First, Integrate.
So, int (2x - xe^x)(-xe^x + secx)dx
= int (-2x^2e^x -x^2e^(2x) + 2xsec x - xe^xsecx)dx
= -2int x^2e^xdx -int x^2e^(2x)dx +2 int xsecx dx -int xe^xsecxdx
Now We have to Integrate By Parts.
Assume I_1 = -2 int x^2e^xdx,
I_2 = -int x^2e^(2x)dx
I_3 = 2int x sec xdx
I_4 = -intxe^xsecxdx
Now The Problem is,
I_1 and I_2 can be found, but I_3 and I_4 can't be expressed by elementary functions. These integrals are undeterminable.
So, Get the I_1 first.
I_1 = -2[x^2inte^xdx - int {d/dx(x^2)inte^x dx}dx]
= -2 [x^2e^x - 2 intxe^xdx]
= -2[x^2e^x - 2{x inte^xdx - int (d/dx(x) int e^xdx)dx}]
= -2[x^2e^x - 2{xe^x - inte^xdx}]
= -2[x^2e^x - 2{xe^x -e^x}]
= -2[x^2e^x - 2xe^x +2e^x]
= -2e^x[x^2 - 2x + 2]
Now, I_2.
I_2 = -2[x^2inte^(2x)dx - int {d/dx(x^2)inte^(2x) dx}dx]
= -2 [1/2x^2e^(2x) - 2 intxe^(2x)dx]
= -2[1/2x^2e^(2x) - 2{x inte^(2x)dx - int (d/dx(x) int e^(2x)dx)dx}]
= -2[1/2x^2e^(2x) - 2{1/2xe^(2x) - inte^(2x)dx}]
= -2[1/2x^2e^(2x) - 2{1/2xe^(2x) -1/2e^(2x)}]
= -2[1/2x^2e^(2x) - xe^(2x) +e^(2x)]
= -2e^(2x)[1/2x^2 - x + 1]
So, The Entire Integral is :
int(2x - xe^x)(-xe^x + sec x) dx
= -2e^x[x^2 - 2x + 2] -2e^(2x)[1/2x^2 - x + 1] + 2 intxsecxdx - intxe^xsecxdx
Hope this helps, but It won't, most probably.