For what $a$ and $b$ are there explicit expressions for $I(a, b) =\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b} $?

The following is trivial due to expansion of digamma (formula 14 of this page) $$\sum _{n=0}^{\infty } \frac{1}{(a+n) (b+n)}=\frac{\psi ^{(0)}(a)-\psi ^{(0)}(b)}{a-b}$$ Thus OP's integral equals to $$I(a,b)=\sum _{n=0}^{\infty } \frac{a b}{(a b n+b) (a b n+a)}=\frac{\psi ^{(0)}\left(\frac{1}{b}\right)-\psi ^{(0)}\left(\frac{1}{a}\right)}{a-b}$$ Due to Gauss's digamma theorem (formula 11 of link above) for all $a,b\in \mathbb Q$ the integral can be expressed in terms of log and trig functions (even square roots, when $a,b$ are small). For instance $$I(3,5)=\frac{1}{2} \pi \sqrt{\frac{1}{6} \left(-\sqrt{3+\frac{6}{\sqrt{5}}}+\frac{3}{\sqrt{5}}+2\right)}+\frac{1}{8} \log \left(\frac{3125}{729}\right)+\frac{1}{4} \sqrt{5} \coth ^{-1}\left(\sqrt{5}\right)$$


May be, we could start with $$I(a, b)=\int_0^1 \int_0^1 \dfrac{dx\,dy}{1-x^ay^b}=\int_0^1 \, _2F_1\left(1,\frac{1}{b};\frac{1}{b}+1;x^a\right)\,dx$$ and consider $$J_a=\int_0^1 \, _2F_1\left(1,\frac{1}{b};\frac{1}{b}+1;x^a\right)\,dx$$ and discarding the ones which contain logarithms of complex arguments, we can obtain a few of them $$J_1=\frac{\psi \left(\frac{1}{b}\right)+\gamma }{1-b}$$ $$J_2=\frac{\psi \left(\frac{1}{b}\right)+\gamma +\log (4)}{2-b}$$ $$J_3=\frac{6 \psi \left(\frac{1}{b}\right)+\sqrt{3} \pi +6 \gamma +9 \log (3)}{18-6 b}$$ $$J_4=\frac{2 \left(\psi \left(\frac{1}{b}\right)+\gamma \right)+\pi +\log (64)}{8-2 b}$$ $$J_6=\frac{2 \psi \left(\frac{1}{b}\right)+\sqrt{3} \pi +2 \gamma +\log (432)}{12-2 b}$$ $$J_8=\frac{2 \psi \left(\frac{1}{b}\right)+\sqrt{2} \pi +\pi +2 \gamma +\log (256)+2 \sqrt{2} \coth ^{-1}\left(\sqrt{2}\right)}{16-2 b}$$ $$J_{12}=\frac{2 \psi \left(\frac{1}{b}\right)+\sqrt{3} \pi +2 \pi +2 \gamma +2 \sqrt{3} \log \left(2+\sqrt{3}\right)+\log (1728)}{24-2 b}$$

For the particular case where $b=a$, the result seems to be quite simpler $$K_a=\int_0^1 \, _2F_1\left(1,\frac{1}{a};\frac{1}{a}+1;x^a\right)\,dx=\frac 1 {a^2}\psi ^{(1)}\left(\frac{1}{a}\right)$$

Tags:

Polynomials

Special Functions

Definite Integrals

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