\[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx \]
Optimal antiderivative \[ \frac {-x -\ln \! \left (2\right )}{2 x +\frac {2 \ln \left (2\right )}{\left ({\mathrm e}^{x^{2}}+x \right )^{2} x}+4} \]
command
Integrate[(-4*x^6 + 2*x^6*Log[2] + E^(4*x^2)*(-4*x^2 + 2*x^2*Log[2]) + E^(3*x^2)*(-16*x^3 + 8*x^3*Log[2]) + (-4*x^3 - 3*x^2*Log[2])*Log[4] + E^(2*x^2)*(-24*x^4 + 12*x^4*Log[2] + (-2*x - 4*x^3 + (-1 - 4*x^2)*Log[2])*Log[4]) + E^x^2*(-16*x^5 + 8*x^5*Log[2] + (-6*x^2 - 4*x^4 + (-4*x - 4*x^3)*Log[2])*Log[4]))/(16*x^6 + 16*x^7 + 4*x^8 + E^(4*x^2)*(16*x^2 + 16*x^3 + 4*x^4) + E^(3*x^2)*(64*x^3 + 64*x^4 + 16*x^5) + (8*x^3 + 4*x^4)*Log[4] + Log[4]^2 + E^(2*x^2)*(96*x^4 + 96*x^5 + 24*x^6 + (8*x + 4*x^2)*Log[4]) + E^x^2*(64*x^5 + 64*x^6 + 16*x^7 + (16*x^2 + 8*x^3)*Log[4])),x]
Mathematica 13.1 output
\[ \int \frac {-4 x^6+2 x^6 \log (2)+e^{4 x^2} \left (-4 x^2+2 x^2 \log (2)\right )+e^{3 x^2} \left (-16 x^3+8 x^3 \log (2)\right )+\left (-4 x^3-3 x^2 \log (2)\right ) \log (4)+e^{2 x^2} \left (-24 x^4+12 x^4 \log (2)+\left (-2 x-4 x^3+\left (-1-4 x^2\right ) \log (2)\right ) \log (4)\right )+e^{x^2} \left (-16 x^5+8 x^5 \log (2)+\left (-6 x^2-4 x^4+\left (-4 x-4 x^3\right ) \log (2)\right ) \log (4)\right )}{16 x^6+16 x^7+4 x^8+e^{4 x^2} \left (16 x^2+16 x^3+4 x^4\right )+e^{3 x^2} \left (64 x^3+64 x^4+16 x^5\right )+\left (8 x^3+4 x^4\right ) \log (4)+\log ^2(4)+e^{2 x^2} \left (96 x^4+96 x^5+24 x^6+\left (8 x+4 x^2\right ) \log (4)\right )+e^{x^2} \left (64 x^5+64 x^6+16 x^7+\left (16 x^2+8 x^3\right ) \log (4)\right )} \, dx \]
Mathematica 12.3 output
\[ \frac {2 \log ^3(4)+4 x \log ^3(4)+18 x^2 \log ^3(4)-2 e^{2 x^2} x \log (4) \left (16 x^7 (-4+\log (4))+88 x^8 (-4+\log (4))+48 x^9 (-4+\log (4))+8 x^{10} (-4+\log (4))+(-4+\log (4)) \log (4)+2 x (-4+\log (4)) \log (4)+9 x^2 (-4+\log (4)) \log (4)+2 x^6 \left (188-55 \log (4)+2 \log ^2(4)\right )+4 x^3 \left (-16-8 \log (4)+3 \log ^2(4)\right )+4 x^5 \left (52-29 \log (4)+4 \log ^2(4)\right )+4 x^4 \left (-24-14 \log (4)+5 \log ^2(4)\right )\right )-4 e^{x^2} x^2 \log (4) \left (16 x^7 (-4+\log (4))+88 x^8 (-4+\log (4))+48 x^9 (-4+\log (4))+8 x^{10} (-4+\log (4))+(-4+\log (4)) \log (4)+2 x (-4+\log (4)) \log (4)+9 x^2 (-4+\log (4)) \log (4)+2 x^6 \left (188-55 \log (4)+2 \log ^2(4)\right )+4 x^3 \left (-16-8 \log (4)+3 \log ^2(4)\right )+4 x^5 \left (52-29 \log (4)+4 \log ^2(4)\right )+4 x^4 \left (-24-14 \log (4)+5 \log ^2(4)\right )\right )+4 x^4 \log ^2(4) \left (16-2 \log ^2(4)+\log (4) (9+\log (16))\right )-4 x^8 \log (4) \left (104-(105+16 \log (2)) \log (4)+16 \log ^2(4)+\log (64)\right )-4 x^{11} (-4+\log (4)) (35 \log (4)+3 \log (64))-8 x^{12} \left (-63 \log (4)+12 \log ^2(4)+5 \log (64)\right )-4 x^{13} \left (-85 \log (4)+4 \log ^2(4)+23 \log (64)\right )-2 x^9 \log (4) \left (376+8 \log ^2(4)+25 \log (64)-\log (4) (233+\log (256))\right )+x^7 \left (192 \log (4)-54 \log ^3(4)-\log ^2(16) \log (64)+4 \log ^2(4) (36+\log (8192))\right )+16 x^{10} \left (8 \log ^2(4)-\log ^2(64)+\log (65536)\right )+x^3 \log ^2(4) \left (40-3 \log ^2(4)+\log (4) (14+\log (64))+\log (65536)\right )+4 x^6 \log (4) \left (32+9 \log ^2(4)-\log (4) (31+\log (67108864))\right )+x^5 \log ^2(4) \left (-32-4 \log ^2(4)+\log (4) \log (256)+\log (268435456)\right )}{4 \log (4) \left (4 x^3+2 x^4+2 e^{2 x^2} x (2+x)+4 e^{x^2} x^2 (2+x)+\log (4)\right ) \left (16 x^7+88 x^8+48 x^9+8 x^{10}+\log (4)+9 x^2 \log (4)+4 x^4 (6+5 \log (4))+x \log (16)+4 x^3 (4+\log (64))+x^6 (-94+\log (256))+4 x^5 (-13+\log (256))\right )} \]