New posts in closed-form

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

sequences-and-series summation gamma-function closed-form

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

calculus definite-integrals closed-form conjectures elliptic-integrals

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

calculus integration logarithms improper-integrals closed-form

A closed form of $\sum_{n=1}^{\infty}(-1)^{n}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)$?

calculus integration sequences-and-series closed-form

Integral $\int_0^{\large\frac{\pi}{4}}\left(\frac{1}{\log(\tan(x))}+\frac{1}{1-\tan(x)}\right)dx$

calculus integration definite-integrals closed-form

Integrals of integer powers of dilogarithm function

integration definite-integrals closed-form riemann-zeta polylogarithm

Closed form of $\sum\limits^\infty_{n=1} \left( {(-1)^n}\left( 1+n\ln(\frac{2n-1}{2n+1}) \right) \right)$

sequences-and-series fourier-series closed-form

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

calculus integration logarithms improper-integrals closed-form

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

calculus sequences-and-series special-functions closed-form hypergeometric-function

Find closed form for $1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, \ldots$

sequences-and-series closed-form

Closed form of $\frac{e^{-\frac{\pi}{5}}}{1+\frac{e^{-\pi}}{1+\frac{e^{-2\pi}}{1+\frac{e^{-3\pi}}{1+\ddots}}}}$

complex-analysis closed-form continued-fractions modular-function

Looking for a proof of Cleo's result for ${\large\int}_0^\infty\operatorname{Ei}^4(-x)\,dx$

calculus integration definite-integrals special-functions closed-form

Prove ${\large\int}_{-1}^1\frac{dx}{\sqrt[3]{9+4\sqrt5\,x}\ \left(1-x^2\right)^{2/3}}=\frac{3^{3/2}}{2^{4/3}5^{5/6}\pi }\Gamma^3\left(\frac13\right)$

calculus definite-integrals special-functions closed-form hypergeometric-function

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

calculus integration improper-integrals closed-form harmonic-numbers

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

special-functions gamma-function closed-form conjectures hypergeometric-function

How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?

calculus complex-analysis integration definite-integrals closed-form

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

calculus integration definite-integrals logarithms closed-form

Conjectural closed-form of $\int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$

integration definite-integrals closed-form zeta-functions polylogarithm

Closed-forms for several tough integrals

calculus integration definite-integrals improper-integrals closed-form

How to compute $\int_0^\infty \frac{1}{(1+x^{\varphi})^{\varphi}}\,dx$?

calculus integration definite-integrals closed-form golden-ratio