These identities, and many more, follow from a theorem in Integrals of polylogarithmic functions, recurrence relations, and associated Euler sums, For example, There are also variations with $\log x$ factor, such as


$c_{n,m}=0$ if $n$ is even; for $n$ odd some experimentation indicates it has the form $$c_{2n+1,m}=\frac{1}{\pi^3}\bigl(a_{n,m}-\tfrac{7}{8}(2n+1)(2m+1)\zeta(3)\bigr),\;\;\text{with}\;\;a_{n,m}=a_{m,n}\in\mathbb{Q}\geq 0.$$ I have not found a closed-form expression for the coefficients $a_{n,m}$, they are quite unwieldy. Some values of $a_{n,m}$ for small $...

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