Question: I cannot figure out how to solve this integral with the commonly known methods of integration. The different online integral calculators that I have tried fail to solve it.

$ \int \sin(a x) ((b-x)^2-c^2)^{-3/2} dx $

where *a, b* and *c* are constants.

Is there a solution? Thanks.

Answer: 
Unlike derivatives, not every elementary function has an elementary antiderivative. Elementary meaning polynomials, trigonometric functions, logarithmic functions, exponential functions and their inverse functions.

One example is $$\int e^{-x^2} dx$$

which cannot be represented by any elementary function. I am pretty certain your function cannot be represented in elementary terms.
The problems with your integral are the unknown parameters $a, b, c$.
Assuming they are all real, it's basically impossible to solve the general integral due to their unknown nature indeed.

Even the very trivial case in which $a = b = c = 1$ brings lots of problems.
There are special cases, for example:

**Special case 1: when $a = b = 1$, $c = 0$**

The solution reads
$$\frac{(x-1)^2 \sin (1) (-\text{Ci}(1-x))+(x-1)^2 \cos (1) \text{Si}(1-x)-\sin (x)-x \cos (x)+\cos (x)}{2 (x-1) \sqrt{(x-1)^2}}$$

Where special functions Integral Sine and Integral Cosine pop out.

**Special case2: $a = 1$, $b = c = 0$**

The solutions reads

$$-\frac{x^2 \text{Si}(x)+\sin (x)+x \cos (x)}{2 x \sqrt{x^2}}$$

It's always hard when there are too many unknown parameters in the integrand function.

In any case, the number of integrals without a solution is huge! You know, to differentiate is easy, but to integrate is art.