D’Alembert’s formula

Recall that the general Cauchy problem (or IVP) for the wave equation on $\mathbb{R}$ is
$$u_{t\, t}=c^2u_{x\, x},u(x,0)=f(x),\text{ and }\,u_t(x,0)=g(x)\text{ for }x\in \mathbb{R}.$$
This can be solved using d'Alembert's formula:
$$u(x,t)=\frac{1}{2}(f(x+c t)+f(x-c t))+\frac{1}{2c}\int _{x-c t}^{x+c t}g(s)ds.$$

Apply d'Alembert's formula to solve the wave problem
$$u_{t\, t}=c^2u_{x\, x},u(x,0)=e^{-x^2}\cos(10x),\text{ and }\,u_t(x,0)=e^{-x^2}\sin(10x)\text{ for }x\in \mathbb{R}.$$

Here's what I have:

In this problem $f(x) = e^{-x^2} \cos(10x)$ and $g(x) = e^{-x^2}\sin(10x)$. Applying the formula,

$$u(x,t) = \frac{1}{2} \left[ e^{-(x+ct)^2}\cos\left(10(x+ct)\right) + e^{-(x-ct)^2}\cos\left(10(x-ct)\right) \right]
+\frac{1}{2c}\int_{x-ct}^{x+ct} e^{-s^2}\sin(10s) \, ds.$$

I am slightly unsure as to whether there is a further step, as I do not know if there is a way to simplify the first half of the solution, and Mathematica tells me the second half evaluates to an expression in terms of the "erfi" function. What do you guys think?