New posts in closed-form

Closed form for $I=\int_{0}^{\infty}\frac{x^n}{x^2+u^2}\tanh(x) \, dx$

integration definite-integrals contour-integration closed-form complex-integration

Difficult trigonometric integral. [Solved]

integration trigonometry definite-integrals closed-form

Prove ${_2F_1}\left({{\tfrac16,\tfrac23}\atop{\tfrac56}}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

calculus special-functions closed-form hypergeometric-function

Closed form of a generating function $\sum _{n=1}^\infty x^{n^2}$

combinatorics number-theory generating-functions closed-form

Alternate proof for weighted alternating shifted central binomial sum relation

analysis asymptotics closed-form

Improper integral of a rational function:$\int_0^\infty \frac{5t^6}{1+t^{10}}dt$

calculus integration definite-integrals improper-integrals closed-form

Prove $\int_0^1 \frac{4\cos^{-1}x}{\sqrt{2x-x^2}}\,dx=\frac{8}{9\sqrt{\pi}}\left(9\Gamma(3/4)^2{}_4F_3(\cdots)+\Gamma(5/4)^2{}_4F_3(\cdots)\right)$

definite-integrals closed-form gamma-function hypergeometric-function trigonometric-integrals

Prove that $\int_0^1\frac{\operatorname{Li}_3(1-z)}{\sqrt{z(1-z)}}\mathrm dz=-\frac{\pi^3}{3}\log 2+\frac{4\pi}3\log^3 2+2\pi\zeta(3)$

integration definite-integrals closed-form polylogarithm

Closed form of $\int_0^1\int_0^1\int_0^1\frac{\left(1-x^y\right)\left(1-x^z\right)\ln x}{(1-x)^3}\,\mathrm dx\;\mathrm dy\;\mathrm dz$

calculus integration definite-integrals improper-integrals closed-form

Closed form for $\int_0^1...\int_0^1\frac{1}{\left(1+\sqrt{1+x_1^2+...+x_n^2}\right)^{n+1}}\;dx_1...dx_n$

integration multivariable-calculus definite-integrals closed-form

Is there a closed form expression for the eigenvectors of a 2x2 matrix?

linear-algebra matrices eigenvalues-eigenvectors closed-form wolfram-alpha

Closed form expression for continued fraction

real-analysis sequences-and-series closed-form continued-fractions

Prime pattern found in the integral $\int_0^\pi \exp\left(\frac{\cos t}{8}\right)\cos\left(\frac{\sin t}{8}\right) \cos(nt )dt$

integration sequences-and-series prime-numbers closed-form

Find the closed form of $u_{n+1}=a_nu_n+b_n$

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

Computing an indefinite integral: $\int \frac{2n!\sin x + x^n }{e^x + \sin x + \cos x + P_n (x)}\, dx $

calculus real-analysis integration indefinite-integrals closed-form

Closed form of $\displaystyle\int_{0}^{\pi/4}\int_{\pi/2}^{\pi}\frac{(\cos x-\sin x)^{y-2}}{(\cos x+\sin x)^{y+2}}\, dy\, dx$

calculus integration multivariable-calculus definite-integrals closed-form

Integral $\int_0^\infty \frac{\ln(1+x+x^2)}{1+x^2}dx$

integration definite-integrals closed-form

Closed-form of $\int\limits_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

calculus integration definite-integrals closed-form polylogarithm

Is $\frac{1}{\pi}\int_{0}^{\infty} \Gamma(\sigma +xi)^2\,\Gamma(\sigma-xi)^2 \,dx = \frac{\Gamma(2\,\sigma)^4}{\Gamma(4\,\sigma)}$?

integration number-theory definite-integrals closed-form gamma-function

Is there a closed form for $\sum_{n=0}^{\infty}{2^{n+1}\over {2n \choose n}}\cdot\left({2n-1\over 2n+1}\right)^2?$

sequences-and-series binomial-coefficients closed-form