There is an integral having a following form : $\int f(x) f(ax-b) dx$ . Is there any general way/substitution that makes this kind of integrals easy to solve? Particularly, my functions are shifted hyperbolic secants: $\int_{-\infty}^{+\infty} sech(x) sech(ax-b) dx$. I have tried to calculate it with the standard methods: integration by parts but it did not work out. I don't see a reason (and possibility) to use more advanced methods like Feynaman technique(yes, I am desperate) , since it leads to even more complicated integrals. i dont know what substitution would be useful here, since the difference in arguments of sech(x) makes such an approach ineffective. The substitutions like $t = e^{ax}$ lead to, e.g. following integral: $$\int_{-\infty}^{+\infty} \frac{2a t^{1/a}}{(t^{2/a}+1)(t^{-2}e^b + e^{-b})} dt $$, that is still hard to solve. Can somehow the residue theorem be applied here ? In this case I have a problem with determination of poles and their degrees.
I will appreciate for any suggestions.