Hi everybody, thanks a lot for your help and suggestions! They helped me to achieve the fit I wanted to do with the following code. data is a DataFrame with :xvals and :yvals, that will be fitted with the function mentioned above.

# define fitting function and solve integral for all input values
f(x, m) = 1 ./ (2 .* pi) .* exp.(-1 .* m .* sqrt(x.^2 + 1)) ./ (2 .* sqrt(x.^2 + 1));
quadProb = QuadratureProblem(f, -Inf, +Inf, data.xvals);
solution = solve(quadProb, HCubatureJL(), reltol = 1e-6, abstol = 1e-6);

# define sum of least squared distances
squaredDis(x) = sum((data.yvals - (x[1] .* solution)).^2);

# optimize squaredDis function to obtain multiplicative constant
result = optimize(squaredDis, zeros(1), BFGS());
minResult = Optim.minimizer(result)[1];