writing out the denominator, but this also isn't that helpful.
I was wondering if there were general ways to tackle these kind of problems and/or if there are any tips for this integral ?
The vectors $\vec{a}$ and $\vec{b}$ are 3D, this can for example be seen by the fact that the factor $4\pi$ follows from the 3D solid angle $$ 4\pi=\int\limits_0^{2\pi}\mathrm{d}\varphi\int\limits_{0}^\pi\sin(\theta)\mathrm{d}\theta. $$