integrate by parts
f = t, df = dt,
dg = sinh(mt)dt
g = cosh(mt)/m
put them together
tcosh(mt)/m - /m * ∫cosh(mt)dt
u = mt, du = mdt
tcosh(mt)/m - /m^2 * ∫cosh(u)dt
integral of cosh(u) = sinh(u)
tcosh(mt)/m - sinh(u)/m^2 + c1
substitute back in the u (mt)
tcosh(mt)/m - sinh(mt)/m^2 + c1
You can simplify this
((mt*cosh(mt)-sinh(mt)))/m^2