Could you possibly use the exact value of the indefinite integral to help trying to find your solution
In[1]:= lmean = 8814/10^4;
lc = 5613/10^4;
Af = 12/10;
u = 1/10;
a = 10813/10^4;
b = 12/10;
p = 1;
q = 25294/10^4;
lct[t_] := lc*(1 - A*Tan[t])/(Exp[u*t]);
f[l_] := a*b*l^(b - 1)*Exp[-a*l^b];
g[t_] := ((Sin[t])^(2*p - 1)*(Cos[t])^(2*q - 1))/(Integrate[(Sin[w])^(2*p - 1)*(Cos[w])^(2*q-1), {w,0,Pi/2}]);
FullSimplify[Integrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], l], Assumptions -> 0 <= t < 2 Pi && 0 <= l]
Out[12]= -(1/2407281094647)126470000 E^(-((10813 l^(6/5))/10000) + t/10) l^(4/5) Cos[t]^(10147/2500)
(150000 + 97317 l^(6/5) + 100000 E^((10813 l^(6/5))/10000)ExpIntegralE[1/3, (10813 l^(6/5))/10000]) Sin[t]
In[13]:= FullSimplify[Integrate[f[l]*g[t]*(l/lmean)*(l/(2*lc))*Exp[u*t], t, l],
Assumptions -> 0 <= t < 2 Pi && 0 <= l]
Out[13]= -(1/(Cos[t]^(2353/2500)))(2470117187500/892485605516233971387177524702941121272167 -
(1788364843750 I)/68652738885864151645167501900226240097859) E^(-((10813 l^(6/5))/10000) +
(1/10 - I) t) (1 + E^(2 I t)) l^(4/5) (150000 + 97317 l^(6/5) + 100000 E^((10813 l^(6/5))/10000)
ExpIntegralE[1/3, (10813 l^(6/5))/10000]) (1243209104158084056000 E^(I t)
Hypergeometric2F1[2647/5000 - I/20, 1, 7353/5000 - I/20, -E^(2 I t)] - (250 + 2353 I)
(51882557652852180070 Cos[t] + 26149933547065465035 Cos[3 t] + 5233573221864243007 Cos[5 t] -
250 (19936133122065382 Sin[t] + 7069109 (485993763 Sin[3 t] + 58539109 Sin[5 t]))))
If I include A=1 then I get a ConditionalExpression depending on (1 - Tan[t])^(2/5)>0 for the exact definite integral for l. For a few other values of A I get a string of "PolynomialGCD::lrgexp: Exponent is out of bounds for function PolynomialGCD." and no result.