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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1
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1answer
57 views

How to find algebraic simplification for recurrence relation with closed-form solution, specifically for the Lucas-Lehmer primality test

I have a question based on the section Proof of correctness in the article Lucas-Lehmer primality test, see following link. ...
18
votes
3answers
399 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 ...
19
votes
3answers
587 views

Integral $\int_0^\infty x^2\,e^{-x^2}\operatorname{erf}(x)\,\log(x)\,dx$

I need to evaluate this integral: $$I=\int_0^\infty x^2\,e^{-x^2}\operatorname{erf}(x)\,\log(x)\,dx\tag1$$ I tried to do this in Mathematica and it returned a result of the form ...
24
votes
2answers
274 views

Need help with $\int_0^\infty e^{-x}\ln\ln\left(e^x+\sqrt{e^{2x}-1}\right)\,dx$

I need help with this integral: $$\int_0^\infty e^{-x}\ln\ln\left(e^x+\sqrt{e^{2x}-1}\right)\,dx\approx0.20597312051214...$$ Is it possible to evaluated it in a closed form?
10
votes
2answers
337 views

Integral $S_\ell(r) = \int_0^{\pi}\int_{\phi}^{\pi}\frac{(1+ r \cos \psi)^{\ell+1}}{(1+ r \cos \phi)^\ell} \rm d\psi \ \rm d\phi $

Is there a closed form for $|r|<1$ and $\ell>0$ integer? The solution for the special cases $\ell=2$ and $4$ would also be interesting if the general case is not available. Integrating ...
1
vote
3answers
208 views

Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Is there a closed form for $n>0$ integer, $m\neq 0$ integer, and $|r|<1$ real?
18
votes
1answer
539 views

Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

Another integral similar to my previous question: $$\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$$ Could you suggets how to evaluate ...
19
votes
1answer
388 views

Derivative of the Meijer G-function with respect to one of its parameters

Are there any approaches that allow to find a derivative of the Meijer G-function with respect to one of its parameters in a closed form (or at least numerically with a high precision and in ...
21
votes
2answers
699 views

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

Could you suggest any ideas how to evaluate this integral? Is there a closed-form result? $$\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$$
19
votes
1answer
257 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
21
votes
3answers
393 views

Integral $\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$

I need to evaluate this integral to a high precision: $$\large I=\int_0^\infty{_1F_2}\left(\begin{array}{c}\tfrac12\\1,\tfrac32\end{array}\middle|-x\right)\frac{dx}{1+4\,x}$$ Symbolic integration in ...
23
votes
3answers
461 views

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

Is it possible to evaluate in a closed form integrals containing a squared hypergeometric function, like in this example? ...
0
votes
1answer
45 views

The closed form of a sum of mod(k,m) where k goes from 1 to a arbitrary number.

Is there a closed form for $\sum_{n=0}^{C} mod(n,m)$ for arbitrary integers m ?
7
votes
1answer
216 views

Simplification of $G_{2,4}^{4,2}\left(\frac18,\frac12\middle|\begin{array}{c}\frac12,\frac12\\0,0,\frac12,\frac12\\\end{array}\right)$

In this post Cleo gives a misterious result containing the following generalized Meijer G-function: ...
22
votes
2answers
703 views

Closed form for $\int_0^\infty\frac{\sin x\,\cdot\,\operatorname{Ci}x-\cos x\,\cdot\,\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx$

Is it possible to find a closed form for this integral? $$\mathcal{S}=\int_0^\infty\frac{\sin x\cdot\operatorname{Ci}x-\cos x\cdot\operatorname{Si}x}{\sqrt{16\,x^2+1}}dx,$$ where $\operatorname{Ci}x$ ...
8
votes
0answers
82 views

Derivatives of the Struve functions $H_\nu(x)$, $L_\nu(x)$ and other related functions w.r.t. their index $\nu$

There are some known formulae for derivatives of the Bessel functions $J_\nu(x),\,$$Y_\nu(x),\,$$K_\nu(x),\,$$I_\nu(x)\,$with respect to their index $\nu$ for certain values of $\nu$, e.g. ...
5
votes
1answer
200 views

closed form for $\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx$

How can I find a closed form for $$\int_0^{\infty}\log^n\left(\frac{e^x}{e^x-1}\right)dx, n\in\mathbb{N}$$
0
votes
2answers
91 views

How can I compute this sum of binomial

Is there any way to compute the following sum: $\displaystyle{ \sum_{\ell = {n + 1 \over{\vphantom{\LARGE A}2}}}^{n}{n \choose \ell}5^{n - \ell}}$ where $n$ is odd. Thank you.
0
votes
0answers
70 views

Close formula for triple sum binomial coefficient

I need to compute the following sum or to find a lower and upper bound that limit the sum: $\sum_{l=\frac{n+1}{2}}^n \binom{n}{l} \sum_{t=0}^{n-l} \binom{l}{t} 2^{l-t} \sum_{m=t}^{n-l} \binom{n-l}{m} ...
34
votes
4answers
1k 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 ...
29
votes
1answer
797 views

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

Regarding this problem, I conjectured that $$ I(r, s) = \int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx = 4 \pi ...
25
votes
1answer
310 views

Conjectured closed form for $\int_0^1x^{2\,q-1}\,K(x)^2dx$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind

I am interested in a general closed-form formula for integrals of the following form: $$\mathcal{J}_q=\int_0^1x^{2\,q-1}\,K(x)^2dx,\tag0$$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ ...
4
votes
1answer
180 views

Closed form for a fixed point of the exponential function?

Let $$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} $$ denote the exponential function, which is defined on the entire complex plane. There is a fixed point of this function at $w= a+bi$ where $a \approx ...
26
votes
2answers
1k views

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

This is somewhat similar to my previous question: Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$ Is it possible to find a closed form ...
9
votes
0answers
129 views

A closed form for $\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx,\ a\notin\mathbb{Z}^+$

Let $$I(a)=\int_0^\infty\left(\frac{2^{-x}-3^{-x}}x\right)^adx.$$ $I(a)$ has closed form representations for all $a\in\mathbb{Z}^+$. Is there any algebraic (or at least period) ...
25
votes
2answers
388 views

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

I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ...
27
votes
2answers
954 views

Integral $\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$I=\int_0^{\pi/2}\arctan^2\left(\frac{6\sin x}{3+\cos 2x}\right)\mathrm dx$$
20
votes
2answers
320 views

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

I need to evaluate the following integral: $$\int_0^1\frac{\ln x}{x^2+1}\cdot\ln\left(\frac{3\,x^2+1}{x^2+3}\right)dx.$$ Could you suggest how to find a closed form for it? I am not sure if there is ...
25
votes
3answers
547 views

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

There is a curious known integral: $$\int_0^1\frac{\ln\left(1+x^{2+\sqrt{3\vphantom{\large3}}}\right)}{1+x}dx=\frac{\pi^2}{12}\left(1-\sqrt{3\vphantom{\large3}}\right)+\ln ...
10
votes
1answer
275 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 ...
140
votes
5answers
35k 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 ...
0
votes
0answers
73 views

Closed form for the following sum

I found this sum in an old math problems book and it asks me to find its closed form. And for the life of me I can't find. Here it is ...
28
votes
3answers
1k views

Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$$
17
votes
1answer
298 views

A closed form for $\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx=\int_0^\infty\ln x\cdot\ln\left(1+\frac1{e^{-x}+e^x}\right)dx$$ I tried to ...
28
votes
1answer
749 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
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 ...
22
votes
3answers
1k views

Integral $\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}\mathrm dx$

I need your assistance with evaluating the integral $$\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}dx$$ I tried manual integration by parts, but it seemed to only ...
8
votes
1answer
270 views

Indefinite integral $\int \arcsin \left(k\sin x\right) dx$

It would take too long to explain the context reasonably well - but in short, this integral, or rather its equivalent $$\int\frac{x\cos x\,dx}{\sqrt{1-k^2\sin^2x}},\qquad 0<k<1$$ is related to ...
1
vote
1answer
152 views

Closed form for general recursive function

Does a closed form exists for general recursive functions? my guess is not, but what types can be solved or what are the constraints on a recursive function so it has a closed form, what are some ...
0
votes
1answer
202 views

Find the probability that splitting the unit interval into three random segments results in the sides of a triangle.

I found the question here. The precise problem is The unit interval is broken at two randomly chosen points along its length. Show that the probability that the lengths of the resulting three ...
21
votes
3answers
495 views

A closed form for $\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx$

I need to a evaluate the following integral $$I=\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx.$$ Both Mathematica and Maple failed to evaluate it in a closed form, and lookups of the ...
0
votes
1answer
52 views

What will be the closed formula for the following recursive function?

What will be the closed formula for the following recursive function? F(n) = F(n/2) +1 if n is even F(n) = F(n-1) + 1 if n is odd F(1) = 0 How do we generate closed formula for such ...
13
votes
1answer
176 views

How to calclulate a derivate of a hypergeometric function w.r.t. one of its parameters?

Is it possible to take a derivative of a hypergeometric function w.r.t. one of its parameters and express it in a closed form? I am particularly interested in this case: ...
21
votes
1answer
274 views

Integral $\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\frac{1}{\sqrt[3]{x}}\left(1+\log\frac{1+e^{x-1}}{1+e^x}\right)dx$$
21
votes
2answers
474 views

Conjecture: $\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\big(\!\frac{2x}{1+x}\!\big)\,dx\stackrel ?=\frac\pi{5\sqrt2}$

$$\int_0^1\frac{3x^3-2x}{(1+x)\sqrt{1-x}}K\left(\frac{2x}{1+x}\right)\,dx\stackrel ?=\frac\pi{5\sqrt2}$$ The integral above comes from the evaluation of the integral ...
0
votes
1answer
45 views

A closed form for $\sum_{i=1}^{n} \prod_{k=1}^{i+2} (3k+2)$

I need to calculate the following expression. Is there any explanation to convert this expression into normal expression without those letters for sum and the product? Just normal expression. $$ ...
14
votes
1answer
311 views

Further our knowledge of a certain class of integral involving logarithms.

$\newcommand{\limitp}{\alpha}\newcommand{\innerp}{\beta}$I am fascinated by definite integrals. Exploring math.stackexchange, I have found many interesting integrals of the form $$ ...
41
votes
1answer
1k views

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

$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. ...
8
votes
2answers
391 views

Does the sequence $1,-1,1,1,-1,1,1,1,-1,1,1,1,1,-1,1,1,1,1,1,-1,\ldots$ have a closed form?

Question : Can we represent the following sequence $\{a_n\}\ (n\ge 0)$ as a closed form?$$a_n : 1,-1,1,1,-1,1,1,1,-1,1,1,1,1,-1,1,1,1,1,1,-1,\ldots$$ Suppose that there exist ${(i+1)}$ $1_s$ ...
46
votes
2answers
1k views

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

Let $$\alpha=\sqrt{6}\ \sqrt{12+7\,\sqrt3}-3\,\sqrt3-6.\tag1$$ Note that $\alpha$ is the unique positive root of the polynomial equation $$\alpha^4+24\,\alpha^3+18\,\alpha^2-27=0.\tag2$$ Now consider ...