Integrand size = 250, antiderivative size = 33 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{2 \left (e^3-2 x\right )^2-2 x+\frac {4}{1+x \log (4)-\frac {1}{\log (x)}}} \] Output:
exp((exp(3)-2*x)^2-x+2/(1-1/ln(x)+2*x*ln(2)))^2
\[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=\int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx \] Input:
Integrate[(E^((2*(-E^6 + x + 4*E^3*x - 4*x^2 + (2 + E^6 - x - 4*E^3*x + 4* x^2 + (E^6*x - x^2 - 4*E^3*x^2 + 4*x^3)*Log[4])*Log[x]))/(-1 + (1 + x*Log[ 4])*Log[x]))*(-4 - 2*x - 8*E^3*x + 16*x^2 + (4*x + 16*E^3*x - 32*x^2 + (4* x^2 + 16*E^3*x^2 - 32*x^3)*Log[4])*Log[x] + (-2*x - 8*E^3*x + 16*x^2 + (-4 *x - 4*x^2 - 16*E^3*x^2 + 32*x^3)*Log[4] + (-2*x^3 - 8*E^3*x^3 + 16*x^4)*L og[4]^2)*Log[x]^2))/(x + (-2*x - 2*x^2*Log[4])*Log[x] + (x + 2*x^2*Log[4] + x^3*Log[4]^2)*Log[x]^2),x]
Output:
Integrate[(E^((2*(-E^6 + x + 4*E^3*x - 4*x^2 + (2 + E^6 - x - 4*E^3*x + 4* x^2 + (E^6*x - x^2 - 4*E^3*x^2 + 4*x^3)*Log[4])*Log[x]))/(-1 + (1 + x*Log[ 4])*Log[x]))*(-4 - 2*x - 8*E^3*x + 16*x^2 + (4*x + 16*E^3*x - 32*x^2 + (4* x^2 + 16*E^3*x^2 - 32*x^3)*Log[4])*Log[x] + (-2*x - 8*E^3*x + 16*x^2 + (-4 *x - 4*x^2 - 16*E^3*x^2 + 32*x^3)*Log[4] + (-2*x^3 - 8*E^3*x^3 + 16*x^4)*L og[4]^2)*Log[x]^2))/(x + (-2*x - 2*x^2*Log[4])*Log[x] + (x + 2*x^2*Log[4] + x^3*Log[4]^2)*Log[x]^2), 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.
| \(\displaystyle \int \frac {\left (16 x^2+\left (-32 x^2+\left (-32 x^3+16 e^3 x^2+4 x^2\right ) \log (4)+16 e^3 x+4 x\right ) \log (x)+\left (16 x^2+\left (16 x^4-8 e^3 x^3-2 x^3\right ) \log ^2(4)+\left (32 x^3-16 e^3 x^2-4 x^2-4 x\right ) \log (4)-8 e^3 x-2 x\right ) \log ^2(x)-8 e^3 x-2 x-4\right ) \exp \left (\frac {2 \left (-4 x^2+\left (4 x^2+\left (4 x^3-4 e^3 x^2-x^2+e^6 x\right ) \log (4)-4 e^3 x-x+e^6+2\right ) \log (x)+4 e^3 x+x-e^6\right )}{(x \log (4)+1) \log (x)-1}\right )}{\left (-2 x^2 \log (4)-2 x\right ) \log (x)+\left (x^3 \log ^2(4)+2 x^2 \log (4)+x\right ) \log ^2(x)+x} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {\left (16 x^2+\left (-32 x^2+\left (-32 x^3+16 e^3 x^2+4 x^2\right ) \log (4)+16 e^3 x+4 x\right ) \log (x)+\left (16 x^2+\left (16 x^4-8 e^3 x^3-2 x^3\right ) \log ^2(4)+\left (32 x^3-16 e^3 x^2-4 x^2-4 x\right ) \log (4)-8 e^3 x-2 x\right ) \log ^2(x)+\left (-2-8 e^3\right ) x-4\right ) \exp \left (\frac {2 \left (-4 x^2+\left (4 x^2+\left (4 x^3-4 e^3 x^2-x^2+e^6 x\right ) \log (4)-4 e^3 x-x+e^6+2\right ) \log (x)+4 e^3 x+x-e^6\right )}{(x \log (4)+1) \log (x)-1}\right )}{\left (-2 x^2 \log (4)-2 x\right ) \log (x)+\left (x^3 \log ^2(4)+2 x^2 \log (4)+x\right ) \log ^2(x)+x}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (16 x^2+\left (-32 x^2+\left (-32 x^3+16 e^3 x^2+4 x^2\right ) \log (4)+16 e^3 x+4 x\right ) \log (x)+\left (16 x^2+\left (16 x^4-8 e^3 x^3-2 x^3\right ) \log ^2(4)+\left (32 x^3-16 e^3 x^2-4 x^2-4 x\right ) \log (4)-8 e^3 x-2 x\right ) \log ^2(x)+\left (-2-8 e^3\right ) x-4\right ) \exp \left (\frac {2 \left (-4 x^2+\left (4 x^2+\left (4 x^3-4 e^3 x^2-x^2+e^6 x\right ) \log (4)-4 e^3 x-x+e^6+2\right ) \log (x)+\left (1+4 e^3\right ) x-e^6\right )}{(x \log (4)+1) \log (x)-1}\right )}{x (-x \log (4) \log (x)-\log (x)+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (8 x^3 \log ^2(4)+x^2 \log (4) \left (16-\log (4)-4 e^3 \log (4)\right )+x \left (8-8 e^3 \log (4)-\log (16)\right )-4 e^3-1-\log (16)\right ) \exp \left (\frac {2 \left (-4 x^2+\left (4 x^2+\left (4 x^3-4 e^3 x^2-x^2+e^6 x\right ) \log (4)-4 e^3 x-x+e^6+2\right ) \log (x)+\left (1+4 e^3\right ) x-e^6\right )}{(x \log (4)+1) \log (x)-1}\right )}{(x \log (4)+1)^2}-\frac {4 \left (x^2 \log ^2(4)+x \log (64)+1\right ) \exp \left (\frac {2 \left (-4 x^2+\left (4 x^2+\left (4 x^3-4 e^3 x^2-x^2+e^6 x\right ) \log (4)-4 e^3 x-x+e^6+2\right ) \log (x)+\left (1+4 e^3\right ) x-e^6\right )}{(x \log (4)+1) \log (x)-1}\right )}{x (x \log (4)+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}-\frac {8 \log (4) \exp \left (\frac {2 \left (-4 x^2+\left (4 x^2+\left (4 x^3-4 e^3 x^2-x^2+e^6 x\right ) \log (4)-4 e^3 x-x+e^6+2\right ) \log (x)+\left (1+4 e^3\right ) x-e^6\right )}{(x \log (4)+1) \log (x)-1}\right )}{(x \log (4)+1)^2 (x \log (4) \log (x)+\log (x)-1)}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (8 x^3 \log ^2(4)+x^2 \log (4) \left (16-\log (4)-4 e^3 \log (4)\right )+x \left (8-8 e^3 \log (4)-\log (16)\right )-4 e^3-1-\log (16)\right ) x^{\frac {2 \left (4 x^3 \log (4)+x^2 \left (4-\log (4)-4 e^3 \log (4)\right )-x \left (1+4 e^3-e^6 \log (4)\right )+e^6+2\right )}{x \log (4) \log (x)+\log (x)-1}} \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right )}{(x \log (4)+1)^2}+\frac {4 \left (x^2 \left (-\log ^2(4)\right )-x \log (64)-1\right ) x^{\frac {8 x^3 \log (4)+8 x^2 \left (1-\frac {1}{4} \left (1+4 e^3\right ) \log (4)\right )-x \log (4) \log (x)-2 x \left (1+4 e^3-e^6 \log (4)\right )-\log (x)+5 \left (1+\frac {2 e^6}{5}\right )}{x \log (4) \log (x)+\log (x)-1}} \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right )}{(x \log (4)+1)^2 (-x \log (4) \log (x)-\log (x)+1)^2}+\frac {8 \log (4) x^{\frac {2 \left (4 x^3 \log (4)+x^2 \left (4-\log (4)-4 e^3 \log (4)\right )-x \left (1+4 e^3-e^6 \log (4)\right )+e^6+2\right )}{x \log (4) \log (x)+\log (x)-1}} \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right )}{(x \log (4)+1)^2 (-x \log (4) \log (x)-\log (x)+1)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 x^{\frac {8 x^3 \log (4)+2 x^2 \left (4-\log (4)-4 e^3 \log (4)\right )-2 x \left (1+4 e^3-e^6 \log (4)\right )+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (8 x^3 \log ^2(4)-\frac {2 \left (x^2 \log ^2(4)+x \log (64)+1\right )}{x (x \log (4) \log (x)+\log (x)-1)^2}-x^2 \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right )-x \left (-8+8 e^3 \log (4)+\log (16)\right )-\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}-4 e^3-1-\log (16)\right ) \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right )}{(x \log (4)+1)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {\exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) x^{-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}} \left (-8 \log ^2(4) x^3-\log (4) \left (16-\log (4)-4 e^3 \log (4)\right ) x^2-\left (8-8 e^3 \log (4)-\log (16)\right ) x-\frac {4 \log (4)}{-x \log (4) \log (x)-\log (x)+1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(-x \log (4) \log (x)-\log (x)+1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {\exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) x^{-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}} \left (-8 \log ^2(4) x^3-\log (4) \left (16-\log (4)-4 e^3 \log (4)\right ) x^2-\left (8-8 e^3 \log (4)-\log (16)\right ) x-\frac {4 \log (4)}{-x \log (4) \log (x)-\log (x)+1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(-x \log (4) \log (x)-\log (x)+1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) \log (4) x^{-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) \left (1+4 e^3+\log (16)\right ) x^{-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{1-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}}}{(\log (4) x+1)^2}+\frac {\exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{2-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}}}{(\log (4) x+1)^2}-\frac {8 \exp \left (\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{-x \log (4) \log (x)-\log (x)+1}\right ) \log ^2(4) x^{3-\frac {2 \left (4 \log (4) x^3+\left (4-\log (4)-4 e^3 \log (4)\right ) x^2-\left (1+4 e^3-e^6 \log (4)\right ) x+e^6+2\right )}{-x \log (4) \log (x)-\log (x)+1}}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -2 \int \frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}} \left (-8 \log ^2(4) x^3+\log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^2+\left (-8+8 e^3 \log (4)+\log (16)\right ) x+\frac {4 \log (4)}{x \log (4) \log (x)+\log (x)-1}+\log (16)+4 e^3+1+\frac {2 \left (\log ^2(4) x^2+\log (64) x+1\right )}{(x \log (4) \log (x)+\log (x)-1)^2 x}\right )}{(\log (4) x+1)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {4 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (1+4 e^3+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}}}{(\log (4) x+1)^2}+\frac {2 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (\log ^2(4) x^2+\log (64) x+1\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}-1}}{(\log (4) x+1)^2 (x \log (4) \log (x)+\log (x)-1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \left (-8+8 e^3 \log (4)+\log (16)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+1}}{(\log (4) x+1)^2}+\frac {\exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log (4) \left (-16+\log (4)+4 e^3 \log (4)\right ) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+2}}{(\log (4) x+1)^2}-\frac {8 \exp \left (-\frac {2 \left (4 x^2-\left (1+4 e^3\right ) x+e^6\right )}{x \log (4) \log (x)+\log (x)-1}\right ) \log ^2(4) x^{\frac {8 \log (4) x^3+2 \left (4-\log (4)-4 e^3 \log (4)\right ) x^2-2 \left (1+4 e^3-e^6 \log (4)\right ) x+2 \left (2+e^6\right )}{x \log (4) \log (x)+\log (x)-1}+3}}{(\log (4) x+1)^2}\right )dx\) |
Input:
Int[(E^((2*(-E^6 + x + 4*E^3*x - 4*x^2 + (2 + E^6 - x - 4*E^3*x + 4*x^2 + (E^6*x - x^2 - 4*E^3*x^2 + 4*x^3)*Log[4])*Log[x]))/(-1 + (1 + x*Log[4])*Lo g[x]))*(-4 - 2*x - 8*E^3*x + 16*x^2 + (4*x + 16*E^3*x - 32*x^2 + (4*x^2 + 16*E^3*x^2 - 32*x^3)*Log[4])*Log[x] + (-2*x - 8*E^3*x + 16*x^2 + (-4*x - 4 *x^2 - 16*E^3*x^2 + 32*x^3)*Log[4] + (-2*x^3 - 8*E^3*x^3 + 16*x^4)*Log[4]^ 2)*Log[x]^2))/(x + (-2*x - 2*x^2*Log[4])*Log[x] + (x + 2*x^2*Log[4] + x^3* Log[4]^2)*Log[x]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(32)=64\).
Time = 44.62 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.61
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {2 \left (2 \left (x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}+4 x^{3}-x^{2}\right ) \ln \left (2\right )+{\mathrm e}^{6}-4 x \,{\mathrm e}^{3}+4 x^{2}-x +2\right ) \ln \left (x \right )-2 \,{\mathrm e}^{6}+8 x \,{\mathrm e}^{3}-8 x^{2}+2 x}{2 x \ln \left (2\right ) \ln \left (x \right )+\ln \left (x \right )-1}}\) | \(86\) |
| risch | \({\mathrm e}^{-\frac {2 \left (8 \ln \left (2\right ) \ln \left (x \right ) {\mathrm e}^{3} x^{2}-8 x^{3} \ln \left (2\right ) \ln \left (x \right )-2 \ln \left (2\right ) \ln \left (x \right ) {\mathrm e}^{6} x +2 x^{2} \ln \left (2\right ) \ln \left (x \right )+4 x \,{\mathrm e}^{3} \ln \left (x \right )-4 x^{2} \ln \left (x \right )-{\mathrm e}^{6} \ln \left (x \right )+x \ln \left (x \right )-4 x \,{\mathrm e}^{3}+4 x^{2}-2 \ln \left (x \right )+{\mathrm e}^{6}-x \right )}{2 x \ln \left (2\right ) \ln \left (x \right )+\ln \left (x \right )-1}}\) | \(99\) |
Input:
int(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*ln(2)^2+2*(-16*x^2*exp(3)+32*x^3-4*x^ 2-4*x)*ln(2)-8*x*exp(3)+16*x^2-2*x)*ln(x)^2+(2*(16*x^2*exp(3)-32*x^3+4*x^2 )*ln(2)+16*x*exp(3)-32*x^2+4*x)*ln(x)-8*x*exp(3)+16*x^2-2*x-4)*exp(((2*(x* exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*ln(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2)*ln(x )-exp(3)^2+4*x*exp(3)-4*x^2+x)/((2*x*ln(2)+1)*ln(x)-1))^2/((4*x^3*ln(2)^2+ 4*x^2*ln(2)+x)*ln(x)^2+(-4*x^2*ln(2)-2*x)*ln(x)+x),x,method=_RETURNVERBOSE )
Output:
exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*ln(2)+exp(3)^2-4*x*exp(3)+4*x^ 2-x+2)*ln(x)-exp(3)^2+4*x*exp(3)-4*x^2+x)/(2*x*ln(2)*ln(x)+ln(x)-1))^2
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (34) = 68\).
Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.42 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{\left (-\frac {2 \, {\left (4 \, x^{2} - 4 \, x e^{3} - {\left (4 \, x^{2} - 4 \, x e^{3} + 2 \, {\left (4 \, x^{3} - 4 \, x^{2} e^{3} - x^{2} + x e^{6}\right )} \log \left (2\right ) - x + e^{6} + 2\right )} \log \left (x\right ) - x + e^{6}\right )}}{{\left (2 \, x \log \left (2\right ) + 1\right )} \log \left (x\right ) - 1}\right )} \] Input:
integrate(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x ^3-4*x^2-4*x)*log(2)-8*x*exp(3)+16*x^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32* x^3+4*x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4)* exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x ^2-x+2)*log(x)-exp(3)^2+4*x*exp(3)-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/( (4*x^3*log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x),x, algorithm="fricas")
Output:
e^(-2*(4*x^2 - 4*x*e^3 - (4*x^2 - 4*x*e^3 + 2*(4*x^3 - 4*x^2*e^3 - x^2 + x *e^6)*log(2) - x + e^6 + 2)*log(x) - x + e^6)/((2*x*log(2) + 1)*log(x) - 1 ))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (29) = 58\).
Time = 0.84 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.64 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{\frac {2 \left (- 4 x^{2} + x + 4 x e^{3} + \left (4 x^{2} - 4 x e^{3} - x + \left (8 x^{3} - 8 x^{2} e^{3} - 2 x^{2} + 2 x e^{6}\right ) \log {\left (2 \right )} + 2 + e^{6}\right ) \log {\left (x \right )} - e^{6}\right )}{\left (2 x \log {\left (2 \right )} + 1\right ) \log {\left (x \right )} - 1}} \] Input:
integrate(((4*(-8*x**3*exp(3)+16*x**4-2*x**3)*ln(2)**2+2*(-16*x**2*exp(3)+ 32*x**3-4*x**2-4*x)*ln(2)-8*x*exp(3)+16*x**2-2*x)*ln(x)**2+(2*(16*x**2*exp (3)-32*x**3+4*x**2)*ln(2)+16*x*exp(3)-32*x**2+4*x)*ln(x)-8*x*exp(3)+16*x** 2-2*x-4)*exp(((2*(x*exp(3)**2-4*x**2*exp(3)+4*x**3-x**2)*ln(2)+exp(3)**2-4 *x*exp(3)+4*x**2-x+2)*ln(x)-exp(3)**2+4*x*exp(3)-4*x**2+x)/((2*x*ln(2)+1)* ln(x)-1))**2/((4*x**3*ln(2)**2+4*x**2*ln(2)+x)*ln(x)**2+(-4*x**2*ln(2)-2*x )*ln(x)+x),x)
Output:
exp(2*(-4*x**2 + x + 4*x*exp(3) + (4*x**2 - 4*x*exp(3) - x + (8*x**3 - 8*x **2*exp(3) - 2*x**2 + 2*x*exp(6))*log(2) + 2 + exp(6))*log(x) - exp(6))/(( 2*x*log(2) + 1)*log(x) - 1))
Time = 1.41 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=e^{\left (8 \, x^{2} - 8 \, x e^{3} - 2 \, x + \frac {4 \, \log \left (x\right )}{{\left (2 \, x \log \left (2\right ) + 1\right )} \log \left (x\right ) - 1} + 2 \, e^{6}\right )} \] Input:
integrate(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x ^3-4*x^2-4*x)*log(2)-8*x*exp(3)+16*x^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32* x^3+4*x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4)* exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x ^2-x+2)*log(x)-exp(3)^2+4*x*exp(3)-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/( (4*x^3*log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x),x, algorithm="maxima")
Output:
e^(8*x^2 - 8*x*e^3 - 2*x + 4*log(x)/((2*x*log(2) + 1)*log(x) - 1) + 2*e^6)
Exception generated. \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x ^3-4*x^2-4*x)*log(2)-8*x*exp(3)+16*x^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32* x^3+4*x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4)* exp(((2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x ^2-x+2)*log(x)-exp(3)^2+4*x*exp(3)-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/( (4*x^3*log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{4194304,[0,20,16,0,1]%%%}+%%%{46137344,[0,19,15,0,1]%%%}+% %%{229638
Time = 5.12 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.18 \[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=x^{\frac {2\,\left ({\mathrm {e}}^6-x-4\,x\,{\mathrm {e}}^3-2\,x^2\,\ln \left (2\right )+8\,x^3\,\ln \left (2\right )+4\,x^2-8\,x^2\,{\mathrm {e}}^3\,\ln \left (2\right )+2\,x\,{\mathrm {e}}^6\,\ln \left (2\right )+2\right )}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{\frac {2\,x}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{-\frac {8\,x^2}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{\frac {8\,x\,{\mathrm {e}}^3}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}}\,{\mathrm {e}}^{-\frac {2\,{\mathrm {e}}^6}{\ln \left (x\right )+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-1}} \] Input:
int(-(exp((2*(x - exp(6) + 4*x*exp(3) + log(x)*(exp(6) - x - 4*x*exp(3) + 2*log(2)*(x*exp(6) - 4*x^2*exp(3) - x^2 + 4*x^3) + 4*x^2 + 2) - 4*x^2))/(l og(x)*(2*x*log(2) + 1) - 1))*(2*x - log(x)*(4*x + 2*log(2)*(16*x^2*exp(3) + 4*x^2 - 32*x^3) + 16*x*exp(3) - 32*x^2) + 8*x*exp(3) - 16*x^2 + log(x)^2 *(2*x + 2*log(2)*(4*x + 16*x^2*exp(3) + 4*x^2 - 32*x^3) + 8*x*exp(3) + 4*l og(2)^2*(8*x^3*exp(3) + 2*x^3 - 16*x^4) - 16*x^2) + 4))/(x - log(x)*(2*x + 4*x^2*log(2)) + log(x)^2*(x + 4*x^3*log(2)^2 + 4*x^2*log(2))),x)
Output:
x^((2*(exp(6) - x - 4*x*exp(3) - 2*x^2*log(2) + 8*x^3*log(2) + 4*x^2 - 8*x ^2*exp(3)*log(2) + 2*x*exp(6)*log(2) + 2))/(log(x) + 2*x*log(2)*log(x) - 1 ))*exp((2*x)/(log(x) + 2*x*log(2)*log(x) - 1))*exp(-(8*x^2)/(log(x) + 2*x* log(2)*log(x) - 1))*exp((8*x*exp(3))/(log(x) + 2*x*log(2)*log(x) - 1))*exp (-(2*exp(6))/(log(x) + 2*x*log(2)*log(x) - 1))
\[ \int \frac {e^{\frac {2 \left (-e^6+x+4 e^3 x-4 x^2+\left (2+e^6-x-4 e^3 x+4 x^2+\left (e^6 x-x^2-4 e^3 x^2+4 x^3\right ) \log (4)\right ) \log (x)\right )}{-1+(1+x \log (4)) \log (x)}} \left (-4-2 x-8 e^3 x+16 x^2+\left (4 x+16 e^3 x-32 x^2+\left (4 x^2+16 e^3 x^2-32 x^3\right ) \log (4)\right ) \log (x)+\left (-2 x-8 e^3 x+16 x^2+\left (-4 x-4 x^2-16 e^3 x^2+32 x^3\right ) \log (4)+\left (-2 x^3-8 e^3 x^3+16 x^4\right ) \log ^2(4)\right ) \log ^2(x)\right )}{x+\left (-2 x-2 x^2 \log (4)\right ) \log (x)+\left (x+2 x^2 \log (4)+x^3 \log ^2(4)\right ) \log ^2(x)} \, dx=\int \frac {\left (\left (4 \left (-8 x^{3} {\mathrm e}^{3}+16 x^{4}-2 x^{3}\right ) \mathrm {log}\left (2\right )^{2}+2 \left (-16 x^{2} {\mathrm e}^{3}+32 x^{3}-4 x^{2}-4 x \right ) \mathrm {log}\left (2\right )-8 x \,{\mathrm e}^{3}+16 x^{2}-2 x \right ) \mathrm {log}\left (x \right )^{2}+\left (2 \left (16 x^{2} {\mathrm e}^{3}-32 x^{3}+4 x^{2}\right ) \mathrm {log}\left (2\right )+16 x \,{\mathrm e}^{3}-32 x^{2}+4 x \right ) \mathrm {log}\left (x \right )-8 x \,{\mathrm e}^{3}+16 x^{2}-2 x -4\right ) \left ({\mathrm e}^{\frac {\left (2 \left (x \left ({\mathrm e}^{3}\right )^{2}-4 x^{2} {\mathrm e}^{3}+4 x^{3}-x^{2}\right ) \mathrm {log}\left (2\right )+\left ({\mathrm e}^{3}\right )^{2}-4 x \,{\mathrm e}^{3}+4 x^{2}-x +2\right ) \mathrm {log}\left (x \right )-\left ({\mathrm e}^{3}\right )^{2}+4 x \,{\mathrm e}^{3}-4 x^{2}+x}{\left (2 \,\mathrm {log}\left (2\right ) x +1\right ) \mathrm {log}\left (x \right )-1}}\right )^{2}}{\left (4 \mathrm {log}\left (2\right )^{2} x^{3}+4 \,\mathrm {log}\left (2\right ) x^{2}+x \right ) \mathrm {log}\left (x \right )^{2}+\left (-4 \,\mathrm {log}\left (2\right ) x^{2}-2 x \right ) \mathrm {log}\left (x \right )+x}d x \] Input:
int(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x^3-4*x ^2-4*x)*log(2)-8*x*exp(3)+16*x^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32*x^3+4* x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4)*exp((( 2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2 )*log(x)-exp(3)^2+4*x*exp(3)-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/((4*x^3 *log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x),x)
Output:
int(((4*(-8*x^3*exp(3)+16*x^4-2*x^3)*log(2)^2+2*(-16*x^2*exp(3)+32*x^3-4*x ^2-4*x)*log(2)-8*x*exp(3)+16*x^2-2*x)*log(x)^2+(2*(16*x^2*exp(3)-32*x^3+4* x^2)*log(2)+16*x*exp(3)-32*x^2+4*x)*log(x)-8*x*exp(3)+16*x^2-2*x-4)*exp((( 2*(x*exp(3)^2-4*x^2*exp(3)+4*x^3-x^2)*log(2)+exp(3)^2-4*x*exp(3)+4*x^2-x+2 )*log(x)-exp(3)^2+4*x*exp(3)-4*x^2+x)/((2*x*log(2)+1)*log(x)-1))^2/((4*x^3 *log(2)^2+4*x^2*log(2)+x)*log(x)^2+(-4*x^2*log(2)-2*x)*log(x)+x),x)