A "closed form expression" is any representation of a mathematical expression in terms of "known" functions, "known" usually being replaced with "elementary".

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21
votes
2answers
6k views

What does closed form solution usually mean?

This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by Qiaochu Yuan. If an equation $f(x)=0$ has no closed form ...
16
votes
2answers
686 views

Alternating harmonic sum $\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k$

How to analytically prove $$\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k=-\frac{11\pi^4}{360}+\frac{\ln^42-\pi^2\ln^22}{12}+2\mathrm{Li}_4\left(\frac12\right)+\frac{7\ln 2}{4}\zeta(3) $$ As O.L answer ...
153
votes
5answers
37k views

Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The ...
17
votes
3answers
575 views

Closed Form for the Imaginary Part of $\text{Li}_3\Big(\frac{1+i}2\Big)$

$\qquad\qquad$ Is there any closed form expression for the imaginary part of $~\text{Li}_3\bigg(\dfrac{1+i}2\bigg)$ ? Motivation: We already know that ...
17
votes
5answers
959 views

Find a closed form for this sequence: $a_{n+1} = a_n + a_n^{-1}$

Today, we had a math class, where we had to show, that $a_{100} > 14$ for $$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$ Apart from this task, I asked my self: Is there a closed form for this ...
11
votes
5answers
414 views

Closed form for $n$th derivative of exponential of $f$

What is the closed form for: $$\frac{\partial^n}{\partial x^n}\exp(f(x))=\exp(f(x))\cdot[????]$$
56
votes
3answers
4k views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( ...
54
votes
6answers
2k views

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx\tag1$$ I suspect it might exist because there are similar integrals having closed forms: ...
20
votes
3answers
487 views

Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n}$

Please help me to find a closed form for the sum $$\sum_{n=1}^\infty\frac{(-1)^n n^4 H_n}{2^n},$$ where $H_n$ are harmonic numbers: $$H_n=\sum_{k=1}^n\frac{1}{k}=\frac{\Gamma'(n+1)}{n!}+\gamma.$$
26
votes
4answers
868 views

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

I want to find a closed form for this integral: $$I=\int_0^1\frac{\ln^2x}{\sqrt{x^2-x+1}}dx\tag1$$ Mathematica and Maple cannot evaluate it directly, and I was not able to find it in tables. A numeric ...
26
votes
4answers
670 views

A sum containing harmonic numbers $\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n}$

I'm trying to find a closed form for the following sum $$\sum_{n=1}^\infty\frac{H_n}{n^3\,2^n},$$ where $H_n=\displaystyle\sum_{k=1}^n\frac{1}{k}$ is a harmonic number. Could you help me with it?
17
votes
2answers
523 views

Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.

What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
23
votes
3answers
455 views

$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2)$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
23
votes
6answers
590 views

Computing $ \int_0^\infty \frac{\log x}{\exp x} \ dx $

I know that $$ \int_0^\infty \frac{\log x}{\exp x} = -\gamma $$ where $ \gamma $ is the Euler-Mascheroni constant, but I have no idea how to prove this. The series definition of $ \gamma $ leads me ...
25
votes
3answers
536 views

Integral of Combination Log and Inverse Trig Function

Does the following integral have a closed-form ?: \begin{equation} \int_{0}^{1}{\ln\left(\,x\,\right) \over 1 + x}\,\arccos\left(\,x\,\right) \,{\rm d}x \end{equation} This integral has been ...
86
votes
2answers
3k views

How to prove $\int_0^1\tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$

How can one prove that $$\int_0^1 \tan^{-1}\left[\frac{\tanh^{-1}x-\tan^{-1}x}{\pi+\tanh^{-1}x-\tan^{-1}x}\right]\frac{dx}{x}$$ $$=\frac{\pi}{8}\ln\frac{\pi^2}{8}?$$
36
votes
4answers
2k views

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx.$$ I do not have a strong reason to be sure it exists, but I ...
38
votes
2answers
1k views

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

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...
30
votes
1answer
813 views

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$$
29
votes
2answers
1k views

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

I'm struggling with this integral $$I=\int_0^1\frac{1-x^2+\left(1+x^2\right)\ln x}{\left(x+x^2\right)\ln^3x}dx.\tag1$$ Mathematica could not evaluate it in a closed form. Its numeric value is ...
11
votes
1answer
346 views

What would qualify as a valid reason to believe there is a closed form?

I noticed that almost every non-homework-level integral posted on this site prompts somebody to ask "Do you have any reason to believe there is a closed form?" (some recent examples here and here) I ...
14
votes
2answers
309 views

$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$

Is there any closed-form representation for the following integral? $$\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx,$$ where $\mathrm{sech}\,x$ is the hyperbolic secant, ...
20
votes
1answer
484 views

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also ...
13
votes
3answers
322 views

$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$

I'm looking for a way to prove $$\sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{(-1)^m E_{2m}\pi^{2m+1}}{4^{m+1}(2m)!}$$ Since ...
27
votes
3answers
583 views

Prove $\displaystyle \int_{0}^{\pi/2} \ln \left(x^{2} + (\ln\cos x)^2 \right) \, dx=\pi\ln\ln2 $

How to prove $$ \int_{0}^{\pi/2}\ln\left(\,x^{2} + \ln^{2}\left(\,\cos\left(\,x\,\right)\,\right) \,\right)\,{\rm d}x\ =\ \pi\ln\left(\,\ln\left(\, 2\,\right)\,\right) $$ I don't know how to ...
19
votes
1answer
662 views

How can I find $\sum\limits_{n=0}^{\infty}\left(\frac{(-1)^n}{2n+1}\sum\limits_{k=0}^{2n}\frac{1}{2n+4k+3}\right)$?

prove that$$\sum_{n=0}^{\infty}\left(\frac{(-1)^n}{2n+1}\sum_{k=0}^{2n}\frac{1}{2n+4k+3}\right)=\frac{3\pi}{8}\log(\frac{1+\sqrt5}{2})-\frac{\pi}{16}\log5 $$ This problem, I think use ...
18
votes
4answers
884 views

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

I am trying to prove that $$\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx = \frac{\pi^3}{16}-3G\log 2 \tag{1}$$ where $G$ is Catalan's Constant. I was able to express it in terms of ...
18
votes
2answers
468 views

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

I am interested in finding a general formula for the following integral: $$\int_0^\infty J_\nu(x)^3dx,\tag1$$ where $J_\nu(x)$ is the Bessel function of the first kind: $$J_\nu(x)=\sum ...
9
votes
3answers
252 views

Harmonic Numbers series I

Can it be shown that \begin{align} \sum_{n=1}^{\infty} \binom{2n}{n} \ \frac{H_{n+1}}{n+1} \ \left(\frac{3}{16}\right)^{n} = \frac{5}{3} + \frac{8}{3} \ \ln 2 - \frac{8}{3} \ \ln 3 \end{align} where ...
21
votes
3answers
2k views

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

One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
13
votes
2answers
403 views

A closed form of $\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$

This integral has been bugging me since yesterday: $$\int_0^1\frac{\ln\ln\left({1}/{x}\right)}{x^2-x+1}\mathrm dx$$ I've tried substitution $y={1}/{x}$ and $e^y={1}/{x}$, but those didn't help ...
7
votes
4answers
217 views

How to evaluate $\int_0^\infty \frac{1}{x^n+1} dx$

Noticed that the integral $$\int_0^\infty \frac{1}{x^n+1} dx$$ is often approached with partial fraction decomposition, but this gets increasingly ugly as $n$ gets bigger. Is there a neat trick to do ...
10
votes
1answer
192 views

Proving that $\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: ...
3
votes
2answers
586 views

Solving for the closed term solution of a third order recurrence relation with real constant coefficients

How would you solve for the closed term form of $a(n)$ given the general form of the third order linear homogenous recurrence relation with real constant coefficients. ...
37
votes
3answers
1k views

An integral involving Fresnel integrals $\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$

I need to calculate the following integral: $$\int_0^\infty \left(\left(2\ S(x)-1\right)^2+\left(2\ C(x)-1\right)^2\right)^2 x\ \mathrm dx,$$ where $$S(x)=\int_0^x\sin\frac{\pi z^2}{2}\mathrm dz,$$ ...
33
votes
2answers
2k views

Integral $\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^1\frac{\arctan^2x}{\sqrt{1-x^2}}\mathrm dx$$ It also can be represented as $$I=\int_0^{\pi/4}\frac{\phi^2}{\cos \phi\,\sqrt{\cos ...
23
votes
2answers
443 views

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

Is it possible to evaluate the following integral in a closed form? $$\int_0^\infty\frac{\sqrt[\phi]{x}\ \arctan x}{\left(x^\phi+1\right)^2}dx,$$ where $\phi$ is the golden ratio: ...
26
votes
2answers
804 views

Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?

This post of Boris Bukh mentions amazing Gustav Herglotz's integral $$\int_0^1\frac{\ln\left(1+t^{\,4\,+\,\sqrt{\vphantom{\large A}\,15\,}\,}\right)}{1+t}\ \mathrm ...
25
votes
3answers
767 views

A conjectured closed form of $\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx$

Consider the following integral: $$\mathcal{I}=\int_0^\infty\frac{x-1}{\sqrt{2^x-1}\ \ln\left(2^x-1\right)}dx.$$ I tried to evaluate $\mathcal{I}$ in a closed form (both manually and using ...
22
votes
2answers
466 views

How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?

I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$, where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$. Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ ...
25
votes
1answer
564 views

Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$

This question came up in the process of finding solution to another problem. Eventually, the problem was solved avoiding calculation of this sum, but it looks quite interesting on its own. Is there a ...
25
votes
1answer
495 views

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

I need help with calculating this integral: $$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx,$$ where ...
23
votes
3answers
549 views

Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$

Here is a challenging one maybe some would like a go at. Show that: ...
11
votes
1answer
200 views

Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$

I need help with calculating this sum: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
20
votes
4answers
1k views

Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$

I need to solve $$ \int_0^{\Large\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx $$ I tried to use symmetric properties of the trigonometric functions as is commonly used to compute $$ ...
26
votes
3answers
1k views

Integrating $\int_0^ex^{1/x}\ \mathrm dx$

Compute $$\int_0^ex^{1/x}\;\mathrm dx.$$ There is an analytical anti-derivative found in this answer. How does one compute this? Using the anti-derivative approach we have $$\int x^{1/x}\;\mathrm d ...
19
votes
3answers
420 views

Closed form for integral $\int_{0}^{\pi} \left[1 - r \cos\left(\phi\right)\right]^{-n} \phi \,{\rm d}\phi$

Is there a closed form for $$I_n =\int_{0}^{\pi} \frac{\phi}{(1 - r \cos\phi)^n} \,{\rm d}\phi $$ for $\left\vert\,r\,\right\vert < 1$ real and $n > 0$ integer ? The solution to this integral ...
18
votes
3answers
181 views

Integrals of integer powers of dilogarithm function

I'm interested in evaluating integrals of positive integer powers of the dilogarithm function. I'd like to see the general case tackled if possible, or barring that then as many particular cases as ...
9
votes
2answers
196 views

Evaluation of a dilogarithmic integral

Problem. Prove that the following dilogarithmic integral has the indicated value: $$\int_{0}^{1}\mathrm{d}x ...
6
votes
4answers
326 views

How to compute $\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$

$$\int_{-\infty}^\infty\exp\left(-\frac{(x^2-13x-1)^2}{611x^2}\right)\ dx$$ WolframAlpha gives a numerical answer of $43.8122$, which appears to be $\sqrt{611\pi}$. And playing with that, it seems ...