New posts in closed-form

Integrate $\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$

calculus trigonometry special-functions definite-integrals closed-form

Closed form for $\int_0^\infty\left(\int_0^1\frac1{\sqrt{1-y^2}\sqrt{1+x^2\,y^2}}\mathrm dy\right)^3\mathrm dx.$

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

Closed form for $n$th derivative of exponential of $f$

derivatives closed-form partial-derivative

Evaluating $\int_0^1 \frac{\log x \log \left(1-x^4 \right)}{1+x^2}dx$

real-analysis integration definite-integrals closed-form

Evaluate $\int_{\pi}^{\infty}\left(x^{2}-\sin\left(x\right)-1\right)^{-1}dx=?$

trigonometry improper-integrals closed-form

Closed form for improper definite integral involving trig functions and exponentials?

definite-integrals closed-form

Prove $\int_0^{\infty }\frac1{\sqrt{x}}\left(\frac{\cos(\pi x^2)}{\sinh (\pi x)}-\frac1{\pi x}\right)dx=\frac{1}{\sqrt{2}}\zeta(\frac{1}{2})$

integration complex-analysis definite-integrals closed-form riemann-zeta

A closed form for $\int_0^\infty\frac{\ln(x+4)}{\sqrt{x\,(x+3)\,(x+4)}}dx$

calculus integration definite-integrals logarithms closed-form

Closed form for $\sum_{n=1}^\infty\frac{1}{2^n\left(1+\sqrt[2^n]{2}\right)}$

calculus sequences-and-series closed-form

Integral $\int_0^\infty\frac{\ln\left(\sqrt{x+1\vphantom{x^0}}-1\right)\,\ln\left(\sqrt{x^{-1}+1}+1\right)}{(x+1)^{3/2}}dx$

calculus integration logarithms improper-integrals closed-form

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

calculus integration logarithms improper-integrals closed-form

Integral of Combination Log and Inverse Trig Function

calculus integration definite-integrals improper-integrals closed-form

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

calculus integration definite-integrals improper-integrals closed-form

Integral $\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}\mathrm dx$

calculus integration logarithms improper-integrals closed-form

Evaluating $\int_0^{\frac\pi2}\frac{\ln{(\sin x)}\ \ln{(\cos x})}{\tan x}\ dx$

calculus integration definite-integrals improper-integrals closed-form

Curious about an empirically found continued fraction for tanh

number-theory closed-form continued-fractions hyperbolic-functions

Compute an integral containing a product of powers of logarithms.

integration closed-form

Integral ${\large\int}_0^\infty\frac{dx}{\sqrt[4]{7+\cosh x}}$

calculus integration definite-integrals closed-form hyperbolic-functions

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

calculus integration definite-integrals logarithms closed-form

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

integration definite-integrals logarithms closed-form polylogarithm