Airy functions

>    f := (x - AiryAi(x) * AiryAi(1,x)) / (x^2 - AiryAi(x)^2);

f := (x-AiryAi(x)*AiryAi(1,x))/(x^2-AiryAi(x)^2)

>    pmint(f,x);

1/2*ln(x+AiryAi(x))+1/2*ln(x-AiryAi(x))

Maple has trouble with such integrands

>    int(f,x);

int((x-AiryAi(x)*AiryAi(1,x))/(x^2-AiryAi(x)^2),x)

>    f := x^2*AiryAi(x);

f := x^2*AiryAi(x)

>    pmint(f,x);

-AiryAi(x)+AiryAi(1,x)*x

Compare with Maple's answer

>    int(f,x);

1/2*3^(5/6)/Pi*(2/27*Pi*3^(1/2)/GAMMA(2/3)*x^3*hypergeom([1],[2/3, 2],1/9*x^3)-2/9*x^(5/2)*3^(1/6)*Pi/(x^(3/2))^(1/3)*BesselI(4/3,2/3*x^(3/2)))