TITLE: The integral $\ J(m,n):=\int_0^1 \frac{x^m}{x^n+1}dx$
QUESTION [5 upvotes]: Here 
Is there a general formula for $I(m,n)$?
I asked for a general formula for $\ I(m,n):=\int_0^{\infty} \frac{x^m}{x^n+1}dx$.

Is the formula $$I(m,n)=\frac{\pi}{n\sin((m+1)\frac{\pi}{n})}$$ true for every pair $\ (m,n)\ $ of real numbers with $\ 0\le m\le n-2\ $ and not just for non-negative integers with $\ 0\le m\le n-2\ $ ?

I wondered whether there is a similar closed form for
$$J(m,n):=\int_0^1 \frac{x^m}{x^n+1}dx$$
I figured out that $$I(m,n)=J(m,n)+J(n-2-m,n)$$ holds for $0\le m\le n-2$.
For $\ m=0\ $, the first few values are
$$J(0,0)=\frac{1}{2}$$ $$J(0,1)=\ln(2)$$ $$J(0,2)=\frac{\pi}{4}$$ $$J(0,3)=\frac{2\sqrt{3}\pi+\ln(64)}{18}$$ $$J(0,4)=\frac{\pi+2\coth^{-1}(\sqrt{2})}{4\sqrt{2}}$$ $$J(0,5)=\frac{1}{5}\sqrt{\frac{1}{10}(5+\sqrt{5})}\pi+\frac{1}{20}(\ln(16)+2\sqrt{5}\coth^{-1}(\frac{3}{\sqrt{5}}))$$ $$J(0,6)=\frac{\pi+\sqrt{3}\ln(2+\sqrt{3})}{6}$$

Is there a general formula for $\ J(m,n)\ $ and does it hold for all pairs $\ (m,n)\ $ of real numbers with $\ 0\le m\le n-2$ ?

REPLY [2 votes]: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{{#1}}\,}
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$\ds{\mrm{J}\pars{m,n} \equiv \int_{0}^{1}{x^{m} \over x^{n} + 1}\,\dd x:\ ?}$.

\begin{align}
\mrm{J}\pars{m,n} & \equiv \int_{0}^{1}{x^{m} \over x^{n} + 1}\,\dd x =
\int_{0}^{1}{x^{m} - x^{m + n} \over 1 - x^{2n}}\,\dd x\quad
\pars{\begin{array}{l}
\mbox{Multiply numerator an denominator}
\\
 \mbox{by}\ds{\quad 1 - x^{n}}
\end{array}}
\\[5mm] &
\stackrel{x^{2n}\ \mapsto\ x}{=}\,\,\,
{1 \over 2n}\int_{0}^{1}{x^{\pars{m + 1}/\pars{2n} - 1} -
x^{\pars{m + n + 1}/\pars{2n} - 1} \over 1 - x}\,\dd x
\\[5mm] & =
\bbox[#ffe,15px,border:1px dotted navy]{\ds{{1 \over 2n}\bracks{%
\Psi\pars{m + n + 1 \over 2n} - \Psi\pars{m + 1 \over 2n}}}}\qquad
\pars{~\Psi:\ Digamma\ Function~}
\end{align}
We used
the identity $\color{#000}{\mathbf{6.3.22}}$: $\ds{\Psi\pars{z} + \gamma =
\int_{0}^{1}{1 - t^{z - 1} \over 1 - t}\,\dd t\,,\ \Re\pars{z} > 0}$. $\ds{\gamma}$ is the Euler-Mascheroni Constant.