Wolfram alpha is useless

Let $\displaystyle u=11x+12$ , this turns our integral to

$\displaystyle \int_{67}^{78} 11 \dfrac{\ln(u)}{u^{2}-12u+5226} du$

Now, let $\displaystyle x=\dfrac{5226}{u}$ , we will get

$\displaystyle \int_{67}^{78} 11 \dfrac{\ln(u)}{u^{2}-12u+5226} du=\dfrac{11\ln(5226)}{2}\int_{67}^{78} \dfrac{1}{u^{2}-12u+5226} du$

The latter integral is now easy to find,

$\displaystyle \dfrac{1}{\sqrt{5082}} \arctan(\dfrac{x-12}{\sqrt{5082}})$

Substituting jn the boundary of integration,

$\displaystyle \arctan(\dfrac{6}{\sqrt{42}})- \arctan(\dfrac{5}{\sqrt{42}})= \arctan(\dfrac{\sqrt{42}}{72})$

Plugging all these in to get,

$\displaystyle \dfrac{\ln(5226)}{2\sqrt{42}} \arctan(\dfrac{\sqrt{42}}{72})$

So, the sum is then $\displaystyle 5226+2+42+72= \boxed{5342}$