More Integrals for the Exponential function

Because I have written some code to extend the integrator to do integrals for
the special functions I have recognized that it is necessary to have more
integrals for the exponential function too.

With the help of the Incomplete Gamma function and the Exponential Integrals a
lot of additional integrals can be implemented. Thus I have collected about 700
integrals involving the exponential function. Most of these integrals Maxima can
not solve.

Now I try to implement step by step the missing integrals. Most of the integrals
can be represented by very general pattern. Here are the first pattern for the
integrand I have implemented to extend Maxima:

a^(b*(z^r)^p+d)
z^v*a^(b*z^r+d)
(a*z+b)^p*%e^(c*z+d)
z^n*d^(a*z^2+b*z+c)

Here are some new results for the exponential function:

integrate(a^(b*z^r),z)
-gamma_incomplete(1/r,-log(a)*b*z^r)*z/(b^(1/r)*r*(-log(a)*z^r)^(1/r))

integrate(a^(b*(z^r)^p),z)
-gamma_incomplete(1/(p*r),-log(a)*b*(z^r)^p)*z
 /(b^(1/(p*r))*p*r*(-log(a)*(z^r)^p)^(1/(p*r)))

integrate(%e^(a*z^r),z)
-gamma_incomplete(1/r,-a*z^r)*z/(a^(1/r)*r*(-z^r)^(1/r))

integrate(z^(v-1)*a^(b*z),z)
-gamma_incomplete(v,-log(a)*b*z)*z^v/(b^v*(-log(a)*z)^v)

integrate(z^(v-1)*%e^(a*sqrt(z)+c),z)
-2*%e^c*gamma_incomplete(2*v,-a*sqrt(z))/(a^(2*v)*(-1)^(2*v))

integrate(%e^(c*z)/(a*z+b),z)
-expintegral_e(1,-c*(a*z+b)/a)*%e^-(b*c/a)/a


An example of an additional integral because of more integrals for the
exponential function:

integrate(log(x)^k,x);
-gamma_incomplete(k+1,-log(x))*(-log(x))^(-k-1)*log(x)^(k+1)


Because I am working on 700 integrals as examples the work is a bit tedious. But
I think at the end it is worth to have as general and as many results as
possible for the exponential function.

I try to do the implementation in a way that we have no changes to the
testsuite. Naturally some examples will give now results and not the expected
noun form.

With the implementation of new integrals for the exponential function and the
extension of the integrator for special functions we should get a lot of more
integrals in Maxima.

Dieter Kaiser