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Compute $\int_0^{\pi/4}\frac{(1-x^2)\ln(1+x^2)+(1+x^2)-(1-x^2)\ln(1-x^2)}{(1-x^4)(1+x^2)} x\exp(\frac{x^2-1}{x^2+1}) dx$
July 5th, 2020
calculus
closed-form
definite-integrals
integration
real-analysis
Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$
August 30th, 2020
closed-form
logarithms
calculus
integration
definite-integrals
Evaluate $ \int_{0}^{\pi/2}\frac{1+\tanh x}{1+\tan x}dx $
August 1st, 2022
closed-form
definite-integrals
complex-analysis
integration
calculus
Conjecture ${\large\int}_0^\infty\left[\frac1{x^4}-\frac1{2x^3}+\frac1{12\,x^2}-\frac1{\left(e^x-1\right)x^3}\right]dx=\frac{\zeta(3)}{8\pi^2}$
August 1st, 2022
experimental-mathematics
closed-form
definite-integrals
integration
calculus
How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?
March 19th, 2020
calculus
sequences-and-series
recurrence-relations
closed-form
harmonic-numbers
What is $\, _4F_3\left(1,1,1,\frac{3}{2};\frac{5}{2},\frac{5}{2},\frac{5}{2};1\right)$?
May 4th, 2021
closed-form
calculus
sequences-and-series
hypergeometric-function
Integrate $\int_0^\pi\frac{3\cos x+\sqrt{8+\cos^2 x}}{\sin x}x\ \mathrm dx$
August 1st, 2022
calculus
trigonometry
special-functions
definite-integrals
closed-form
Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
March 28th, 2022
calculus
integration
definite-integrals
contour-integration
closed-form
Closed Form for the Imaginary Part of $\text{Li}_3\Big(\frac{1+i}2\Big)$
May 8th, 2020
integration
closed-form
polylogarithm
Closed form for $\int_0^\infty\arctan\Bigl(\frac{2\pi}{x-\ln\,x+\ln(\frac\pi2)}\Bigr)\frac{dx}{x+1}$
March 8th, 2020
calculus
integration
definite-integrals
logarithms
closed-form
Conjectured value of a harmonic sum $\sum_{n=1}^\infty\left(H_n-\,2H_{2n}+H_{4n}\right)^2$
August 1st, 2022
calculus
sequences-and-series
closed-form
conjectures
harmonic-numbers
Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$
February 28th, 2022
calculus
integration
definite-integrals
closed-form
hyperbolic-functions
How to evaluate $\sum_{k=3}^{\infty} \frac{\ln (k)}{k^2 - 4}$?
July 24th, 2021
calculus
sequences-and-series
analysis
definite-integrals
closed-form
Proving that $\int_0^1 \frac{\arctan x}{x}\ln\left(\frac{1+x^2}{(1-x)^2}\right)dx=\frac{\pi^3}{16}$
September 9th, 2021
harmonic-numbers
closed-form
logarithms
integration
definite-integrals
Solving for the closed term solution of a third order recurrence relation with real constant coefficients
October 27th, 2020
recurrence-relations
closed-form
An example of Risch algorithm for integrating $y$, $F(x,y)=0$.
June 30th, 2020
integration
closed-form
Finding a Linear Recurrence Relation
April 15th, 2020
sequences-and-series
discrete-mathematics
closed-form
recurrence-relations
How to solve homogeneous linear recurrence relations with constant coefficients?
August 1st, 2022
faq
fibonacci-numbers
closed-form
recurrence-relations
sequences-and-series
Is the Risch-algorithm more powerful than the usual integration methods?
August 1st, 2022
calculus
integration
closed-form
elementary-functions
How to integrate $\frac{dx}{(x-p)\sqrt {(x-p)(x-q)}} $?
August 1st, 2022
closed-form
integration
calculus
Do all systems of differential equations have a solution?
August 1st, 2022
chaos-theory
closed-form
ordinary-differential-equations
No closed form for the partial sum of ${n\choose k}$ for $k \le K$?
August 1st, 2022
binomial-coefficients
closed-form
combinatorics
Functions with no closed-form derivative
August 1st, 2022
calculus
derivatives
closed-form
Is there a closed form expression for the first half of the Binomial series?
January 12th, 2020
binomial-coefficients
closed-form
generating-functions
sequences-and-series
General formula for the higher order derivatives of composition with exponential function
August 1st, 2022
closed-form
derivatives
exponential-function
Infinite summation formula for modified Bessel functions of first kind
August 1st, 2022
definite-integrals
bessel-functions
closed-form
sequences-and-series
Is it possible to find a closed-form expression for $f(n)$?
August 13th, 2020
modular-arithmetic
number-theory
discrete-logarithms
closed-form
discrete-mathematics
Evaluate the sum $\sum_{k=1}^nk^2\, 2^{-k}$
June 6th, 2022
closed-form
summation
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