Finding the Integral of A Composition of Logarithm, exp, and Trigometric Functions

Question

I have the function $$f(x)=\ln \left(e^{5+6\left(\sum_{k=0}^{100} \frac{\sin \left(k\pi x\right)}k \right)}+e \right)$$ One, is it possible to integrate this without "special functions" (functions like $|x|$ are OK, but not functions like $\mathbb{erf}(x)$)? If so, how? If not, why?

Answer 1

So, you want to have a closed form expression of $$I_p=\int \log\Big[e+\exp\left(5+6\sum_{k=0}^p \frac{\sin (\pi k x)}{k} \right) \Big]\,dx$$

$$I_0=\int\log \left(e+e^{5+6 \pi x}\right)\,dx=x-\frac{\text{Li}_2\left(-e^{6 \pi x+4}\right)}{6 \pi }$$ where already appears a special function.

$$I_1=\int\log \left(e+e^{5+6 (\pi x+\sin (\pi x))}\right)\,dx$$ does not show any antiderivative.