I'm following a Electrodynamics course and I'm currently stack in the following calculation:
$\int{ \frac{ \vec{n} \times (\vec{n}-\vec{\beta}(t)) \times \vec{\beta'}(t) }{(1-\vec{n} \cdot \vec{\beta}(t) )^2 } dt}$
that the book calculates as
$\int{ \frac{ \vec{n} \times (\vec{n}-\vec{\beta}(t)) \times \vec{\beta'}(t) }{(1-\vec{n} \cdot \vec{\beta}(t) )^2 } dt}= \frac{ \vec{n} \times \vec{n} \times \vec{\beta}(t) }{(1-\vec{n} \cdot \vec{\beta}(t)) }$
I have been unable to reproduce that calculus without proving the reverse ( that the derivative of the right side is the integrand of the left side). I would like to calculate the integral without using a coordinate basis if possible. Some details of the notation:
$\frac{d \vec{\beta}(t)}{d t} = \vec{\beta'}(t)$
$\vec{n}$ is a constant vector that does not depend of the integration variable t.
This is from The Physics of Synchrotron Radiation by Albert Hofmann, section 2.8. The Fourier transform of the radiation field, page 36.