$$a=i \leadsto \int _0^ \infty lnxln( \sqrt{1+ x^{4} } ) e^{iArctan \frac{1}{ x^{4} } } dx= \frac{ \pi }{2} \sqrt{i}lni- \pi \sqrt{i} = \pi e^{i \frac{ \pi }{4} } [i \frac{ \pi }{2} \frac{1}{2}-1 ]= \pi ( \frac{1}{ \sqrt{2} } - i\frac{1}{ { \sqrt{2} } } )[ \frac{i \pi -4}{4} ]= \frac{ \pi }{4 \sqrt{2} }(i \pi -4- \pi -4i) $$
