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I also tried the following code:

reset()
forget()

var('x,y,z,t, x1,y1,z1,t1, x2,y2,z2,t2, x3,y3,z3,t3')   # my real variables to be substituted in the function arguments
R0=[x,y,z,t]
R1=[x1,y1,z1,t1]

U_0=[]
U_1=[]

for i in range(4):
    U_0.append(function('U%s0' %i, *R0))
    U_1.append(function('U%s1' %i, *R0+R1))


var('td,xd1,xd2,x_B1')
It = lambda f: integral(f(t=td),td,0,t)     #or use mathematica's algorithm using:  ,algorithm='mathematica_free'
                                            #or use: ,algorithm='sympy' 
Ixx = lambda f: integral(integral(f(x=xd1),xd1,x_B1,xd2),xd2,x_B1,x)

eq=[]
assume(t>0)
for i in range(4):
    #eq.append(It(U_1[i])==It(U_0[i](x=R0[i],y=x+z)*U_1[i])\      # it was working while *U_1[i] had not been included
    #                         +Ixx(U_0[i]))

    #eq.append(It(U_1[i])==integral((U_0[i](x=R0[i],y=x+z)*U_1[i])(t=td),td,0,t)\    #not using "It()" also didn't help
    #                         +Ixx(U_0[i]))

    eq.append(It(U_1[i])==integral(U_0[i](x=R0[i],y=x+z,t=td)*U_1[i](t=td),td,0,t)\
                          +Ixx(U_0[i]))                                              # again here we have problem!

# the problem is with integration code of sage or Maxima it seems. If so maybe I should leave sage to Maple or Mathematica ?

    show(eq[i])

which shows that seemingly the problem is with the integral() function in sage (or Maxima?) rather than the defined integral operators and this destroys all my hopes to use this interesting open source math software, maybe I should leave Sage and use either Maple or mathematica?