New posts in closed-form

A Challenging Integral $\int_0^{\frac{\pi}{2}}\log \left( x^2+\log^2(\cos x)\right)dx$

analysis integration definite-integrals closed-form

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

calculus integration closed-form conjectures elliptic-functions

A closed form of the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

calculus sequences-and-series closed-form euler-mascheroni-constant stieltjes-constants

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

special-functions logarithms definite-integrals improper-integrals closed-form

On the integral $\int_0^1\frac{dx}{\sqrt[4]x\ \sqrt{1-x}\ \sqrt[4]{1-x\,\gamma^2}}=\frac{1}{N}\,\frac{2\pi}{\sqrt{2\gamma}}$

integration definite-integrals closed-form elliptic-curves experimental-mathematics

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

calculus integration special-functions definite-integrals closed-form

Evaluating $Q = \sqrt{1!+\sqrt{2!+\sqrt{3!+\dots}}}$

radicals closed-form nested-radicals

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

calculus integration definite-integrals improper-integrals closed-form

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

calculus integration definite-integrals logarithms closed-form

A closed-form expression for the integral $\int_0^\infty\text{Ci}^{3}(x) \, \mathrm dx$

calculus integration definite-integrals special-functions closed-form

Closed-form of $\int_0^1 \frac{\ln^2(x)}{\sqrt{x(a-bx)}}\,dx$

calculus integration definite-integrals closed-form

Need help with $\int_0^\pi\arctan^2\left(\frac{\sin x}{2+\cos x}\right)dx$

calculus integration trigonometry definite-integrals closed-form

Integral $\int_0^\infty\text{Li}_2\left(e^{-\pi x}\right)\arctan x\,dx$

calculus integration definite-integrals closed-form polylogarithm

Closed-form of $\int_0^\infty \frac{1}{\left(a+\cosh x\right)^{1/n}} \, dx$ for $a=0,1$

calculus integration definite-integrals closed-form hyperbolic-functions

Integral $\int_0^1\frac{\log(1-x)}{\sqrt{x-x^3}}dx$

calculus integration definite-integrals improper-integrals closed-form

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

integration sequences-and-series definite-integrals closed-form

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

sequences-and-series numerical-methods approximation closed-form

Expressing the integral $\int_{0}^{1}\frac{\mathrm{d}x}{\sqrt{\left(1-x^3\right)\left(1-a^6x^3\right)}}$ in terms of elliptic integrals

integration definite-integrals closed-form hypergeometric-function elliptic-integrals

Conjectured closed form for $\sum_{n=-\infty}^\infty\frac{1}{\cosh\pi n+\frac{1}{\sqrt{2}}}$

calculus sequences-and-series analysis number-theory closed-form

Family of definite integrals involving Dedekind eta function of a complex argument, $\int_0^{\infty} \eta^k(ix)dx$

integration definite-integrals special-functions closed-form theta-functions