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Infiniticism's user avatar
Infiniticism
  • Member for 5 years, 3 months
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  • Militaireetmari
22 votes

What is $\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right)$?

19 votes
Accepted

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

19 votes
Accepted

Evaluate hypergeometric $_6F_5\left(\{\frac12\}_3,\{1\}_3;\{\frac32\}_5;1\right)$

8 votes

Prove: $\int_0^2 \frac{dx}{\sqrt{1+x^3}}=\frac{\Gamma\left(\frac{1}{6}\right)\Gamma\left(\frac{1}{3}\right)}{6\Gamma\left(\frac{1}{2}\right)}$

8 votes
Accepted

How to calculate $\int_{-\infty}^{+\infty}\Gamma(x+yi)\Gamma(x-yi) \, dy$?

8 votes
Accepted

Proving $\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }\log \left| 1+e^{i x}+e^{i y}+e^{i z}\right| dxdydz=28 \pi \zeta (3)$

7 votes

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

7 votes

Is there a closed form for $\sum_{n=1}^\infty\frac{2^{2n}H_n}{n^3{2n\choose n}}?$

6 votes

How to evaluate $\int_0^{\pi/2} x\ln^2(\sin x)\textrm{d}x$ in a different way?

6 votes

How to evaluate $\int _0^1\frac{\ln ^2\left(1-x\right)\ln ^5\left(1+x\right)}{1+x}\:dx$

6 votes

Proving $\int_{\sqrt{\frac{3}{5}}}^1 \frac{\arctan (x)}{\sqrt{2 x^2-1} \left(3 x^2-1\right)} \, dx=\frac{3 \pi ^2}{160}$

6 votes
Accepted

Integral $\int_{0}^{1} \frac{\ln^2 x \ln^2 (1+x)\ln^2(1-x)}{x^2}dx$

6 votes

Another beautiful integral (Part 2)

6 votes
Accepted

Evaluate $\int_0^{\infty } \log \left(\frac{a^2}{x^2}+1\right) \log \left(\frac{b^2}{x^2}+1\right) \log \left(\frac{c^2}{x^2}+1\right) \, dx$

5 votes
Accepted

A twisted hypergeometric series $\sum_{n=1}^\infty\frac{H_n}{n}\left(\frac{(2n)!}{4^n(n!)^2}\right)^2$

5 votes

Evaluate $\int_0^1 x^{a-1}(1-x)^{b-1}\operatorname{Li}_3(x) \, dx$

5 votes

Evaluate$ \int _0^{\infty} \frac{\operatorname{sech}^{k}(x)}{\cosh(2\pi/k)-\cos(2x)}\,dx $ for $k=1,2$

5 votes

Counterexample to Tonelli's theorem

4 votes
Accepted

Evaluate $\int^1_0 x^a (1-x)^b \operatorname{Li}_2 (x)\, \mathrm dx$

4 votes
Accepted

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} $?

4 votes
Accepted

Closed form of $\int_0^1(\ln(1-x)\ln(1+x)\ln(x))^2\,dx$

4 votes
Accepted

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

4 votes
Accepted

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

4 votes
Accepted

Evaluate $\int_0^1 \frac{\log ^2(x+1) \log \left(x^2+1\right)}{x^2+1} dx$

4 votes
Accepted

The calculation does not use complex variable function theory $\int _0^{2\pi }e^{\cos \left(x\right)}\cos \left(\sin \left(x\right)\right)dx$

4 votes

Sums of the form $S_k=\sum_{n\geq 1}\frac{1}{\sinh^{2k}(n \pi)}$ and the residue theorem.

3 votes

Integral residue $\int_{0}^{\infty} {\frac{\cos\left(x\right)}{x^{2} + 2x + 4}}\,{\rm d}x. $

3 votes
Accepted

Challenging integral: $\int_0^{\pi/2}x^2\frac{\ln(\sin x)}{\sin x}dx$

3 votes

Calculate $\int_0^{1/2}\, x\sin(\pi x)\Gamma(\frac{d-1}{2} + x)\Gamma(\frac{d-1}{2} - x) \mathrm dx$ for even $d\ge 4$

3 votes
Accepted

Inverse Fourier transform of $F(w)=\frac{1-\sum_{i=1}^n{a_ie^{-{r_k}{b_i}|w|}}}{|w|}$