Use tanh-sinh algorithm to integrate and improve differenate precision

[BenderBlog]

1. tanh-sinh algorithm

Introduction at here, ported code from here. Testing use case from here.

Almost all test cases passed, except 22 of them with larger but considerable errors , and 8 of them cannot be handled both xcas and kalker:P

With this method, functions with one or two singular points can be integrated. For example:

Original Kalk

>> integral(-1,1,(1-x^2)^(-1/2),dx)
≈ Not defined.

With this

>> integral(-1,1,(1-x^2)^(-1/2),dx)
≈ 3.1415926469 ≈ π

2. differentiation percision

I changed the code with the formula I found from here, and changed almost everything to KalkValue.

The original code, too confusion:

>> f(x)=x^4
>> f(3)
= 81
>> f'(3)
≈ 108
>> f''(3)
≈ 108
>> f'''(3)
≈ 38.99999
>> f''''(3)
≈ 14
>> f'''''(3)
≈ 24 749 993 000 000.001119983 ≈ 2.4749993×10^13

The changed code:

>> f(x)=x^4
>> f(3)
= 81
>> f'(3)
≈ 108
>> f''(3)
≈ 108
>> f'''(3)
≈ 72
>> f''''(3)
≈ 24
>> f'''''(3)
≈ 0 

[PaddiM8]

Thank you! This is great