Calculating of difficult integrals in python

[enitorc]

I have some problems in my scientific research with numerical modeling in Python. I do my research in Jupyter notebook (Anaconda).
I have a difficult integral that includes other integrals: Big integral[Integral]https://i.stack.imgur.com/fE3kS.jpg[/img]
Where S(x) and E(x) are integrals too: S-integral [S-integral]https://i.stack.imgur.com/lBasi.jpg[/img] E-integral [E-integral]https://i.stack.imgur.com/N3CNL.jpg[/img]
And S' is a derivative S(x) with respect to x.
My to-do list includes:
1.To make a numerical model of s-integral:

def s(x):
    def spec(y,x):
        y2=z**2
        x2=x**2
        k=np.sqrt(y2+x2)
        e=np.exp(k)
        return (k+y2/(3*k))*y2/(e+1)
    i=quad(spec, 0, np.inf, args=(x))
    return 1+(45/2pi**4)*i[0]

This part is certainly correct. (When I tried x=0 then s(x)=2.75)

To Make a numerical model of E-integral. (I think I have right solution although I don't have any confirmation in book.)

To get a derivative of S(x) with respect to x. How Can I do it without manual calculation? It is a problem because integrand is rather big for numerical methods (or I don't know how I can do it).

After uniting this functions I have to calculate indefinite integral, but when I do it manually, jupyter gives a NaN massive with big amount of warnings such as overflow encountered in exp, in double scalars and slowly convergent integrals.

---

I'm soory for URL's:
Main int:

S int:

E-int:

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Warnings:

Error:
C:\Users\Enitorc\Anaconda3\lib\site-packages\ipykernel\__main__.py:40: RuntimeWarning: overflow encountered in double_scalars
C:\Users\Enitorc\Anaconda3\lib\site-packages\ipykernel\__main__.py:39: RuntimeWarning: overflow encountered in exp
C:\Users\Enitorc\Anaconda3\lib\site-packages\ipykernel\__main__.py:40: RuntimeWarning: invalid value encountered in double_scalars
C:\Users\Enitorc\Anaconda3\lib\site-packages\scipy\integrate\quadpack.py:356: IntegrationWarning: The occurrence of roundoff error is detected, which prevents 
  the requested tolerance from being achieved.  The error may be 
  underestimated.
  warnings.warn(msg, IntegrationWarning)