From the second line, you have in each integral, but they were in terms of u.

I think its just a typo.

Sorry, but I did not check the long division step.

but for the integral

You can:

1- Rewrite and use partial fraction method.

2- Use a "hyperbolic" substitution.

3- evaluate it as an infinite series, since where .

Also for the original integral itself, you can do it as an infinite series.

its easy to prove that where .