Integrand size = 255, antiderivative size = 31 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{e^{2 x^4} \left (2-\frac {1}{x}+2 x\right )^2 (-\log (2)+\log (x))^2} \] Output:
exp((ln(x)-ln(2))^2*(2*x-1/x+2)^2*exp(x^4)^2)
\[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=\int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx \] Input:
Integrate[(E^((E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[2]^2 + E^(2*x^4)*(- 2 + 8*x - 16*x^3 - 8*x^4)*Log[2]*Log[x] + E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x ^4)*Log[x]^2)/x^2)*(E^(2*x^4)*((-2 + 8*x - 16*x^3 - 8*x^4)*Log[2] + (-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8)*Log[2]^2) + E^(2*x^4)*(2 - 8*x + 16*x^3 + 8*x^4 + (4 - 8*x - 16*x^3 - 32*x^4 + 64*x^5 - 128*x^7 - 6 4*x^8)*Log[2])*Log[x] + E^(2*x^4)*(-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64 *x^7 + 32*x^8)*Log[x]^2))/x^3,x]
Output:
Integrate[(E^((E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[2]^2 + E^(2*x^4)*(- 2 + 8*x - 16*x^3 - 8*x^4)*Log[2]*Log[x] + E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x ^4)*Log[x]^2)/x^2)*(E^(2*x^4)*((-2 + 8*x - 16*x^3 - 8*x^4)*Log[2] + (-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8)*Log[2]^2) + E^(2*x^4)*(2 - 8*x + 16*x^3 + 8*x^4 + (4 - 8*x - 16*x^3 - 32*x^4 + 64*x^5 - 128*x^7 - 6 4*x^8)*Log[2])*Log[x] + E^(2*x^4)*(-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64 *x^7 + 32*x^8)*Log[x]^2))/x^3, x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (e^{2 x^4} \left (32 x^8+64 x^7-32 x^5+16 x^4+8 x^3+4 x-2\right ) \log ^2(x)+e^{2 x^4} \left (\left (-8 x^4-16 x^3+8 x-2\right ) \log (2)+\left (32 x^8+64 x^7-32 x^5+16 x^4+8 x^3+4 x-2\right ) \log ^2(2)\right )+e^{2 x^4} \left (8 x^4+16 x^3+\left (-64 x^8-128 x^7+64 x^5-32 x^4-16 x^3-8 x+4\right ) \log (2)-8 x+2\right ) \log (x)\right ) \exp \left (\frac {e^{2 x^4} \left (4 x^4+8 x^3-4 x+1\right ) \log ^2(x)+e^{2 x^4} \left (4 x^4+8 x^3-4 x+1\right ) \log ^2(2)+e^{2 x^4} \left (-8 x^4-16 x^3+8 x-2\right ) \log (2) \log (x)}{x^2}\right )}{x^3} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-2 x^2-2 x+1\right ) (\log (2)-\log (x)) \left (-8 x^6 \log (2)-8 x^5 \log (2)+4 x^4 \log (2)-x^2 (\log (4)-2)+\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log (x)+2 x-1-\log (2)\right ) \exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right )}{x^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-2 x^2-2 x+1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) (\log (2)-\log (x)) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-(2-\log (4)) x^2-2 x-\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log (x)+\log (2)+1\right )}{x^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-2 x^2-2 x+1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) (\log (2)-\log (x)) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-(2-\log (4)) x^2-2 x-\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log (x)+\log (2)+1\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-2 x^2-2 x+1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-2 x^2-2 x+1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-2 x^2-2 x+1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (\left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)-\left (16 \log (2) x^6+16 \log (2) x^5-8 \log (2) x^4+2 (-1+\log (4)) x^2-2 x+\log (4)+1\right ) \log (x)+\log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4+(-2+\log (4)) x^2-2 x+\log (2)+1\right )\right )}{x^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (-\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (2 x^2+2 x-1\right ) \left (8 x^6+8 x^5-4 x^4+2 x^2+1\right ) \log ^2(x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \left (-16 \log (2) x^6-16 \log (2) x^5+8 \log (2) x^4+2 (1-\log (4)) x^2+2 x-\log (4)-1\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (2 x^2+2 x-1\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-2 x^2-2 x+1\right ) \log (2) \left (8 \log (2) x^6+8 \log (2) x^5-4 \log (2) x^4-2 (1-\log (2)) x^2-2 x+\log (2)+1\right )}{x^3}\right )dx\) |
Input:
Int[(E^((E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[2]^2 + E^(2*x^4)*(-2 + 8* x - 16*x^3 - 8*x^4)*Log[2]*Log[x] + E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Lo g[x]^2)/x^2)*(E^(2*x^4)*((-2 + 8*x - 16*x^3 - 8*x^4)*Log[2] + (-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8)*Log[2]^2) + E^(2*x^4)*(2 - 8*x + 16*x^3 + 8*x^4 + (4 - 8*x - 16*x^3 - 32*x^4 + 64*x^5 - 128*x^7 - 64*x^8) *Log[2])*Log[x] + E^(2*x^4)*(-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8)*Log[x]^2))/x^3,x]
Output:
$Aborted
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06
\[{\mathrm e}^{\frac {{\mathrm e}^{2 x^{4}} \left (2 x^{2}+2 x -1\right )^{2} \left (-\ln \left (x \right )+\ln \left (2\right )\right )^{2}}{x^{2}}}\]
Input:
int(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*ln(x)^2+((-64*x^ 8-128*x^7+64*x^5-32*x^4-16*x^3-8*x+4)*ln(2)+8*x^4+16*x^3-8*x+2)*exp(x^4)^2 *ln(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*ln(2)^2+(-8*x^4-16*x^3+8 *x-2)*ln(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*ln(x)^2+(-8*x ^4-16*x^3+8*x-2)*ln(2)*exp(x^4)^2*ln(x)+(4*x^4+8*x^3-4*x+1)*ln(2)^2*exp(x^ 4)^2)/x^2)/x^3,x)
Output:
exp(exp(2*x^4)*(2*x^2+2*x-1)^2*(-ln(x)+ln(2))^2/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\left (\frac {{\left (4 \, x^{4} + 8 \, x^{3} - 4 \, x + 1\right )} e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 2 \, {\left (4 \, x^{4} + 8 \, x^{3} - 4 \, x + 1\right )} e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + {\left (4 \, x^{4} + 8 \, x^{3} - 4 \, x + 1\right )} e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x^{2}}\right )} \] Input:
integrate(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*log(x)^2+( (-64*x^8-128*x^7+64*x^5-32*x^4-16*x^3-8*x+4)*log(2)+8*x^4+16*x^3-8*x+2)*ex p(x^4)^2*log(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*log(2)^2+(-8*x^ 4-16*x^3+8*x-2)*log(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*lo g(x)^2+(-8*x^4-16*x^3+8*x-2)*log(2)*exp(x^4)^2*log(x)+(4*x^4+8*x^3-4*x+1)* log(2)^2*exp(x^4)^2)/x^2)/x^3,x, algorithm="fricas")
Output:
e^(((4*x^4 + 8*x^3 - 4*x + 1)*e^(2*x^4)*log(2)^2 - 2*(4*x^4 + 8*x^3 - 4*x + 1)*e^(2*x^4)*log(2)*log(x) + (4*x^4 + 8*x^3 - 4*x + 1)*e^(2*x^4)*log(x)^ 2)/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (26) = 52\).
Time = 1.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\frac {\left (- 8 x^{4} - 16 x^{3} + 8 x - 2\right ) e^{2 x^{4}} \log {\left (2 \right )} \log {\left (x \right )} + \left (4 x^{4} + 8 x^{3} - 4 x + 1\right ) e^{2 x^{4}} \log {\left (x \right )}^{2} + \left (4 x^{4} + 8 x^{3} - 4 x + 1\right ) e^{2 x^{4}} \log {\left (2 \right )}^{2}}{x^{2}}} \] Input:
integrate(((32*x**8+64*x**7-32*x**5+16*x**4+8*x**3+4*x-2)*exp(x**4)**2*ln( x)**2+((-64*x**8-128*x**7+64*x**5-32*x**4-16*x**3-8*x+4)*ln(2)+8*x**4+16*x **3-8*x+2)*exp(x**4)**2*ln(x)+((32*x**8+64*x**7-32*x**5+16*x**4+8*x**3+4*x -2)*ln(2)**2+(-8*x**4-16*x**3+8*x-2)*ln(2))*exp(x**4)**2)*exp(((4*x**4+8*x **3-4*x+1)*exp(x**4)**2*ln(x)**2+(-8*x**4-16*x**3+8*x-2)*ln(2)*exp(x**4)** 2*ln(x)+(4*x**4+8*x**3-4*x+1)*ln(2)**2*exp(x**4)**2)/x**2)/x**3,x)
Output:
exp(((-8*x**4 - 16*x**3 + 8*x - 2)*exp(2*x**4)*log(2)*log(x) + (4*x**4 + 8 *x**3 - 4*x + 1)*exp(2*x**4)*log(x)**2 + (4*x**4 + 8*x**3 - 4*x + 1)*exp(2 *x**4)*log(2)**2)/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (29) = 58\).
Time = 0.64 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.61 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\left (4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 8 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 16 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x} + \frac {8 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x^{2}} - \frac {2 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x^{2}}\right )} \] Input:
integrate(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*log(x)^2+( (-64*x^8-128*x^7+64*x^5-32*x^4-16*x^3-8*x+4)*log(2)+8*x^4+16*x^3-8*x+2)*ex p(x^4)^2*log(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*log(2)^2+(-8*x^ 4-16*x^3+8*x-2)*log(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*lo g(x)^2+(-8*x^4-16*x^3+8*x-2)*log(2)*exp(x^4)^2*log(x)+(4*x^4+8*x^3-4*x+1)* log(2)^2*exp(x^4)^2)/x^2)/x^3,x, algorithm="maxima")
Output:
e^(4*x^2*e^(2*x^4)*log(2)^2 - 8*x^2*e^(2*x^4)*log(2)*log(x) + 4*x^2*e^(2*x ^4)*log(x)^2 + 8*x*e^(2*x^4)*log(2)^2 - 16*x*e^(2*x^4)*log(2)*log(x) + 8*x *e^(2*x^4)*log(x)^2 - 4*e^(2*x^4)*log(2)^2/x + 8*e^(2*x^4)*log(2)*log(x)/x - 4*e^(2*x^4)*log(x)^2/x + e^(2*x^4)*log(2)^2/x^2 - 2*e^(2*x^4)*log(2)*lo g(x)/x^2 + e^(2*x^4)*log(x)^2/x^2)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (29) = 58\).
Time = 0.51 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.61 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\left (4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 8 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 16 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x} + \frac {8 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x^{2}} - \frac {2 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x^{2}}\right )} \] Input:
integrate(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*log(x)^2+( (-64*x^8-128*x^7+64*x^5-32*x^4-16*x^3-8*x+4)*log(2)+8*x^4+16*x^3-8*x+2)*ex p(x^4)^2*log(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*log(2)^2+(-8*x^ 4-16*x^3+8*x-2)*log(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*lo g(x)^2+(-8*x^4-16*x^3+8*x-2)*log(2)*exp(x^4)^2*log(x)+(4*x^4+8*x^3-4*x+1)* log(2)^2*exp(x^4)^2)/x^2)/x^3,x, algorithm="giac")
Output:
e^(4*x^2*e^(2*x^4)*log(2)^2 - 8*x^2*e^(2*x^4)*log(2)*log(x) + 4*x^2*e^(2*x ^4)*log(x)^2 + 8*x*e^(2*x^4)*log(2)^2 - 16*x*e^(2*x^4)*log(2)*log(x) + 8*x *e^(2*x^4)*log(x)^2 - 4*e^(2*x^4)*log(2)^2/x + 8*e^(2*x^4)*log(2)*log(x)/x - 4*e^(2*x^4)*log(x)^2/x + e^(2*x^4)*log(2)^2/x^2 - 2*e^(2*x^4)*log(2)*lo g(x)/x^2 + e^(2*x^4)*log(x)^2/x^2)
Time = 2.85 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.00 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}{x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}{x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}{x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}{x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}}{x^{\frac {2\,{\mathrm {e}}^{2\,x^4}\,\ln \left (2\right )\,\left (4\,x^4+8\,x^3-4\,x+1\right )}{x^2}}} \] Input:
int((exp((exp(2*x^4)*log(2)^2*(8*x^3 - 4*x + 4*x^4 + 1) + exp(2*x^4)*log(x )^2*(8*x^3 - 4*x + 4*x^4 + 1) - exp(2*x^4)*log(2)*log(x)*(16*x^3 - 8*x + 8 *x^4 + 2))/x^2)*(exp(2*x^4)*log(x)^2*(4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x ^7 + 32*x^8 - 2) - exp(2*x^4)*(log(2)*(16*x^3 - 8*x + 8*x^4 + 2) - log(2)^ 2*(4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8 - 2)) + exp(2*x^4)*log( x)*(16*x^3 - log(2)*(8*x + 16*x^3 + 32*x^4 - 64*x^5 + 128*x^7 + 64*x^8 - 4 ) - 8*x + 8*x^4 + 2)))/x^3,x)
Output:
(exp((exp(2*x^4)*log(2)^2)/x^2)*exp(-(4*exp(2*x^4)*log(2)^2)/x)*exp(4*x^2* exp(2*x^4)*log(2)^2)*exp(8*x*exp(2*x^4)*log(x)^2)*exp((exp(2*x^4)*log(x)^2 )/x^2)*exp(-(4*exp(2*x^4)*log(x)^2)/x)*exp(4*x^2*exp(2*x^4)*log(x)^2)*exp( 8*x*exp(2*x^4)*log(2)^2))/x^((2*exp(2*x^4)*log(2)*(8*x^3 - 4*x + 4*x^4 + 1 ))/x^2)
Time = 5.59 (sec) , antiderivative size = 192, normalized size of antiderivative = 6.19 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=\frac {e^{\frac {4 e^{2 x^{4}} \mathrm {log}\left (x \right )^{2} x^{4}+8 e^{2 x^{4}} \mathrm {log}\left (x \right )^{2} x^{3}+e^{2 x^{4}} \mathrm {log}\left (x \right )^{2}+8 e^{2 x^{4}} \mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x +4 e^{2 x^{4}} \mathrm {log}\left (2\right )^{2} x^{4}+8 e^{2 x^{4}} \mathrm {log}\left (2\right )^{2} x^{3}+e^{2 x^{4}} \mathrm {log}\left (2\right )^{2}}{x^{2}}}}{e^{\frac {4 e^{2 x^{4}} \mathrm {log}\left (x \right )^{2} x +8 e^{2 x^{4}} \mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{4}+16 e^{2 x^{4}} \mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{3}+2 e^{2 x^{4}} \mathrm {log}\left (x \right ) \mathrm {log}\left (2\right )+4 e^{2 x^{4}} \mathrm {log}\left (2\right )^{2} x}{x^{2}}}} \] Input:
int(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*log(x)^2+((-64*x ^8-128*x^7+64*x^5-32*x^4-16*x^3-8*x+4)*log(2)+8*x^4+16*x^3-8*x+2)*exp(x^4) ^2*log(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*log(2)^2+(-8*x^4-16*x ^3+8*x-2)*log(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*log(x)^2 +(-8*x^4-16*x^3+8*x-2)*log(2)*exp(x^4)^2*log(x)+(4*x^4+8*x^3-4*x+1)*log(2) ^2*exp(x^4)^2)/x^2)/x^3,x)
Output:
e**((4*e**(2*x**4)*log(x)**2*x**4 + 8*e**(2*x**4)*log(x)**2*x**3 + e**(2*x **4)*log(x)**2 + 8*e**(2*x**4)*log(x)*log(2)*x + 4*e**(2*x**4)*log(2)**2*x **4 + 8*e**(2*x**4)*log(2)**2*x**3 + e**(2*x**4)*log(2)**2)/x**2)/e**((4*e **(2*x**4)*log(x)**2*x + 8*e**(2*x**4)*log(x)*log(2)*x**4 + 16*e**(2*x**4) *log(x)*log(2)*x**3 + 2*e**(2*x**4)*log(x)*log(2) + 4*e**(2*x**4)*log(2)** 2*x)/x**2)