Integrand size = 206, antiderivative size = 22 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{4 \left (-5 x+\log (4)+\left (x+x^2\right )^2 \log (x)\right )^2} \]
\[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=\int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx \]
Integrate[E^(100*x^2 - 40*x*Log[4] + 4*Log[4]^2 + 4*(-10*x^3 - 20*x^4 - 10 *x^5 + (2*x^2 + 4*x^3 + 2*x^4)*Log[4])*Log[x] + 4*(x^4 + 4*x^5 + 6*x^6 + 4 *x^7 + x^8)*Log[x]^2)*(200*x - 40*x^2 - 80*x^3 - 40*x^4 + (-40 + 8*x + 16* x^2 + 8*x^3)*Log[4] + (-120*x^2 - 312*x^3 - 168*x^4 + 48*x^5 + 32*x^6 + 8* x^7 + (16*x + 48*x^2 + 32*x^3)*Log[4])*Log[x] + (16*x^3 + 80*x^4 + 144*x^5 + 112*x^6 + 32*x^7)*Log[x]^2),x]
Integrate[E^(100*x^2 - 40*x*Log[4] + 4*Log[4]^2 + 4*(-10*x^3 - 20*x^4 - 10 *x^5 + (2*x^2 + 4*x^3 + 2*x^4)*Log[4])*Log[x] + 4*(x^4 + 4*x^5 + 6*x^6 + 4 *x^7 + x^8)*Log[x]^2)*(200*x - 40*x^2 - 80*x^3 - 40*x^4 + (-40 + 8*x + 16* x^2 + 8*x^3)*Log[4] + (-120*x^2 - 312*x^3 - 168*x^4 + 48*x^5 + 32*x^6 + 8* x^7 + (16*x + 48*x^2 + 32*x^3)*Log[4])*Log[x] + (16*x^3 + 80*x^4 + 144*x^5 + 112*x^6 + 32*x^7)*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.
| \(\displaystyle \int \left (-40 x^4-80 x^3-40 x^2+\left (8 x^3+16 x^2+8 x-40\right ) \log (4)+\left (32 x^7+112 x^6+144 x^5+80 x^4+16 x^3\right ) \log ^2(x)+\left (8 x^7+32 x^6+48 x^5-168 x^4-312 x^3-120 x^2+\left (32 x^3+48 x^2+16 x\right ) \log (4)\right ) \log (x)+200 x\right ) \exp \left (100 x^2+4 \left (-10 x^5-20 x^4-10 x^3+\left (2 x^4+4 x^3+2 x^2\right ) \log (4)\right ) \log (x)+4 \left (x^8+4 x^7+6 x^6+4 x^5+x^4\right ) \log ^2(x)-40 x \log (4)+4 \log ^2(4)\right ) \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (-x^2 (x+1)^2 \log (x)+5 x-\log (4)\right ) \left (-x^3-2 x^2-2 \left (2 x^2+3 x+1\right ) x \log (x)-x+5\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2^{3-80 x} x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \left (2 x^3 (2 x+1) (x+1)^3 \log ^2(x)-\left (x^3+2 x^2+x-5\right ) (5 x-\log (4))+x (x+1) \left (x^5+3 x^4+3 x^3-24 x^2+x (\log (256)-15)+\log (16)\right ) \log (x)\right ) \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2^{4-80 x} (x+1)^3 (2 x+1) \log ^2(x) x^{3-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )-2^{3-80 x} \left (x^3+2 x^2+x-5\right ) (5 x-\log (4)) x^{-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )+2^{3-80 x} (x+1) \left (x^5+3 x^4+3 x^3-24 x^2-15 x \left (1-\frac {8 \log (2)}{15}\right )+\log (16)\right ) \log (x) x^{1-8 x^2 (x+1)^2 (5 x-\log (4))} \exp \left (4 \left (x^4 (x+1)^4 \log ^2(x)+25 x^2+\log ^2(4)\right )\right )\right )dx\) |
Int[E^(100*x^2 - 40*x*Log[4] + 4*Log[4]^2 + 4*(-10*x^3 - 20*x^4 - 10*x^5 + (2*x^2 + 4*x^3 + 2*x^4)*Log[4])*Log[x] + 4*(x^4 + 4*x^5 + 6*x^6 + 4*x^7 + x^8)*Log[x]^2)*(200*x - 40*x^2 - 80*x^3 - 40*x^4 + (-40 + 8*x + 16*x^2 + 8*x^3)*Log[4] + (-120*x^2 - 312*x^3 - 168*x^4 + 48*x^5 + 32*x^6 + 8*x^7 + (16*x + 48*x^2 + 32*x^3)*Log[4])*Log[x] + (16*x^3 + 80*x^4 + 144*x^5 + 112 *x^6 + 32*x^7)*Log[x]^2),x]
3.13.51.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(23)=46\).
Time = 42.80 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95
| method | result | size |
| risch | \(1048576^{-4 x} x^{8 x^{2} \left (1+x \right )^{2} \left (2 \ln \left (2\right )-5 x \right )} {\mathrm e}^{4 x^{8} \ln \left (x \right )^{2}+16 x^{7} \ln \left (x \right )^{2}+24 x^{6} \ln \left (x \right )^{2}+16 x^{5} \ln \left (x \right )^{2}+4 x^{4} \ln \left (x \right )^{2}+16 \ln \left (2\right )^{2}+100 x^{2}}\) | \(87\) |
| parallelrisch | \({\mathrm e}^{4 \left (x^{8}+4 x^{7}+6 x^{6}+4 x^{5}+x^{4}\right ) \ln \left (x \right )^{2}+\left (8 \left (2 x^{4}+4 x^{3}+2 x^{2}\right ) \ln \left (2\right )-40 x^{5}-80 x^{4}-40 x^{3}\right ) \ln \left (x \right )+16 \ln \left (2\right )^{2}-80 x \ln \left (2\right )+100 x^{2}}\) | \(87\) |
int(((32*x^7+112*x^6+144*x^5+80*x^4+16*x^3)*ln(x)^2+(2*(32*x^3+48*x^2+16*x )*ln(2)+8*x^7+32*x^6+48*x^5-168*x^4-312*x^3-120*x^2)*ln(x)+2*(8*x^3+16*x^2 +8*x-40)*ln(2)-40*x^4-80*x^3-40*x^2+200*x)*exp((x^8+4*x^7+6*x^6+4*x^5+x^4) *ln(x)^2+(2*(2*x^4+4*x^3+2*x^2)*ln(2)-10*x^5-20*x^4-10*x^3)*ln(x)+4*ln(2)^ 2-20*x*ln(2)+25*x^2)^4,x,method=_RETURNVERBOSE)
((1/1048576)^x)^4*(x^(2*x^2*(1+x)^2*(2*ln(2)-5*x)))^4*exp(4*x^8*ln(x)^2+16 *x^7*ln(x)^2+24*x^6*ln(x)^2+16*x^5*ln(x)^2+4*x^4*ln(x)^2+16*ln(2)^2+100*x^ 2)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.73 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{\left (4 \, {\left (x^{8} + 4 \, x^{7} + 6 \, x^{6} + 4 \, x^{5} + x^{4}\right )} \log \left (x\right )^{2} + 100 \, x^{2} - 80 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2} - 8 \, {\left (5 \, x^{5} + 10 \, x^{4} + 5 \, x^{3} - 2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \left (2\right )\right )} \log \left (x\right )\right )} \]
integrate(((32*x^7+112*x^6+144*x^5+80*x^4+16*x^3)*log(x)^2+(2*(32*x^3+48*x ^2+16*x)*log(2)+8*x^7+32*x^6+48*x^5-168*x^4-312*x^3-120*x^2)*log(x)+2*(8*x ^3+16*x^2+8*x-40)*log(2)-40*x^4-80*x^3-40*x^2+200*x)*exp((x^8+4*x^7+6*x^6+ 4*x^5+x^4)*log(x)^2+(2*(2*x^4+4*x^3+2*x^2)*log(2)-10*x^5-20*x^4-10*x^3)*lo g(x)+4*log(2)^2-20*x*log(2)+25*x^2)^4,x, algorithm=\
e^(4*(x^8 + 4*x^7 + 6*x^6 + 4*x^5 + x^4)*log(x)^2 + 100*x^2 - 80*x*log(2) + 16*log(2)^2 - 8*(5*x^5 + 10*x^4 + 5*x^3 - 2*(x^4 + 2*x^3 + x^2)*log(2))* log(x))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (22) = 44\).
Time = 0.39 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.95 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{100 x^{2} - 80 x \log {\left (2 \right )} + 4 \left (- 10 x^{5} - 20 x^{4} - 10 x^{3} + \left (4 x^{4} + 8 x^{3} + 4 x^{2}\right ) \log {\left (2 \right )}\right ) \log {\left (x \right )} + 4 \left (x^{8} + 4 x^{7} + 6 x^{6} + 4 x^{5} + x^{4}\right ) \log {\left (x \right )}^{2} + 16 \log {\left (2 \right )}^{2}} \]
integrate(((32*x**7+112*x**6+144*x**5+80*x**4+16*x**3)*ln(x)**2+(2*(32*x** 3+48*x**2+16*x)*ln(2)+8*x**7+32*x**6+48*x**5-168*x**4-312*x**3-120*x**2)*l n(x)+2*(8*x**3+16*x**2+8*x-40)*ln(2)-40*x**4-80*x**3-40*x**2+200*x)*exp((x **8+4*x**7+6*x**6+4*x**5+x**4)*ln(x)**2+(2*(2*x**4+4*x**3+2*x**2)*ln(2)-10 *x**5-20*x**4-10*x**3)*ln(x)+4*ln(2)**2-20*x*ln(2)+25*x**2)**4,x)
exp(100*x**2 - 80*x*log(2) + 4*(-10*x**5 - 20*x**4 - 10*x**3 + (4*x**4 + 8 *x**3 + 4*x**2)*log(2))*log(x) + 4*(x**8 + 4*x**7 + 6*x**6 + 4*x**5 + x**4 )*log(x)**2 + 16*log(2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (23) = 46\).
Time = 0.60 (sec) , antiderivative size = 111, normalized size of antiderivative = 5.05 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{\left (4 \, x^{8} \log \left (x\right )^{2} + 16 \, x^{7} \log \left (x\right )^{2} + 24 \, x^{6} \log \left (x\right )^{2} + 16 \, x^{5} \log \left (x\right )^{2} - 40 \, x^{5} \log \left (x\right ) + 16 \, x^{4} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{4} \log \left (x\right )^{2} - 80 \, x^{4} \log \left (x\right ) + 32 \, x^{3} \log \left (2\right ) \log \left (x\right ) - 40 \, x^{3} \log \left (x\right ) + 16 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 100 \, x^{2} - 80 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2}\right )} \]
integrate(((32*x^7+112*x^6+144*x^5+80*x^4+16*x^3)*log(x)^2+(2*(32*x^3+48*x ^2+16*x)*log(2)+8*x^7+32*x^6+48*x^5-168*x^4-312*x^3-120*x^2)*log(x)+2*(8*x ^3+16*x^2+8*x-40)*log(2)-40*x^4-80*x^3-40*x^2+200*x)*exp((x^8+4*x^7+6*x^6+ 4*x^5+x^4)*log(x)^2+(2*(2*x^4+4*x^3+2*x^2)*log(2)-10*x^5-20*x^4-10*x^3)*lo g(x)+4*log(2)^2-20*x*log(2)+25*x^2)^4,x, algorithm=\
e^(4*x^8*log(x)^2 + 16*x^7*log(x)^2 + 24*x^6*log(x)^2 + 16*x^5*log(x)^2 - 40*x^5*log(x) + 16*x^4*log(2)*log(x) + 4*x^4*log(x)^2 - 80*x^4*log(x) + 32 *x^3*log(2)*log(x) - 40*x^3*log(x) + 16*x^2*log(2)*log(x) + 100*x^2 - 80*x *log(2) + 16*log(2)^2)
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (23) = 46\).
Time = 0.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 5.05 \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=e^{\left (4 \, x^{8} \log \left (x\right )^{2} + 16 \, x^{7} \log \left (x\right )^{2} + 24 \, x^{6} \log \left (x\right )^{2} + 16 \, x^{5} \log \left (x\right )^{2} - 40 \, x^{5} \log \left (x\right ) + 16 \, x^{4} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{4} \log \left (x\right )^{2} - 80 \, x^{4} \log \left (x\right ) + 32 \, x^{3} \log \left (2\right ) \log \left (x\right ) - 40 \, x^{3} \log \left (x\right ) + 16 \, x^{2} \log \left (2\right ) \log \left (x\right ) + 100 \, x^{2} - 80 \, x \log \left (2\right ) + 16 \, \log \left (2\right )^{2}\right )} \]
integrate(((32*x^7+112*x^6+144*x^5+80*x^4+16*x^3)*log(x)^2+(2*(32*x^3+48*x ^2+16*x)*log(2)+8*x^7+32*x^6+48*x^5-168*x^4-312*x^3-120*x^2)*log(x)+2*(8*x ^3+16*x^2+8*x-40)*log(2)-40*x^4-80*x^3-40*x^2+200*x)*exp((x^8+4*x^7+6*x^6+ 4*x^5+x^4)*log(x)^2+(2*(2*x^4+4*x^3+2*x^2)*log(2)-10*x^5-20*x^4-10*x^3)*lo g(x)+4*log(2)^2-20*x*log(2)+25*x^2)^4,x, algorithm=\
e^(4*x^8*log(x)^2 + 16*x^7*log(x)^2 + 24*x^6*log(x)^2 + 16*x^5*log(x)^2 - 40*x^5*log(x) + 16*x^4*log(2)*log(x) + 4*x^4*log(x)^2 - 80*x^4*log(x) + 32 *x^3*log(2)*log(x) - 40*x^3*log(x) + 16*x^2*log(2)*log(x) + 100*x^2 - 80*x *log(2) + 16*log(2)^2)
Timed out. \[ \int e^{100 x^2-40 x \log (4)+4 \log ^2(4)+4 \left (-10 x^3-20 x^4-10 x^5+\left (2 x^2+4 x^3+2 x^4\right ) \log (4)\right ) \log (x)+4 \left (x^4+4 x^5+6 x^6+4 x^7+x^8\right ) \log ^2(x)} \left (200 x-40 x^2-80 x^3-40 x^4+\left (-40+8 x+16 x^2+8 x^3\right ) \log (4)+\left (-120 x^2-312 x^3-168 x^4+48 x^5+32 x^6+8 x^7+\left (16 x+48 x^2+32 x^3\right ) \log (4)\right ) \log (x)+\left (16 x^3+80 x^4+144 x^5+112 x^6+32 x^7\right ) \log ^2(x)\right ) \, dx=\int {\mathrm {e}}^{4\,{\ln \left (x\right )}^2\,\left (x^8+4\,x^7+6\,x^6+4\,x^5+x^4\right )-80\,x\,\ln \left (2\right )-4\,\ln \left (x\right )\,\left (10\,x^3-2\,\ln \left (2\right )\,\left (2\,x^4+4\,x^3+2\,x^2\right )+20\,x^4+10\,x^5\right )+16\,{\ln \left (2\right )}^2+100\,x^2}\,\left (200\,x+{\ln \left (x\right )}^2\,\left (32\,x^7+112\,x^6+144\,x^5+80\,x^4+16\,x^3\right )+\ln \left (x\right )\,\left (2\,\ln \left (2\right )\,\left (32\,x^3+48\,x^2+16\,x\right )-120\,x^2-312\,x^3-168\,x^4+48\,x^5+32\,x^6+8\,x^7\right )+2\,\ln \left (2\right )\,\left (8\,x^3+16\,x^2+8\,x-40\right )-40\,x^2-80\,x^3-40\,x^4\right ) \,d x \]
int(exp(4*log(x)^2*(x^4 + 4*x^5 + 6*x^6 + 4*x^7 + x^8) - 80*x*log(2) - 4*l og(x)*(10*x^3 - 2*log(2)*(2*x^2 + 4*x^3 + 2*x^4) + 20*x^4 + 10*x^5) + 16*l og(2)^2 + 100*x^2)*(200*x + log(x)^2*(16*x^3 + 80*x^4 + 144*x^5 + 112*x^6 + 32*x^7) + log(x)*(2*log(2)*(16*x + 48*x^2 + 32*x^3) - 120*x^2 - 312*x^3 - 168*x^4 + 48*x^5 + 32*x^6 + 8*x^7) + 2*log(2)*(8*x + 16*x^2 + 8*x^3 - 40 ) - 40*x^2 - 80*x^3 - 40*x^4),x)
int(exp(4*log(x)^2*(x^4 + 4*x^5 + 6*x^6 + 4*x^7 + x^8) - 80*x*log(2) - 4*l og(x)*(10*x^3 - 2*log(2)*(2*x^2 + 4*x^3 + 2*x^4) + 20*x^4 + 10*x^5) + 16*l og(2)^2 + 100*x^2)*(200*x + log(x)^2*(16*x^3 + 80*x^4 + 144*x^5 + 112*x^6 + 32*x^7) + log(x)*(2*log(2)*(16*x + 48*x^2 + 32*x^3) - 120*x^2 - 312*x^3 - 168*x^4 + 48*x^5 + 32*x^6 + 8*x^7) + 2*log(2)*(8*x + 16*x^2 + 8*x^3 - 40 ) - 40*x^2 - 80*x^3 - 40*x^4), x)