**Corollary 8.**

$$\begin{aligned} \int\_0^1 \mathbf{x}^a \, \arctan^3(\mathbf{x}) \, d\mathbf{x} &= \sum\_{n \ge 1} \frac{(-1)^{n+1}}{n} \int\_0^1 \mathbf{x}^{2n+a} \arctan^3(\mathbf{x}) \, d\mathbf{x} \\ &= \sum\_{n \ge 1} \frac{(-1)^{n+1}}{4n} \frac{h\_n}{\left(\frac{\pi + H\_{\frac{2n+a-2}{4}} - H\_{\frac{2n+a}{4}}}{\left(2n+a+1\right)}\right) \end{aligned}$$

.

*Various cases follow, for a* = 0, *and using the double argument of the polygamma function*

$$\begin{aligned} \int\_0^1 \arctan^3(x) \, dx &= \sum\_{n\geq 1} \frac{(-1)^{n+1} h\_n}{4n} \left( \frac{\pi + H\_{\frac{2n-2}{4}} - H\_{\frac{2n}{4}}}{2n+1} \right) \\ &= \sum\_{n\geq 1} \frac{(-1)^{n+1} h\_n}{4n\left(2n+1\right)} \left( \pi - 2\log(2) + 2H\_n - 2H\_{\frac{4}{4}} \right) \\ &= \frac{63}{64} \zeta\left(3\right) - \frac{3}{4} \pi G + \frac{\pi^3}{64} + \frac{18}{32} \zeta\left(2\right) \log 2. \end{aligned}$$

 *Manipulating this integral identity gives us the new Euler sums,*

$$\sum\_{n\geq 1} \frac{(-1)^{n+1} \, h\_n H\_{\frac{n}{2}}}{n} = \pi G - \frac{35}{32} \zeta'(3) - \frac{9}{8} \zeta'(2) \log 2,$$

$$\sum\_{n\geq 1} \frac{(-1)^{n+1} \, h\_n \left(H\_{\frac{n}{2}} - H\_n\right)}{2n+1} = \frac{21}{32} \zeta'(3) - \frac{1}{4} \pi G + \frac{3}{4} \pi \log^2(2) - \frac{1}{2} G \log 2,$$

$$\sum\_{n\geq 1} \frac{h\_{2n}}{n(2n+1)} = \log 2 + \frac{\pi}{2} - \frac{3}{4} \zeta'(2) .$$

*For a* = −1,

$$\begin{aligned} \int\_0^1 \frac{1}{x} \arctan^3(x) \, dx &= \sum\_{n \ge 1} \frac{(-1)^{n+1} h\_n}{8n^2} \left( \pi + H\_{\frac{n}{2} - \frac{3}{4}} - H\_{\frac{n}{2} - \frac{1}{4}} \right) \\ &= \frac{9}{8} \zeta'(2) \, G + \frac{1}{1024} \left( \psi^{(3)}\left(\frac{3}{4}\right) - \psi^{(3)}\left(\frac{1}{4}\right) \right) \\ &= \frac{9}{8} \zeta'(2) \, G - \frac{3}{2} \beta'(4) \, . \end{aligned}$$

*where the Dirichlet function β* (4) = ∑ *j*≥1 (−<sup>1</sup>)*j*+<sup>1</sup> (2*j*−<sup>1</sup>)<sup>4</sup> . *Here, we note that*

$$
\psi^{(3)}\left(\frac{3}{4}\right) - \psi^{(3)}\left(\frac{1}{4}\right) = 8\pi^4 - 768\pounds(4) - \left(8\pi^4 + 768\pounds(4)\right) = -1536\pounds(4)\geqslant
$$

*moreover, using the integral identity, we obtain the new Euler sum*

$$\begin{split} \sum\_{n\geq 1} \frac{(-1)^{n+1} h\_n}{n^2} \left( H\_{\frac{n}{2} + \frac{1}{4}} - H\_{\frac{n}{2} - \frac{1}{4}} \right) &= -3\zeta'(2)G - 12\beta(4) + \frac{7}{4}\pi\zeta'(3) + 4\pi\zeta G \\ &- 7\zeta'(3) + 8G - 3\zeta'(2) - 2\pi\log 2. \end{split}$$

*For the quartic*

$$\begin{aligned} \int\_0^1 \frac{1}{x} \arctan^4(x) \, dx &= \sum\_{n \ge 1} \sum\_{j \ge 1} \frac{(-1)^{n+j} \, h\_n h\_j}{j n \, (j+n)} \\ &= \quad \frac{1}{16} \pi^3 G + \frac{93}{32} \zeta(5) + \frac{3}{2} \pi \beta(4) \,, \end{aligned}$$

*and*

$$\begin{split} \int\_{0}^{1} \mathbf{x} \cdot \arctan^{6} \left( \mathbf{x} \right) \, d\mathbf{x} &= \sum\_{n \geq 1} \sum\_{j \geq 1} \sum\_{k \geq 1} \frac{(-1)^{n+j+k+1} h\_{n} h\_{j} h\_{k}}{2jkn \, (j+k+n+1)} \\ &= \frac{15}{32} \pi^{3} G + \frac{20925}{1024} \zeta \left( 5 \right) - \frac{45}{4} \pi \beta \left( 4 \right) - \frac{3}{1024} \pi^{5} \\ &\quad + \frac{945}{4096} \zeta \left( 6 \right) - \frac{675}{256} \zeta \left( 4 \right) \log 2 + \frac{405}{256} \zeta \left( 2 \right) \zeta \left( 3 \right). \end{split}$$

## **5. Conclusions**

In the first three sections of the paper, we treated miscellaneous Euler sums, particularly, the alternating sums. We developed many new Euler type identities. In particular, we have developed some new identities for the Catalan constant, Apery's constant and Euler's famous *ζ* (2) constant. In the fourth section of this paper, we have demonstrated and explored the connection of integrals involving trigonometric and hyperbolic functions with Euler sums. We have evaluated particular integrals related to Euler sums, many of which are not amenable to a mathematical software package. Integrals dealing with powers of arctangent and hyperbolic functions will be further developed in a forthcoming paper.

**Author Contributions:** Conceptualization, A.S. and A.S.N.; methodology, A.S and A.S.N.; software, A.S.N.; validation, A.S.; formal analysis, A.S. and A.S.N.; investigation, A.S. and A.S.N.; writing–original draft preparation, A.S.N.; writing–review and editing, A.S.; visualization, A.S.N.; supervision, A.S.; project administration, A.S. and A.S.N.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
