Difficulty obtaining anti-derivative

Question

This was evidently a question in a recent Calculus 1 exam paper.

$$ \int_{0}^{0.5} \frac{(16x^2-8x+1)\exp(\cos(\pi x)}{\exp(\cos(\pi x)+ \exp(\sin(\pi x)}dx $$

I have several attempt to find an analytical expression for the anti-derivative if the integrand but fail on every occasion. My only resort now is to calculate it numerically. Doing so results in a value 0.11161. Can any one suggest how to obtain the anti-derivative of the integrand or offer any constructive advice ?

Answer 1

\begin{align} & \int_{0}^{0.5} \frac{(16x^2-8x+1)\exp(\cos\pi x)}{\exp(\cos\pi x)+ \exp(\sin\pi x)}dx \\\overset{x\to0.5-x}=& \int_{0}^{0.5} \frac{(16x^2-8x+1)\exp(\sin\pi x)}{\exp(\sin\pi x)+ \exp(\cos\pi x)}dx \\ =& \ \frac12 \int_{0}^{0.5} (16x^2-8x+1)dx=\frac1{12} \end{align}