In Apostol, the case that the partial fraction is presented as $\frac{C}{(u^2 + a^2)^m}$ can be done with the following reduction formula:
$$\int \frac{du}{(u^2 + a^2)^m} = \frac{1}{2a^2(m-1)} \frac{u}{(u^2 + a^2)^{m-1}} + \frac{2m - 3}{2a^2(m-1)} \int \frac{du}{(u^2 + a^2)^{m-1}}$$
The author states that this can be shown using integration by parts. I cannot figure how - i tried taking $dv = (u^2 + a^2)^{m-1}$ or $dv = 1 \Rightarrow v = u$. In case the $m = 2$, the integral can be done with trigonometric substitution. However, this general case is not as clear. Can someone show how this is obtained with integration by parts?
I suggest that you write $$ \frac{1}{(u^2+a^2)^{m-1}} = u\cdot\frac{u}{(u^2+a^2)^m}+\frac{a^2}{(u^2+a^2)^m}, $$ and integrate by parts in the first term in the right-hand side, and then rearrange your terms.