Question: Numerical evaluation of a complex integral solution

Consider a soil moisture transport problem with parameters: a = 9.88\times10^{-5} m/s,  b = -0.0065\ m³/m³, and D = 3.51*10^{-7} m²/s. The length is L=0.08 m, and no-flux (outflow) is assumed, meaning q=0. In this case θ₀= 0.355$ and θL = 0.10.

A solution is expressed as a contour integral involving a complex-valued function V(k,x,t) and evaluated numerically in Maple using parameterized contours k1(r),k2(r),k3(s).

However, when this code is executed, Maple takes an extremely long time and appears to run indefinitely.

Could you suggest an alternative numerical or analytical approach ( that could improve efficiency when evaluating this type of contour integral?

a := 988/10000000;
b := -65/10000;
d := 351/1000000000;
q := 0;
theta0 := 355/1000;
thetaL := 1/10;
L := 8/100;
mu := a*(theta0 + b)/d;
c := a*(thetaL + b)/d;

mu := a*(theta0 + b)/d;
c := a*(thetaL + b)/d;

V := 1/(2*Pi)*exp(-d*k^2*t)*((exp(k*x*I)(mu - k*I) - exp(-mu*L)*exp(k*(L - x)*I)*(mu + L*I))*k*cos(k*L)/(k^2 + mu^2) + c*sin(k*L) - exp(k*L*I)(k*I + c)*sin(k*x)*(1 - exp(-mu*L)*exp(-I*k*L))/(mu + k*I) - (k*cos(k(L - x)) + c*sin(k(L - x)))*(1 - exp(-mu*L)*exp(k*L*I))/(mu - k*I) + 2*I*k*d*(k*cos(k(L - x)) + c*sin(k(L - x)))/(d*k^2 + a*q/d))/(k*cos(k*L) + c*sin(k*L)):

l := 5;
k1 := r -> l + I + r*exp(1/6*I*Pi);
k2 := r -> -l + I + r*exp(5/6*I*Pi);
k3 := s -> s + I;

dk1 := D(k1);
dk2 := D(k2);
dk3 := D(k3);

integrand1 := Re(eval(V, k = k1(r))*dk1(r) + eval(V, k = k2(r))*dk2(r));
integrand3 := Re(eval(V, k = k3(s))*dk3(s));
integrand2 := simplify(evalc(integrand1));

integrand4 := simplify(evalc(integrand3));
approx_u := proc(x, t) local temp1, temp2; temp1 := Int(eval(integrand2, [:-x = x, :-t = t]), r = 0 .. infinity, method = _d01amc); temp2 := Int(eval(integrand4, [:-x = x, :-t = t]), s = -L .. L, method = _d01ajc); evalf(temp1 + temp2); end proc;