I'm trying to integrate the following function analytically (wrt $x$), but with some software (preferably maple): $$ f(x) = \frac{1}{ax^2+bx+c} $$
I look at wikis integration table which states:
For my problem: $(4ac-b^2)<0$, meaning the $\ln()$ integral should work. And indeed, it does work (I integrated it by hand). However, comparing the results to the $\operatorname{artanh}()$ integral did not work for my problem. I did this because some math book (Merziger, Mühlbach et al. 2013) said that the $\ln()$ integral is equal to the $\operatorname{artanh}()$ as long as $(4ac-b^2)<0$ (which, as I said, is true for my problem).
The thing is, I'm trying to integrate the functions using Maple or WolframAlpha. However both these programs will always result in the $\operatorname{artanh}$ integral, which yields the wrong results for my problem. This makes sense, as in wiki, the condition next to the $\operatorname{artanh}()$ integral is not met in my problem. I discovered the wiki integration table only recently, but unfortunately I'm too dumb to compute $\operatorname{arcoth}()$, as it's not implemented in python. (I'm sure I'd find some way to compute it though, if necessary).
Can someone help me find a solution? E.g. is it possible to tell Maple to use the $ln()$ instead of $\operatorname{artanh}()$? (Or some other software?)
You can use the following identity to convert from $\tanh^{-1}x$ to $\log x$ : $$\tanh^{-1}x=\frac{1}{2}\log \left( \frac{1+x}{1-x}\right) $$