I am interested in the integral $$I(t,s)=\int_0^1 \frac{J_0(tx)J_1(t(x+s)) }{x+s}\mathrm{d}x$$ for $t,s\ge 0$.
I found that for $s=0$ there is a closed expression $$I(t,0)=t J_0(t){}^2-J_1(t) J_0(t)+t J_1(t)^2$$ but I am wondering whether for $s>0$ there might be something similar?