Approximate integration

1. The problem statement, all variables and given/known data
This is technically an E&M question, but I've reduced it to a calculus problem. Basically I have to evaluate:

[tex]B_0(\int_{-H_{max}}^{H_{max}}{tanh(\frac{H+H_c}{H_0})dH - \int_{-H_{max}}^{H_{max}}{tanh(\frac{H-H_c}{H_0})dH)[/tex]

Where [tex]H_{max}>>H_C, H_0[/tex].


2. Relevant equations



3. The attempt at a solution

I'm looking at this and I have no idea how to go about approximating this integral...I suppose I could just brute force the integrals and keep all the H_max and stuff, and then later see if i can approximate something...but the expressions are really quite long and I'd like to avoid that if I can. Is there a way?