It's been a while but I think it's: ln(-a/b*cos(bx)+ 1/2cx^2 + dx + k)

The resolution is attached. I belive it is right.

Sean, I think that it is not integrable analytically. You can't express x in term of u in your development so that your method doesn't work.

For c = 0, the integral can be evaluated analytically, here is the result (attached).

I think that you can use Maple or MATLAB to check you results. These types of software are unable to compute your integral in its general form. Hope this helps a bit!

It cannot be integrated by known programs. Even if you do the transformation:

c*x+d=y=>x=(y-d)/c

the integral is still non integrable.

You can expand first the integrand in Taylor series around a suitable point( which?) and then integrate term by term, it is not a difficult task.

From where has that integral arisen?

The "^-1" stands for the inverse function of asin(bx) + cx+d or the ratio 1/(asin(bx)+cx+d) ?

I send as an attachment WolframalphaPro' solution only around 0, which must be multiplied by a^(-1). The solution uses an integration of a geometry serie , wich converges only around 0.

I send as an attachment a solution (as a series) of the integral for big x.

It might be helpful to look at the saddle points of the integrand. Rescaling x and taking out a constant factor, the integrand becomes 1/(A*sin(X)+X+D)=:exp(f(X)). Thus, the saddle points are given by the solutions of D(f)(X)=0, i.e. of cos(X)=-1/A. For abs(A) > 1, the saddlepoints are real, otherwise, they are complex. Observe that there are infinitely many saddlepoints. For large abs(A), the saddlepoints are close to the zeroes of the cosine of X. Thus, by summing the saddlepoint contributions, one obtains an infinite series for the integral. This approximation then can be improved using corrections.

Interestingly, one can also obtain a series expansion of the integral for small A in terms of logarithms, powers and Ci and Si functions.

Of course, all these series are only approximate, and not a closed form solution. However, by looking at the asymptotic regimes - or at the limiting cases, one gets a feeling how the integral behaves semiquantitatively.