Annoying math compilation

The vertices of a 10-sided polygon are so joined to form all possible quadrilaterals such that none of the sides of the polygon coincide with that of the quadrilateral. Let n be the number of all such quadrilaterals. $\frac { n }{ \pi } \int _{ 0 }^{ n\pi }{ { e }^{ \cos { x } }\cos { \left( \sin { x } \right) } dx } = t$ Let $a$ , $b$ , $c$ , and $d$ be positive reals such that $a+b+c+d=\sqrt { t }$ . The maximum value of $\large \frac { ab }{ a+b } +\frac { ac }{ a+c } +\frac { ad }{ a+d } +\frac { bc }{ b+c } +\frac { bd }{ b+d } +\frac { cd }{ c+d } = \frac { p }{ q }$ where p, q are co-prime natural numbers. Find $p+q$