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Closed-form
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Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$
August 1st, 2022
closed-form
hyperbolic-functions
calculus
integration
definite-integrals
Closed form of $\mathscr{R}=\int_0^{\pi/2}\sin^2x\,\ln\big(\sin^2(\tan x)\big)\,\,dx$
July 26th, 2020
closed-form
calculus
real-analysis
integration
Find the closed form of $\sum_{n=1}^{\infty} \frac{H_{ n}}{2^nn^4}$
December 12th, 2020
calculus
integration
sequences-and-series
closed-form
harmonic-numbers
Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^32^n}$
August 1st, 2022
closed-form
sequences-and-series
real-analysis
harmonic-numbers
zeta-functions
Arithmetic-geometric mean of 3 numbers
September 24th, 2020
sequences-and-series
elliptic-integrals
closed-form
reference-request
calculus
Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$
December 30th, 2020
improper-integrals
logarithms
integration
calculus
closed-form
Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$
May 29th, 2020
special-functions
definite-integrals
hypergeometric-function
closed-form
gamma-function
Closed form for $\left(1+\left(\frac{1}{2}+\left(\frac{1}{3}+\left(\frac{1}{4}+\cdots\right)^2\right)^2\right)^2\right)^2$?
March 26th, 2020
closed-form
sequences-and-series
nested-radicals
Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$
August 1st, 2022
special-functions
hypergeometric-function
definite-integrals
calculus
closed-form
Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$
August 1st, 2022
calculus
integration
definite-integrals
closed-form
conjectures
Closed form for $\int_0^1\sqrt{\frac{2-x}{(1-x)\,x}}\,\log\left(\frac{(2-x)\,x}{1-x}\right)dx$
November 24th, 2020
integration
closed-form
logarithms
definite-integrals
calculus
Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,\mathrm{d}x$ have a closed form?
August 1st, 2022
closed-form
special-functions
definite-integrals
integration
What is a closed form for ${\large\int}_0^1\frac{\ln^3(1+x)\,\ln^2x}xdx$?
February 22nd, 2020
polylogarithm
harmonic-numbers
closed-form
logarithms
integration
Are there other cases similar to Herglotz's integral $\int_0^1\frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\ \mathrm dt$?
August 1st, 2022
closed-form
calculus
integration
definite-integrals
logarithms
What does closed form solution usually mean?
August 17th, 2020
closed-form
soft-question
An integral involving Airy functions $\int_0^\infty\frac{x^p}{\operatorname{Ai}^2 x + \operatorname{Bi}^2 x}\mathrm dx$
August 1st, 2022
calculus
integration
special-functions
definite-integrals
closed-form
Infinite Series $\sum\limits_{n=1}^\infty\left(\frac{H_n}n\right)^2$
August 1st, 2022
real-analysis
sequences-and-series
complex-analysis
closed-form
harmonic-numbers
Infinite Series $\sum_{n=1}^\infty\frac{H_n}{n^22^n}$
September 9th, 2020
real-analysis
sequences-and-series
closed-form
zeta-functions
harmonic-numbers
Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$
August 1st, 2022
calculus
integration
definite-integrals
closed-form
hypergeometric-function
What is the solution of $\cos(x)=x$?
August 1st, 2022
transcendental-equations
trigonometry
closed-form
Integral $\int_0^\infty\frac{\operatorname{arccot}\left(\sqrt{x}-2\,\sqrt{x+1}\right)}{x+1}\mathrm dx$
August 1st, 2022
closed-form
improper-integrals
trigonometry
integration
calculus
Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$
August 3rd, 2020
definite-integrals
integration
calculus
closed-form
logarithms
Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$
August 30th, 2020
closed-form
contest-math
logarithms
integration
calculus
Integral $\int_0^\infty\left(x+5\,x^5\right)\operatorname{erfc}\left(x+x^5\right)\,dx$
October 13th, 2020
error-function
closed-form
definite-integrals
integration
calculus
Looking for a proof of Cleo's result for ${\large\int}_0^\infty\operatorname{Ei}^4(-x)\,dx$
April 1st, 2022
calculus
integration
definite-integrals
special-functions
closed-form
$\int_0^1\arctan\,_4F_3\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};\frac{1}{2},\frac{3}{4},\frac{5}{4};\frac{x}{64}\right)\,\mathrm dx$
January 3rd, 2020
hypergeometric-function
closed-form
special-functions
trigonometry
integration
Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
July 29th, 2020
closed-form
fourier-analysis
complex-analysis
sequences-and-series
calculus
Closed form for the integral $\int_{0}^{\infty}\frac{\ln^{2}(x)\ln(1+x)}{(1-x)(x^{2}+1)}dx$
September 16th, 2020
improper-integrals
real-analysis
integration
definite-integrals
closed-form
The entry-level PhD integral: $\int_0^\infty\frac{\sin 3x\sin 4x\sin5x\cos6x}{x\sin^2 x\cosh x}\ dx$
May 19th, 2020
calculus
integration
trigonometry
definite-integrals
closed-form
Integral $\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \right) \, \mathrm dx$
August 1st, 2022
calculus
integration
definite-integrals
closed-form
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