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You are right that Maxima doesn't want delta to be a function, since it's a limit of integration. That was slightly surprising to me.
Anyway, it turns out you can fix this post-hoc.
sage: integralTu.subs(delta=function('delta',x))
1/10*(2*U_0*h_mstar*rho_f*cos(Phi)*delta(x)^4 + 7*T_m*k_f*delta(x)^3 + 3*T_w*k_f*delta(x)^3)*R*U_0*sin(Phi)/(k_f*delta(x)^3)
That might seem a little hackish, but is just fine.
sage: IT = integralTu.subs(delta=function('delta',x))
sage: diff(IT,x)
1/10*(8*U_0*h_mstar*rho_f*cos(Phi)*delta(x)^3*D[0](delta)(x) + 21*T_m*k_f*delta(x)^2*D[0](delta)(x) + 9*T_w*k_f*delta(x)^2*D[0](delta)(x))*R*U_0*sin(Phi)/(k_f*delta(x)^3) - 3/10*(2*U_0*h_mstar*rho_f*cos(Phi)*delta(x)^4 + 7*T_m*k_f*delta(x)^3 + 3*T_w*k_f*delta(x)^3)*R*U_0*sin(Phi)*D[0](delta)(x)/(k_f*delta(x)^4)
By the way, I've updated your question because you needed one more variable defined.
