One small appendix, maxima 5.30 does quite well for this example too:
(%i3) integrate(cosh(a*x^2+c)/x, x);
2 c 2 - c
- gamma_incomplete(0, - a x ) %e - gamma_incomplete(0, a x ) %e
(%o3) -------------------------------------------------------------------
4
It seems that they can keep up for this type of integrands:
(%i12) integrate(cosh(a*x^5+c)/x, x);
5 c 5 - c
- gamma_incomplete(0, - a x ) %e - gamma_incomplete(0, a x ) %e
(%o12) -------------------------------------------------------------------
10
However, I like the Fricas answer better:
(39) -> integrate(cosh(a*x^5+c)/x, x)
5 5 5 5
(Ei(a x ) - Ei(- a x ))sinh(c) + (Ei(a x ) + Ei(- a x ))cosh(c)
(39) ---------------------------------------------------------------
10
A little bit more simplification on Ei and it would be perfect.
Compare to the nice solution of Mathematica, essentially:
1/5*(cosh(c)*Chi(a*x^5)+sinh(c)*Shi(a*x^5))
Oh, and maxima can do the cos case:
(%i13) integrate(cos(a*x^5+c)/x, x);
5
(%o13) ((%i expintegral_e(1, %i (a x + c) - %i c)
5
- %i expintegral_e(1, %i c - %i (a x + c))) sin(c)
5
+ (- expintegral_e(1, %i (a x + c) - %i c)
5
- expintegral_e(1, %i c - %i (a x + c))) cos(c))/10
Don't ask me by which algorithm ...
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