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user97357329
  • Member for 5 years, 5 months
  • Last seen more than a week ago
16 votes
Accepted

Help on this integral $I=\int_0^1 \frac{x \arctan(x)}{1-x^2}\ln\left(\frac{2}{1+x^2}\right) dx$

13 votes

Conjectural closed-form of $\int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$

12 votes

How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

12 votes
Accepted

Evaluating $\int_0^1\frac{\arctan x\ln\left(\frac{2x^2}{1+x^2}\right)}{1-x}dx$

11 votes
Accepted

Compute $\int_0^1\frac{\ln x\operatorname{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

10 votes
Accepted

Two very advanced harmonic series of weight $5$

10 votes

How to prove $\int_0^1 \frac{\arctan^2(x)\ln\left(\frac{x}{(1-x)^2}\right)}x \, \mathrm{d}x=G^2$?

9 votes
Accepted

How to approach $\sum _{n=1}^{\infty } \frac{16^n}{n^4 \binom{2 n}{n}^2}$?

9 votes
Accepted

How to find $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}}{n^3}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_{2n}^{(2)}}{n^2}$ using real methods?

9 votes

Unexpected appearances of $\pi^2 /~6$.

8 votes

Looking for closed-forms of $\int_0^{\pi/4}\ln^2(\sin x)\,dx$ and $\int_0^{\pi/4}\ln^2(\cos x)\,dx$

8 votes

Request for crazy integrals

8 votes
Accepted

How can you approach $\int_0^{\pi/2} x\frac{\ln(\cos x)}{\sin x}dx$

8 votes

Prove: $\int_0^{\infty} \frac{\ln{(1+x)}\arctan{(\sqrt{x})}}{4+x^2} \, \mathrm{d}x = \frac{\pi}{2} \arctan{\left(\frac{1}{2}\right)} \ln{5}$

7 votes
Accepted

General formula for $\int^1_0 x^\alpha \log(1-x)\operatorname{Li}_2 (x)\, \mathrm dx$

7 votes

On the alternating quadratic Euler sum $\sum_{n = 1}^\infty \frac{(-1)^n H_n H_{2n}}{n^2}$

7 votes
Accepted

Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?

7 votes
Accepted

Proving $\sum_{n=1}^{\infty}\binom{2n}{n}^2\frac{4H_{2n}-3H_n}{n2^{4n}}=\zeta(2)$

7 votes
Accepted

Compute $\sum_{n=1}^\infty\frac{H_nH_{2n}}{(2n+1)^3}$

7 votes

Closed-form of $\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)=-\int_0^1\frac{\ln(1+x)\ln^3 x}{1-x}\,dx$

6 votes

Proving that $\int_0^1 \frac{\log^2(x)\tanh^{-1}(x)}{1+x^2}dx=\beta(4)-\frac{\pi^2}{12}G$

6 votes

Evaluate $\int_{0}^{\pi/4}x\ln^{2}(\sin(x))dx$

6 votes

Compute $\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx$

6 votes
Accepted

Evaluating $\int_{0}^{\infty} \frac{\ln^3 (1+x^2)}{1+2x^2} \, dx$ with contour integration

6 votes

A reason for $ 64\int_0^1 \left(\frac \pi 4+\arctan t\right)^2\cdot \log t\cdot\frac 1{1-t^2}\; dt =-\pi^4$ ...

6 votes
Accepted

Integrating $\int_{0}^{1} \frac{\arctan(x)\arctan(x^2)}{x^2} dx$

6 votes

A challenging integral $ -\int_0^1 \frac{\ln(1-x)}{1+x} \operatorname{Li}_2(x) \, \mathrm{d}x $

5 votes

How to prove $\int_{0}^{\infty} \frac{(1-x^2) \, \text{sech}^2\left(\frac{\pi x}{2} \right)}{(1+x^2)^2}\, dx = \frac{\zeta(3)}{\pi}$?

5 votes

Two hard integrals: $\int_{0}^{1}\frac{\log{(x)}\log{(1-x)}\log{(1+x^2)}}{x}dx$ and $\int_{0}^{1}\frac{\log^2{(x)}\log{(1+x^2)}}{1-x}dx$

5 votes
Accepted

Integral $\int_0^1\frac{\operatorname{Li}_2(x^2)}{1-x^2}\left(\frac{\ln(1+x)}{x}-\ln2\right)\ dx$

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