<s>[INST] How prove this $\prod\limits_{cyc}(a^3+2b+\frac{2}{a^2+1})\ge 64$ let $a,b,c>0$ and such
$$abc\ge 1$$
show that
$$\left(a^3+2b+\dfrac{2}{a^2+1}\right)\left(b^3+2c+\dfrac{2}{b^2+1}\right)\left(c^3+2a+\dfrac{2}{c^2+1}\right)\ge 64$$
my try: 
$$\sum_{cyc}\ln{\left(a^3+2b+\dfrac{2}{a^2+1}\right)}\ge 6\ln{2}$$
Then  I can't,Thank you
 [/INST] Hint: first prove that $a^3+\frac{2}{a^2+1} \geq 2a$
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