I have obtained exp1 and exp2 by trying to solve the problem proposed by Professor Coxeter
in AMM 95 (1988) :
Proof of identity
\int_1^6 \frac{\left(\frac{1}{\sqrt{t+3}}+2\right) \sec ^{-1}(t)}{(t+2) \sqrt{t+1}} \, dt=\frac{2 \pi ^2}{15}
by a direct calculation, that is, not relying on some geometry or the computer.