Try multiplying top and bottom by
Here we are again -- let's get cracking on some of these hard integrals again!
Number 358 in my source work:
which has to be demonstrated to be equal to:
I tried a Weierstrass substitution: but ended up with the horrible:
which doesn't look like it's going in the right direction.
I also tried integration by parts, setting both and to but got no further than:
... which looks horrible and I haven't had the courage to take it further down that route.
I tried a few substitutions but they didn't look promising either.
Any hints as to how this may be attacked?
from which I haven't got a clue as to how to get to:
Presumably there are half-angle substitutions to be used, but one supposes there's an easier way to get there without going through this admittedly perfectly legitimate solution.
However, the form of the solution suggests that there may be integrations of secants to be found, as:
but I haven't been able to find it.
The reason I am going through this particular exercise is specifically to validate the results in this source work. Some of them appear to be wrong, and I want to make sure I know precisely where the errors are before I can recommend it as a learning aid.