New posts in closed-form

An incorrect answer for an integral

definite-integrals closed-form nested-radicals integration

Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

definite-integrals logarithms closed-form polylogarithm polygamma

Closed-form of log gamma integral $\int_0^z\ln\Gamma(t)~dt$ for $z =1,\frac12, \frac13, \frac14, \frac16,$ using Catalan's and Gieseking's constant?

definite-integrals logarithms special-functions closed-form gamma-function

Infinite Series $\sum\limits_{n=1}^{\infty}\frac{(-1)^n}{n}\left\lfloor\frac{\log(n)}{\log(2)}\right\rfloor$

real-analysis sequences-and-series summation closed-form euler-mascheroni-constant

Closed-form of $\int_{0}^{\infty} \frac{{\text{Li}}_2^3(-x)}{x^3}\,dx$

calculus integration definite-integrals closed-form polylogarithm

Closed- form of $\int_0^1 \frac{{\text{Li}}_3^2(-x)}{x^2}\,dx$

calculus integration definite-integrals closed-form polylogarithm

Evaluating $\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$

definite-integrals special-functions closed-form

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

calculus sequences-and-series trigonometry closed-form trigonometric-series

How to evaluate $\int_0^\infty\frac{\frac{\pi^2}{6}-\operatorname{Li}_2\left(e^{-x}\right)-\operatorname{Li}_2\left(e^{-\frac{1}{x}}\right)}{x}dx$

calculus definite-integrals improper-integrals closed-form polylogarithm

On the integral $I(a)=\int_0^1\frac{\log(a+t^2)}{1+t^2}\mathrm dt$

integration definite-integrals closed-form

Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{n}\left(e^{-\pi x}\right)}{1+x^2} dx $

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

Prove $_4F_3(1/8,3/8,5/8,7/8;1/4,1/2,3/4;1/2)=\frac{\sqrt{2-\sqrt2+\sqrt{2-\sqrt2}}+\sqrt{2+\sqrt2+\sqrt{2+\sqrt2}}}{2\,\sqrt2}$

calculus closed-form conjectures hypergeometric-function radicals

Evaluate $\int_{-\pi/4}^{\pi/4}\frac{x}{\sin x}\mathrm{d}x$

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

How to find $\int_0^\pi (\log(1 - 2a \cos(x) + a^2))^2 \mathrm{d}x, \quad a>1$?

calculus integration contest-math closed-form

Tough Inverse Fourier Transform

ordinary-differential-equations fourier-analysis special-functions definite-integrals closed-form

Closed-form of $\sum_{n=0}^\infty\;(-1)^n \frac{\left(2-\sqrt{3}\right)^{2n+1}}{(2n+1)^2\quad}$

calculus sequences-and-series summation closed-form

Closed form for $\sum_{k=1}^\infty(\zeta(4k+1)-1)$

sequences-and-series closed-form riemann-zeta parity

How to compute $\int_0^\infty \frac{x^4}{(x^4+ x^2 +1)^3} dx =\frac{\pi}{48\sqrt{3}}$?

calculus integration definite-integrals improper-integrals closed-form

How to compute the monstrous $ \int_0^{\frac{e-1}{e}}{\frac{x(2-x)}{(1-x)}\frac{\log\left(\log\left(1+\frac{x^2}{2-2x}\right)\right)}{2-2x+x^2}dx} $

calculus integration definite-integrals closed-form

Solving $\int_0^\infty\dfrac{dx}{(1+x^n)^n}=1$

calculus integration definite-integrals closed-form