New posts in closed-form

Closed form of $\int_0^\infty \ln \left( \frac{x^2+2kx\cos b+k^2}{x^2+2kx\cos a+k^2}\right) \;\frac{\mathrm dx}{x}$

calculus integration definite-integrals improper-integrals closed-form

Proof $\int_{-\infty}^\infty\frac{dx}{\left(e^x+e^{-x}+e^{ix\sqrt{3}}\right)^2}=\frac{1}{3}$

integration definite-integrals contour-integration closed-form alternative-proof

prove $\ln(1+x^2)\arctan x=-2\sum_{n=1}^\infty \frac{(-1)^n H_{2n}}{2n+1}x^{2n+1}$

real-analysis calculus generating-functions closed-form harmonic-numbers

Express solutions of equation $ \tan x= x $ in closed form

sequences-and-series trigonometry roots closed-form

Is there a closed-form solution for $\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{nm(3n+m)}$?

integration definite-integrals summation closed-form

Infinite Series $1+\frac12-\frac23+\frac14+\frac15-\frac26+\cdots$

calculus real-analysis sequences-and-series summation closed-form

How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?

calculus sequences-and-series recurrence-relations closed-form harmonic-numbers

A closed form for a lot of integrals on the logarithm

real-analysis sequences-and-series definite-integrals closed-form polylogarithm

Conjecture ${\large\int}_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\frac1{12\,x^2}-\frac1{\left(e^x-1\right)x^3}\right]dx=\frac{\zeta(3)}{8\pi^2}$

calculus integration definite-integrals closed-form experimental-mathematics

Closed form for $\int_{-\infty}^\infty\operatorname{sech}(x)\operatorname{sech}(a\, x)\ dx$

calculus integration definite-integrals closed-form hyperbolic-functions

Closed form for $\int_0^{\pi/2}\frac{\sqrt{1+\sin\phi}}{\sqrt{\sin2\phi}\,\sqrt{\sin\phi+\cos\phi}}d\phi$

calculus integration definite-integrals improper-integrals closed-form

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

calculus integration special-functions closed-form hypergeometric-function

Improper Integral $\int\limits_0^1\frac{\ln(x)}{x^2-1}\,dx$

calculus integration definite-integrals improper-integrals closed-form

Evaluating $\int_0^\pi\arctan\bigl(\frac{\ln\sin x}{x}\bigr)\mathrm{d}x$

calculus integration definite-integrals logarithms closed-form

A definite integral $\int_0^\infty\frac{2-\cos x}{\left(1+x^4\right)\,\left(5-4\cos x\right)}dx$

calculus integration trigonometry definite-integrals closed-form

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

calculus integration definite-integrals logarithms closed-form

Does $\int_{-1}^1\frac{\arctan x}{\text{arctanh}\,x}\,\mathrm{d}x$ have a closed form?

integration definite-integrals special-functions closed-form

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

calculus integration definite-integrals logarithms closed-form

Closed form for $\sum_{n=1}^\infty\frac{\psi(n+\frac{5}{4})}{(1+2n)(1+4n)^2}$

sequences-and-series special-functions gamma-function closed-form combinatorics

Integral $\int_0^1\ln\ln\,_3F_2\left(\frac{1}{4},\frac{1}{2},\frac{3}{4};\frac{2}{3},\frac{4}{3};x\right)\,dx$

calculus integration logarithms closed-form hypergeometric-function