Conjecture for the value of $\int_0^1 \frac{1}{1+x^{p}}dx$ While browsing the post Is there any integral for the golden ratio $\phi$?, I came across this nice answer,
$$ \int_0^\infty \frac{1}{1+x^{10}}dx=\frac{\pi\,\phi}5$$
it seems the general form is just

$$p \int_0^\infty \frac{1}{1+x^{p}}dx=\color{blue}{\frac{\pi}{\sin\big(\tfrac{\pi}{p}\big)}}$$

I wondered about
$$\int_0^\color{red}1 \frac{1}{1+x^p}dx=\,?$$
Mathematica could find messy closed-forms for $p=5,7$. After some laborious simplification, 
$$5\int_0^1 \frac{1}{1+x^5}dx=\frac{\pi\sqrt{\phi}}{5^{1/4}}+\ln2+\sqrt{5}\ln\phi$$
Question 1: In general, is it true that for any $p$ ,

$$2p\,\int_0^1 \frac{1}{1+x^p}dx=\color{blue}{\frac{\pi}{\sin\big(\tfrac{\pi}{p}\big)}}+2\ln2-\psi\big(\tfrac{1}{p}\big)+\psi\big(\tfrac{p-1}{2p}\big)+\psi\big(\tfrac{p+1}{2p}\big)-\psi\big(\tfrac{p-1}{p}\big)$$

where $\psi(z)$ is the digamma function? 
Note: The four digammas, implemented in Mathematica as PolyGamma[z], can be expressed as a sum of cosines x logarithms for odd $p=2m+1$. Let $k=\frac{2n-1}{p}\pi$, then,
$$-\psi\big(\tfrac{1}{p}\big)+\psi\big(\tfrac{p-1}{2p}\big)+\psi\big(\tfrac{p+1}{2p}\big)-\psi\big(\tfrac{p-1}{p}\big)=-4\sum_{n=1}^m \cos (k)\ln\big(\sin\tfrac{k}{2}\big)$$
Question 2: For even $p$, can we can also avoid the digamma by using cosines and logarithms?
 A: I'm only going to address Question 1. 
The expression proposed in Question 1 is true. However, it is a little bit too complicated than necessary. A simpler version of the expression is
$$2p\int_0^1 \frac{dx}{1+x^p} = \psi\left(\frac{1}{2p} + \frac12\right) - \psi\left(\frac{1}{2p}\right)$$

From reflection formula, take logarithm and differentiate, we get
$$\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin\pi z} \implies
\psi(z) - \psi(1-z) = \pi\cot\pi z$$
This leads to
$$\frac{\pi}{\sin z} = \pi\cot\frac{\pi z}{2} - \pi\cot\pi z = \psi\left(\frac{z}{2}\right) - \psi\left(1-\frac{z}{2}\right) - \psi(z) + \psi(1-z)
$$
From duplication formula, take logarithm and differentiate, we get
$$\Gamma(z)\Gamma\left(z+\frac12\right) = 2^{1-2z}\sqrt{\pi}\Gamma(2z)
\implies \psi(z) + \psi\left(z + \frac12\right) = -2\log 2 + 2\psi(2z)
$$
Apply these to RHS of Question 1, we can expose RHS to following mess
$$
\left[ \color{red}{\psi\left(\frac{1}{2p}\right)} - \psi\left(1 - \frac{1}{2p}\right) - \color{green}{\psi\left(\frac{1}{p}\right)} + \color{blue}{\psi\left(1 - \frac{1}{p}\right)} \right]
+ \left[ \color{green}{2\psi\left(\frac1p\right)} - \color{red}{\psi\left(\frac{1}{2p}\right)} - \color{magenta}{\psi\left(\frac{p+1}{2p}\right)}\right]\\
- \color{green}{\psi\left(\frac{1}{p}\right)} + \psi\left(\frac{p-1}{2p}\right) + \color{magenta}{\psi\left(\frac{p+1}{2p}\right)} - \color{blue}{\psi\left(\frac{p-1}{p}\right)}
$$
After massive cancellation, we can simplify RHS to
$$
\psi\left(\frac{p-1}{2p}\right) - \psi\left(1 - \frac{1}{2p}\right)
= \psi\left( 1 - \left(\frac{1}{2p} + \frac12\right)\right) - \psi\left(1 - \frac{1}{2p}\right)
= \psi\left(\frac{1}{2p} + \frac12\right) - \psi\left(\frac{1}{2p}\right)
$$
Recall following expansion of digamma function
$$\psi(z) = \frac{1}{z} + \sum_{n=1}^\infty \left(\frac{1}{z+n} - \frac{1}{n}\right)$$
We find
$$\begin{align}
\text{RHS}
&= \frac{1}{\frac{1}{2p}} - \frac{1}{\frac{1}{2p} + \frac12} + \sum_{n=1}^\infty\left(\frac{1}{\frac{1}{2p}+n} - \frac{1}{\frac{1}{2p} + n + \frac12}\right)\\
&= 2\sum_{n=0}^\infty\frac{(-1)^n}{\frac{1}{p}+n}
= 2\sum_{n=0}^\infty\int_0^1 (-1)^n t^{\frac{1}{p}+n-1} dt
= 2\int_0^1 \sum_{n=0}^\infty (-1)^n t^{\frac{1}{p}+n-1} dt\\
&= 2 \int_0^1 \frac{t^{\frac{1}{p}-1}}{1+t} dt
= 2p\int_0^1 \frac{dx}{1+x^p} = \text{LHS}
\end{align}
$$
A: This answer is incomplete as it does not proof all the steps completely: 
Let us start with the partial fraction expansion
$$\tag{1}\frac{p}{1+x^p}=- \sum_{j=1}^p\frac{\omega^{2j-1}}{x-\omega^{2j-1}}$$
with $\omega= e^{i\pi/p}$.
Now, we can integrate and obtain
$$\int_0^1\!dx \frac{p}{1+x^p} = -\sum_{j=1}^p \omega^{2j-1} \log\left(1-\omega^{1-2j}\right). $$
We can simplify this formula a bit taking the real part. This yields
$$\int_0^1\!dx \frac{p}{1+x^p} = \pi\sum_{j=1}^{p} \frac{1-2j+p}{2p} \sin\left(\frac{\pi(1-2j)}{p}\right)+\sum_{j=1}^{p}\cos\left(\frac{\pi(1-2j)}{p}\right) \log\left[2 \sin\left(\frac{\pi(1-2j)}{2p}\right)\right].$$
The first term can be summed explicitly
$$\int_0^1\!dx \frac{p}{1+x^p} =\frac{\pi}{2\sin(\pi/p)}+\sum_{j=1}^{p}\cos\left(\frac{\pi(1-2j)}{p}\right) \log\left[2 \sin\left(\frac{\pi(1-2j)}{2p}\right)\right].$$ The second term can be related to the digamma function.
Formula (1) can be proven by multiplying the expression by $1+x^p$. For the right hand side, we obtain
$$- \sum_{j=1}^p\omega^{2j-1} \prod_{k\neq j} (x-\omega^{2k-1})
= -x^{p-1}\sum_{j=1}^p\omega^{2j-1}+  x^{p-2} \sum_{j=1}^p\omega^{2j-1} \sum_{k,k'\neq j} \omega^{2k-1}\omega^{2k'-1} - \cdots\\+(-1)^p \sum_{j=1}^p\omega^{2j-1} \prod_{k\neq j} \omega^{2k-1}$$
as $1+x^p=\prod_{k=1}^p (x- \omega^{2j-1})$. The result (1) follows from some not so straightforward combinatorics.
A: The following close-form
holds for any integer $p\ge 2$
\begin{align}
\int_0^1 \frac1{1+x^p}dx
= \frac2p \sum_{k=1}^{[\frac p2]} ( \theta_k \sin2\theta_k + \cos2\theta_k \ln \cos\theta_k)
\end{align}
where $\theta_k= \frac{p-2k+1}{2p}\pi $. In particular
\begin{align}
 & \int_0^1 \frac1{1+x^5} dx= \frac{\pi\sqrt{\phi}}{5^{5/4}}+\frac15\ln2+\frac1{\sqrt5}\ln\phi\\
 & \int_0^1 \frac1{1+x^6} dx= \frac\pi6 +\frac1{2\sqrt3}\ln(2+\sqrt3) \\
 & \int_0^1 \frac1{1+x^7} dx 
=\frac\pi{14}\csc \frac{\pi}7-\frac27\left(\frac{\ln\cos\frac{3\pi}7}{\sec\frac{\pi}7} +\frac{\ln\cos\frac{2\pi}7}{\sec\frac{3\pi}7} -\frac{\ln\cos\frac{\pi}7}{\sec\frac{2\pi}7} \right)
\end{align}
