a3001d50-c434-4e5d-b12b-5c9df444eca1 32 4 5 false false false 0 0 decimal

Integration using Runge-Kutta algorithms

x y D x y :

ic a b RK ic a b N D al_rkckadapt N 1 + 2 el :

ε 1015 -^:

N 1000 :

Set "Tools/Options/Calulation/Integrals: accuracy" to 10000.

x y x 10 log x * x atan 1 x 2^- (* / :

00 ε + 1 ε - RK 0.566035 -

x y x 0 ε + 1 ε - int 0.566194 -

x y x atan x arctan ≡ eval x 01 int evalf maple 0.566034 -

x y x atan 1 x 2^- 10 log * 1 x 2^+ 10 log * x 2^/ :

00 ε + 1 ε - RK 0.058847 -

x y x 0 ε + 1 ε - int 0.058885 -

x y x atan x arctan ≡ eval x 01 int evalf rational convert maple 0.058847 -

x y x sin x sec * 1 x x tan * + / x 10 log * :

00 ε + 1 RK 0.094574 -

x y x 0 ε + 1 int 0.094574 -

x y x atan x arctan ≡ eval x 01 int evalf rational convert maple 0.094574 -

x y x atan x 10 log * x / :

00 ε + 1 RK 0.420809 -

x y x 0 ε + 1 int 0.420968 -

x y x atan x arctan ≡ eval x 01 int evalf maple 0.420808 -

x y 1 x 2^- 10 log x atan * x / :

00 ε + 1 ε - RK 0.22338 -

x y x 0 ε + 1 ε - int 0.223505 -

x y x atan x arctan ≡ eval x 01 int evalf maple 0.22338 -

x y x atanh 2^x 2^/ :

00 ε + 1 ε - RK 1.644934

x y x 0 ε + 1 ε - int 1.649394

x y x atanh x arctanh ≡ eval x 01 int evalf maple 4.207199 3.875785 i * -

Maple fails maybe because the symbolic expression is so complicate, that it's numerical evaluation is so difficult at the point that it is useless, or it have a bug for this case:

I x y x atanh x arctanh ≡ eval x 01 int maple :

I 41 - (_k1^* 232_k1 * + 2 / Ψ - 12_k1 * + 2 / Ψ + (* π i π_k1 * tan + (* - (* π_k1 * cos * 12_k1 * + (2^/_k1 0..infinity = π 2^- Sum 4 / -

New versions of maple have options for expecify the numerical procedure for evaluate the definite integral: https://www.maplesoft.com/support/help/Maple/view.aspx?path=evalf/Int

Double integrals

ε 107 -^:

2 f x.1 x.2 y.1 y.2 n.x n.y DInt a b n RK.1 011 mat a b n x y D al_rkckadapt n 1 + 2 el : y u D x v D x y f 11 mat : x.1 x.2 n.x RK.1 21 line : y.1 y.2 n.y RK.1 31 line :

x y f 1 x - 1 x 2^y 2^* - / :

x y f 0101 ε - 2020 DInt 0.540553

x y f x 01 int y 01 int evalf maple 0.540553

x y f 1 x y + (1 x 2^y 2^* - (* / :

x y f 0.5 1 0.5 1 ε - 2020 DInt 0.290638

x y f x 0.5 1 int y 0.5 1 int evalf Re maple 0.290638

References

Maritn's post about Maxima quadrature:
- https://en.smath.com/forum/yaf_postst25506_Smath-cananot-solve-this-integral-No5-while-Mathcad-Prime-can.aspx

Single int: A bug in the odesolvers and its solution:
- https://en.smath.com/forum/yaf_postsm56081_Ode-solvers-implementation.aspx
- https://en.smath.com/forum/yaf_postsm57304_ODE-Solvers.aspx#post57304

Double int:
- https://en.smath.com/forum/yaf_postsm61577_Double-Integration-using-ode-solvers.aspx
- https://en.smath.com/forum/yaf_postsm85273_Mathcad-Toolbox.aspx#post85273

Maple convert/rational:
- https://en.smath.com/forum/yaf_postsm82867_Maple-Tools.aspx#post82867

Alvaro

4 appVersion 1.73.9126.0