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# [CALL] for challenging univariate symbolic integrals

Posted 1 year ago

This is a call for univariate indefinite or definite exact integrals that cannot immediately computed in Mathematica and Wolfram Language. Submitted integrals will be collected for potential new event where they will be discussed and solved. You may refer to our previous event, which is showcased on the Wolfram Community post: https://community.wolfram.com/groups/-/m/t/2821053.
Do you have an integral that cannot be computed directly using Integrate ? Does the following apply to this integral?

## Submission rules DOs

• Univariate (terms in a single variable)
• Exact (use only symbols, integers, or exact fractions).
• Definite integrals (1, 2)
• Indefinite Integrals (1, 2)
• Keep comments directly related to the topics.

## Submission rules DON'Ts

• Don't submit those that Mathematica / Wolfram Language can do directly using Integrate[ ].
• Don't submit numerical integrals (NIntegrate[ ])
• Don't submit decimal approximate numbers in Integrate[ ].
• Don't submit images, only copyable code.

If you have such an integral - please put it in a comment below!

## Examples of submission formats

You can post a formatted code:

Integrate[Gamma[s] z^-s, s]

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Posted 1 year ago
 I tried another very popular book with random integrals with Mathematica that can't solve: Attachments:
Posted 1 year ago
 I opened another book about LaplaceTransform and tried a few examples:  LaplaceTransform[BesselJ[v, a x] BesselI[v, b x], x, s, Assumptions -> {Re[v] > 0, a > 0, b > 0}](*Can't Solve !*) LaplaceTransform[(1 - Exp[-x])^v* Erfc[ a (1 - Exp[-x])], x, s, Assumptions -> {Re[v] > -1, a > 0}](*Can't Solve !*) LaplaceTransform[(1 - Exp[-x])^\[Mu]* Gamma[v, a*(1 - Exp[-x])], x, s, Assumptions -> {Re[\[Mu]] > -1, Re[v] > 0, a > 0}](*Can't Solve !*) LaplaceTransform[(x + a)^n*LegendreP[n, (x - a)/(x + a)], x, s, Assumptions -> {Re[n] > 0, a > 0}](*Can't Solve !*) 
Posted 1 year ago
 I opened a book about MellinTransform and tried a few examples:  MellinTransform[(Sqrt[x^2 + a^2] + a)^\[Rho]/Sqrt[x^2 + a^2]* BesselJ[v, b*x], x, s, Method -> "Integrate", Assumptions -> {b > 0, Re[a] > 0, -Re[v] < Re[s] < 5/2 - Re[\[Rho]]}](*Can't Solve !*) MellinTransform[ Exp[-\[Sigma] Sqrt[x] - \[Omega]*x] Hypergeometric1F1[a, b, \[Omega]*x], x, s, Method -> "Integrate", Assumptions -> {a > 0, b > 0, Re[\[Sigma]] > 0, Re[s] > 0}](*Can't Solve !*) MellinTransform[ 1/(x + \[Sigma])^\[Rho] Hypergeometric2F1[a, b, c, -\[Omega]*x], x, s, Method -> "Integrate", Assumptions -> {Re[\[Rho]] > 0, Re[\[Sigma]] > 0, Re[s] > 0}](*Can't Solve !*) 
Posted 1 year ago
 I would be very interested in seeing a solution for this definite parameter integral: Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x Log[1 - x]^2), {x, 0, 1}, Assumptions -> r >= 0] 
Posted 1 year ago
 Looks like: $$\int_0^1 \frac{(1-x)^r (-2 x+(-2+x) \log (1-x))}{2 x \log ^2(1-x)} \, dx=1+r-r \log (1+r)-\frac{1}{2} \log (2 \pi (1+r))+\text{log\Gamma }(1+r)$$  Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x )*t (1 - x)^ t, {t, 0, Infinity}, {x, 0, 1}, Assumptions -> {r >= 0, t >= 0}] (*1 + r - r Log[1 + r] - 1/2 Log[2 \[Pi] (1 + r)] + LogGamma[1 + r]*) Regards M.I
Posted 1 year ago
 Nice trick ... :)
Posted 1 year ago
 It should be nice to generalize Integrate function to be able to solve this directly: Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x Log[1 - x]^2), {x, 0, 1}, Assumptions -> r >= 0] using (in this specific case) the following identity: 1/Log[1 - x]^2 = Integrate[t (1 - x)^t, {t, 0, Infinity}] if Re[Log[1 - x]] < 0 like Integrate[((1 - x)^r (-2 x + (-2 + x) Log[1 - x]))/(2 x)* t (1 - x)^t, {t, 0, Infinity}, {x, 0, 1}, Assumptions -> r >= 0] 
Posted 1 year ago
 What a nice trick, indeed, I am impressed! Many thanks, Mariusz!
Posted 1 year ago
 I have a site of Table of Define integrals but you have to go through the whole list and see if Integrate can solve. Attachments:
Posted 1 year ago
 Another list for define integrals with attached files. Attachments:
Posted 1 year ago
 Another list for define integrals with attached files.Examples from here. Attachments:
Posted 1 year ago
Posted 1 year ago
 I have a site of Table of indefine integrals but you have to go through the whole list and see if Integrate can solve. Another site which can be very helpful.My list attached below Attachments:
Posted 1 year ago
 2 another indefine integrals:  Integrate[Exp[-a Sqrt[x]]*Exp[-a*Sqrt[ x - b]], x](* 1/2 b (E^(-a Sqrt[b] x) (1/x - (2 + 2 a Sqrt[b] x + a^2 b x^2)/(a^3 b^(3/2))) + a Sqrt[b] ExpIntegralEi[-a Sqrt[b] x])*) Integrate[Gamma[a , x]^2/x^2, x](* -((2 E^-x x^(-1 + a) Gamma[a, x])/(-1 + a)) + Gamma[a, x]^2/(-1 + a) + Gamma[a, x]^2/((-1 + a) x) - (a Gamma[a, x]^2)/((-1 + a) x) + ( 2^(2 - 2 a) Gamma[-1 + 2 a, 2 x])/(-1 + a)*) 
Posted 1 year ago
 Hi Oleg, since a long time I don't come to terms with all the orthogonality relations of polynomial families.Just after the invention of the internet I found your books at Oxfords Blackwells,, ordered the four volumes and paid about 1000 GBP :-( The next days I found Mathenatica 2.0 in Hamburg for 50€ :-). Here is my attempt with Jacobi polynomials (Gradshtey/Ryzhik 7.391) (Notebook attached)Regards Roland Attachments:
Posted 1 year ago
 Here is one: