If t is unknown, then you need to use symbolic tools to do the integration.
syms x t
K = 1/((exp(0.014342/(x * t)) - 1)) * (1/(x^4));
int(1/(x^4*(exp(2066900027383925/(144115188075855872*t*x)) - 1)), x, 0, 1)
MATLAB just returns the integral in the form it was given. I tried Wolfram Alpha, which also agrees it cannot find a solution. There is no assurance that anything you write down has a nice closed form solution.
You have some options. If you had some value for t, then we could write it as:
Kxt = @(x,t) 1./((exp(0.014342./(x * t)) - 1)) .* (1./(x.^4));
foft = @(t) integral(@(x) Kxt(x,t),0,1);
So while no analytical form is available, we can find a numerical solution. I could have used vpaintegral too.
Finally, you could probably write the problem as a series expansion, but then you would need to consider the domain of convergence, etc.