User Vladimir Reshetnikov

106 votes

Collection of surprising identities and equations.

answered Sep 28, 2013 at 4:40

83 votes

Collection of surprising identities and equations.

answered Feb 28, 2014 at 2:00

49 votes

Accepted

How to find $\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15}}}{1+x^{2+\sqrt{3}}}\right)}{\left(1+x^2\right)\log x}\mathrm dx$

answered Mar 7, 2014 at 2:12

46 votes

Accepted

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

answered Sep 29, 2013 at 2:55

42 votes

Accepted

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$

answered Oct 23, 2013 at 18:46

41 votes

Accepted

Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

answered Jul 29, 2015 at 23:05

40 votes

Accepted

What is $(26+15(3)^{1/2})^{1/3}+(26-15(3)^{1/2})^{1/3}$?

answered Apr 26, 2013 at 22:54

40 votes

Accepted

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

answered May 11, 2013 at 23:19

39 votes

Accepted

Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

answered Mar 11, 2014 at 22:37

37 votes

Accepted

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

answered Dec 2, 2015 at 23:45

37 votes

Accepted

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

answered May 19, 2013 at 23:25

33 votes

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

answered Oct 13, 2013 at 21:26

32 votes

Accepted

Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

answered Nov 13, 2013 at 18:39

30 votes

Accepted

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

answered Aug 19, 2014 at 1:13

30 votes

Accepted

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

answered Sep 27, 2013 at 2:47

30 votes

Accepted

Integral $\int_0^\infty\left(x+5\,x^5\right)\operatorname{erfc}\left(x+x^5\right)\,dx$

answered Sep 29, 2013 at 0:57

27 votes

Accepted

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

answered Jun 25, 2013 at 21:14

27 votes

Accepted

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$

answered Aug 29, 2014 at 19:22

26 votes

Accepted

A closed form for $\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx$

answered Sep 26, 2013 at 3:33

23 votes

What are examples of unexpected algebraic numbers of high degree occured in some math problems?

answered May 10, 2013 at 0:25

22 votes

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

answered Oct 10, 2013 at 1:13

22 votes

Accepted

Simplify $7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$

answered Jun 25, 2015 at 18:03

21 votes

Accepted

A nicer closed form? $\int_0^1 \frac{\log (x) \log \left(x^2-x+1\right)}{x^2-x+2} \, dx$

answered Jul 22, 2015 at 1:52

20 votes

Accepted

What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

answered May 28, 2013 at 16:17

20 votes

Do harmonic numbers have a “closed-form” expression?

answered May 5, 2013 at 18:09

20 votes

Accepted

Need help with $\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx$

answered Jan 10, 2016 at 1:57

19 votes

Accepted

Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$

answered May 17, 2013 at 5:25

19 votes

Funny identities

answered Mar 1, 2014 at 21:47

18 votes

Accepted

Does $\int_0^{\infty} \left( p + q W \left( r e^{- s x + t} \right) + u x \right) e^{- x} d x$ have a closed-form expression?

answered May 5, 2013 at 2:07

18 votes

Accepted

Integral $\int_0^\infty\operatorname{arccot}(x)\,\operatorname{arccot}(2x)\,\operatorname{arccot}(5x)\,dx$

answered Dec 3, 2015 at 1:04

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89 Answers

Collection of surprising identities and equations.

Collection of surprising identities and equations.

How to find $\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15}}}{1+x^{2+\sqrt{3}}}\right)}{\left(1+x^2\right)\log x}\mathrm dx$

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$

Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx $

What is $(26+15(3)^{1/2})^{1/3}+(26-15(3)^{1/2})^{1/3}$?

$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$

Integral $\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx$

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

Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$

An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$

Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

A closed form for $\int_0^1{_2F_1}\left(-\frac{1}{4},\frac{5}{4};\,1;\,\frac{x}{2}\right)^2dx$

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

Integral $\int_0^\infty\left(x+5\,x^5\right)\operatorname{erfc}\left(x+x^5\right)\,dx$

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$

A closed form for $\int_0^\infty\frac{e^{-x}\ J_0(x)\ \sin\left(x\,\sqrt[3]{2}\right)}{x}dx$

What are examples of unexpected algebraic numbers of high degree occured in some math problems?

A closed form of $\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx$

Simplify $7\arctan^2\varphi+2\arctan^2\varphi^3-\arctan^2\varphi^5$

A nicer closed form? $\int_0^1 \frac{\log (x) \log \left(x^2-x+1\right)}{x^2-x+2} \, dx$

What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

Do harmonic numbers have a “closed-form” expression?

Need help with $\int_0^\infty\arctan\left(e^{-x}\right)\,\arctan\left(e^{-2x}\right)\,dx$

Find the value of $\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$

Funny identities

Does $\int_0^{\infty} \left( p + q W \left( r e^{- s x + t} \right) + u x \right) e^{- x} d x$ have a closed-form expression?

Integral $\int_0^\infty\operatorname{arccot}(x)\,\operatorname{arccot}(2x)\,\operatorname{arccot}(5x)\,dx$