User Vladimir Reshetnikov

180 votes

5 answers

12k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

asked Oct 11, 2013 at 23:22

101 votes

1 answer

4k views

Arithmetic-geometric mean of 3 numbers

asked May 22, 2016 at 0:39

96 votes

3 answers

6k views

Complexity class of comparison of power towers

asked Jan 21, 2012 at 22:08

95 votes

2 answers

7k views

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

asked Oct 27, 2013 at 5:23

90 votes

2 answers

5k views

Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$

asked Oct 24, 2013 at 20:22

84 votes

1 answer

2k views

Conjectured formula for the Fabius function

asked Jul 5, 2019 at 0:01

82 votes

1 answer

3k views

Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

asked May 18, 2013 at 19:45

80 votes

4 answers

4k views

Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions

asked Sep 21, 2013 at 18:48

80 votes

4 answers

4k views

Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$

asked Oct 10, 2013 at 23:27

69 votes

4 answers

20k views

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

asked May 12, 2015 at 18:32

57 votes

3 answers

2k views

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

asked Aug 22, 2015 at 17:46

55 votes

4 answers

5k views

A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

asked Oct 9, 2013 at 20:56

50 votes

3 answers

3k views

A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$

asked Oct 6, 2013 at 5:16

48 votes

4 answers

7k views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

asked Sep 4, 2014 at 2:15

47 votes

3 answers

1k views

Conjectural closed-form representations of sums, products or integrals

asked May 15, 2013 at 20:41

42 votes

9 answers

7k views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

asked Dec 20, 2015 at 5:47

41 votes

2 answers

2k views

Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

asked Sep 6, 2014 at 18:22

40 votes

1 answer

2k views

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

asked Jul 25, 2014 at 0:54

38 votes

1 answer

1k views

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

asked Nov 27, 2013 at 5:23

38 votes

3 answers

721 views

Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,...$

asked May 24, 2016 at 5:31

36 votes

3 answers

2k views

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

asked Jul 26, 2014 at 22:56

36 votes

4 answers

2k views

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

asked Jun 15, 2015 at 18:39

35 votes

1 answer

964 views

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

asked Apr 26, 2013 at 6:40

34 votes

2 answers

2k views

Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$

asked May 18, 2013 at 0:24

34 votes

0 answers

549 views

An iterative logarithmic transformation of a power series

asked Dec 20, 2021 at 3:33

33 votes

2 answers

1k views

Closed form for improper definite integral involving trig functions and exponentials?

asked Dec 14, 2011 at 0:43

33 votes

3 answers

1k views

How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?

asked May 27, 2013 at 21:26

33 votes

3 answers

1k views

Integral $\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx$

asked Nov 11, 2013 at 23:49

31 votes

2 answers

2k views

Algebraic numbers that cannot be expressed using integers and elementary functions

asked Oct 14, 2013 at 17:56

31 votes

2 answers

7k views

Looking for a proof of Cleo's result for ${\large\int}_0^\infty\operatorname{Ei}^4(-x)\,dx$

asked May 27, 2015 at 22:23

Stack Exchange Network

218 Questions

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

Arithmetic-geometric mean of 3 numbers

Complexity class of comparison of power towers

Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$

Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$

Conjectured formula for the Fabius function

Closed form for $\int_0^\infty\ln\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$

Integrals of $\sqrt{x+\sqrt{\phantom|\dots+\sqrt{x+1}}}$ in elementary functions

Integral $\int_1^\infty\operatorname{arccot}\left(1+\frac{2\pi}{\operatorname{arcoth}x-\operatorname{arccsc}x}\right)\frac{\mathrm dx}{\sqrt{x^2-1}}$

Integrals of the form ${\large\int}_0^\infty\operatorname{arccot}(x)\cdot\operatorname{arccot}(a\,x)\cdot\operatorname{arccot}(b\,x)\ dx$

Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$

A conjectured closed form of $\int\limits_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

A conjectural closed form for $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

Conjectural closed-form representations of sums, products or integrals

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

Prove ${\large\int}_0^\infty\frac{\ln x}{\sqrt{x}\ \sqrt{x+1}\ \sqrt{2x+1}}dx\stackrel?=\frac{\pi^{3/2}\,\ln2}{2^{3/2}\Gamma^2\left(\tfrac34\right)}$

Are elementary and generalized hypergeometric functions sufficient to express all algebraic numbers?

Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,...$

Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$

An iterative logarithmic transformation of a power series

Closed form for improper definite integral involving trig functions and exponentials?

How to prove $\int_0^\infty J_\nu(x)^3dx\stackrel?=\frac{\Gamma(1/6)\ \Gamma(1/6+\nu/2)}{2^{5/3}\ 3^{1/2}\ \pi^{3/2}\ \Gamma(5/6+\nu/2)}$?

Integral $\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx$

Algebraic numbers that cannot be expressed using integers and elementary functions

Looking for a proof of Cleo's result for ${\large\int}_0^\infty\operatorname{Ei}^4(-x)\,dx$