Integrand size = 244, antiderivative size = 26 \[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=e^{\left (4+2 x-\log (3)-\frac {\log (x)}{x \log (5+x)}\right )^2} \] Output:
exp((2*x+4-ln(3)-1/ln(5+x)*ln(x)/x)^2)
\[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=\int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx \] Input:
Integrate[(E^((Log[x]^2 + (-8*x - 4*x^2 + 2*x*Log[3])*Log[x]*Log[5 + x] + (16*x^2 + 16*x^3 + 4*x^4 + (-8*x^2 - 4*x^3)*Log[3] + x^2*Log[3]^2)*Log[5 + x]^2)/(x^2*Log[5 + x]^2))*(-2*x*Log[x]^2 + ((10 + 2*x + 8*x^2 + 4*x^3 - 2 *x^2*Log[3])*Log[x] + (-10 - 2*x)*Log[x]^2)*Log[5 + x] + (-40*x - 28*x^2 - 4*x^3 + (10*x + 2*x^2)*Log[3] + (40*x + 8*x^2 + (-10*x - 2*x^2)*Log[3])*L og[x])*Log[5 + x]^2 + (80*x^3 + 56*x^4 + 8*x^5 + (-20*x^3 - 4*x^4)*Log[3]) *Log[5 + x]^3))/((5*x^3 + x^4)*Log[5 + x]^3),x]
Output:
Integrate[(E^((Log[x]^2 + (-8*x - 4*x^2 + 2*x*Log[3])*Log[x]*Log[5 + x] + (16*x^2 + 16*x^3 + 4*x^4 + (-8*x^2 - 4*x^3)*Log[3] + x^2*Log[3]^2)*Log[5 + x]^2)/(x^2*Log[5 + x]^2))*(-2*x*Log[x]^2 + ((10 + 2*x + 8*x^2 + 4*x^3 - 2 *x^2*Log[3])*Log[x] + (-10 - 2*x)*Log[x]^2)*Log[5 + x] + (-40*x - 28*x^2 - 4*x^3 + (10*x + 2*x^2)*Log[3] + (40*x + 8*x^2 + (-10*x - 2*x^2)*Log[3])*L og[x])*Log[5 + x]^2 + (80*x^3 + 56*x^4 + 8*x^5 + (-20*x^3 - 4*x^4)*Log[3]) *Log[5 + x]^3))/((5*x^3 + x^4)*Log[5 + 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.
| \(\displaystyle \int \frac {\left (\left (-4 x^3-28 x^2+\left (8 x^2+\left (-2 x^2-10 x\right ) \log (3)+40 x\right ) \log (x)+\left (2 x^2+10 x\right ) \log (3)-40 x\right ) \log ^2(x+5)+\left (\left (4 x^3+8 x^2-2 x^2 \log (3)+2 x+10\right ) \log (x)+(-2 x-10) \log ^2(x)\right ) \log (x+5)+\left (8 x^5+56 x^4+80 x^3+\left (-4 x^4-20 x^3\right ) \log (3)\right ) \log ^3(x+5)-2 x \log ^2(x)\right ) \exp \left (\frac {\left (-4 x^2-8 x+2 x \log (3)\right ) \log (x+5) \log (x)+\left (4 x^4+16 x^3+16 x^2+x^2 \log ^2(3)+\left (-4 x^3-8 x^2\right ) \log (3)\right ) \log ^2(x+5)+\log ^2(x)}{x^2 \log ^2(x+5)}\right )}{\left (x^4+5 x^3\right ) \log ^3(x+5)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (\left (-4 x^3-28 x^2+\left (8 x^2+\left (-2 x^2-10 x\right ) \log (3)+40 x\right ) \log (x)+\left (2 x^2+10 x\right ) \log (3)-40 x\right ) \log ^2(x+5)+\left (\left (4 x^3+8 x^2-2 x^2 \log (3)+2 x+10\right ) \log (x)+(-2 x-10) \log ^2(x)\right ) \log (x+5)+\left (8 x^5+56 x^4+80 x^3+\left (-4 x^4-20 x^3\right ) \log (3)\right ) \log ^3(x+5)-2 x \log ^2(x)\right ) \exp \left (\frac {\left (-4 x^2-8 x+2 x \log (3)\right ) \log (x+5) \log (x)+\left (4 x^4+16 x^3+16 x^2+x^2 \log ^2(3)+\left (-4 x^3-8 x^2\right ) \log (3)\right ) \log ^2(x+5)+\log ^2(x)}{x^2 \log ^2(x+5)}\right )}{x^3 (x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2\ 3^{-4 x-8} x^{\frac {-4 x-3 x \log (x+5)-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x-4+\log (3)) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right ) \exp \left (4 x^2+\frac {\log ^2(x)}{x^2 \log ^2(x+5)}+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {3^{-4 x-8} \exp \left (4 x^2+16 x+\log ^2(3)+16+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{-\frac {3 \log (x+5) x+4 x-\log (9)+8}{x \log (x+5)}} (\log (x)-x (2 x-\log (3)+4) \log (x+5)) \left ((x+5) \log (x+5) \left (1-2 x^2 \log (x+5)\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 2 \int \frac {3^{-4 x-8} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{-\frac {3 \log (x+5) x+4 x+8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)-x (2 x-\log (3)+4) \log (x+5)) \left ((x+5) \log (x+5) \left (1-2 x^2 \log (x+5)\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {3^{-4 x-8} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{-\frac {3 \log (x+5) x+4 x+8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {3^{-4 x-8} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{1-\frac {3 \log (x+5) x+4 x+8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{\log (x+5)}-\frac {3^{-4 x-8} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{1-\frac {3 \log (x+5) x+4 x+8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^3(x+5)}+2\ 3^{-4 x-8} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{3-\frac {3 \log (x+5) x+4 x+8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log (x) \left (2 x^3+4 \left (1-\frac {\log (3)}{4}\right ) x^2-\log (x) x+x-5 \log (x)+5\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}}}{(x+5) \log ^2(x+5)}+\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \left (-2 x+4 \left (1-\frac {\log (3)}{4}\right ) \log (x)-4 \left (1-\frac {\log (3)}{4}\right )\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{\log (x+5)}-\frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) \log ^2(x) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+1}}{(x+5) \log ^3(x+5)}+2\ 81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) (2 x-\log (3)+4) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}+3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {81^{-x-2} \exp \left (4 x^2+16 x+16 \left (1+\frac {\log ^2(3)}{16}\right )+\frac {\log ^2(x)}{\log ^2(x+5) x^2}\right ) x^{\frac {-3 \log (x+5) x-4 x-8 \left (1-\frac {\log (3)}{4}\right )}{x \log (x+5)}} (\log (x)+x (-2 x+\log (3)-4) \log (x+5)) \left (-\left ((x+5) \log (x+5) \left (2 x^2 \log (x+5)-1\right )\right )-\log (x) (x+(x+5) \log (x+5))\right )}{(x+5) \log ^3(x+5)}dx\) |
Input:
Int[(E^((Log[x]^2 + (-8*x - 4*x^2 + 2*x*Log[3])*Log[x]*Log[5 + x] + (16*x^ 2 + 16*x^3 + 4*x^4 + (-8*x^2 - 4*x^3)*Log[3] + x^2*Log[3]^2)*Log[5 + x]^2) /(x^2*Log[5 + x]^2))*(-2*x*Log[x]^2 + ((10 + 2*x + 8*x^2 + 4*x^3 - 2*x^2*L og[3])*Log[x] + (-10 - 2*x)*Log[x]^2)*Log[5 + x] + (-40*x - 28*x^2 - 4*x^3 + (10*x + 2*x^2)*Log[3] + (40*x + 8*x^2 + (-10*x - 2*x^2)*Log[3])*Log[x]) *Log[5 + x]^2 + (80*x^3 + 56*x^4 + 8*x^5 + (-20*x^3 - 4*x^4)*Log[3])*Log[5 + x]^3))/((5*x^3 + x^4)*Log[5 + x]^3),x]
Output:
$Aborted
Time = 0.04 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58
\[{\mathrm e}^{\frac {\left (\ln \left (3\right ) \ln \left (5+x \right ) x -2 x^{2} \ln \left (5+x \right )-4 x \ln \left (5+x \right )+\ln \left (x \right )\right )^{2}}{x^{2} \ln \left (5+x \right )^{2}}}\]
Input:
int((((-4*x^4-20*x^3)*ln(3)+8*x^5+56*x^4+80*x^3)*ln(5+x)^3+(((-2*x^2-10*x) *ln(3)+8*x^2+40*x)*ln(x)+(2*x^2+10*x)*ln(3)-4*x^3-28*x^2-40*x)*ln(5+x)^2+( (-2*x-10)*ln(x)^2+(-2*x^2*ln(3)+4*x^3+8*x^2+2*x+10)*ln(x))*ln(5+x)-2*x*ln( x)^2)*exp(((x^2*ln(3)^2+(-4*x^3-8*x^2)*ln(3)+4*x^4+16*x^3+16*x^2)*ln(5+x)^ 2+(2*x*ln(3)-4*x^2-8*x)*ln(x)*ln(5+x)+ln(x)^2)/x^2/ln(5+x)^2)/(x^4+5*x^3)/ ln(5+x)^3,x)
Output:
exp((ln(3)*ln(5+x)*x-2*x^2*ln(5+x)-4*x*ln(5+x)+ln(x))^2/x^2/ln(5+x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.15 \[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=e^{\left (\frac {{\left (4 \, x^{4} + x^{2} \log \left (3\right )^{2} + 16 \, x^{3} + 16 \, x^{2} - 4 \, {\left (x^{3} + 2 \, x^{2}\right )} \log \left (3\right )\right )} \log \left (x + 5\right )^{2} - 2 \, {\left (2 \, x^{2} - x \log \left (3\right ) + 4 \, x\right )} \log \left (x + 5\right ) \log \left (x\right ) + \log \left (x\right )^{2}}{x^{2} \log \left (x + 5\right )^{2}}\right )} \] Input:
integrate((((-4*x^4-20*x^3)*log(3)+8*x^5+56*x^4+80*x^3)*log(5+x)^3+(((-2*x ^2-10*x)*log(3)+8*x^2+40*x)*log(x)+(2*x^2+10*x)*log(3)-4*x^3-28*x^2-40*x)* log(5+x)^2+((-2*x-10)*log(x)^2+(-2*x^2*log(3)+4*x^3+8*x^2+2*x+10)*log(x))* log(5+x)-2*x*log(x)^2)*exp(((x^2*log(3)^2+(-4*x^3-8*x^2)*log(3)+4*x^4+16*x ^3+16*x^2)*log(5+x)^2+(2*x*log(3)-4*x^2-8*x)*log(x)*log(5+x)+log(x)^2)/x^2 /log(5+x)^2)/(x^4+5*x^3)/log(5+x)^3,x, algorithm="fricas")
Output:
e^(((4*x^4 + x^2*log(3)^2 + 16*x^3 + 16*x^2 - 4*(x^3 + 2*x^2)*log(3))*log( x + 5)^2 - 2*(2*x^2 - x*log(3) + 4*x)*log(x + 5)*log(x) + log(x)^2)/(x^2*l og(x + 5)^2))
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (20) = 40\).
Time = 1.44 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.27 \[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=e^{\frac {\left (- 4 x^{2} - 8 x + 2 x \log {\left (3 \right )}\right ) \log {\left (x \right )} \log {\left (x + 5 \right )} + \left (4 x^{4} + 16 x^{3} + x^{2} \log {\left (3 \right )}^{2} + 16 x^{2} + \left (- 4 x^{3} - 8 x^{2}\right ) \log {\left (3 \right )}\right ) \log {\left (x + 5 \right )}^{2} + \log {\left (x \right )}^{2}}{x^{2} \log {\left (x + 5 \right )}^{2}}} \] Input:
integrate((((-4*x**4-20*x**3)*ln(3)+8*x**5+56*x**4+80*x**3)*ln(5+x)**3+((( -2*x**2-10*x)*ln(3)+8*x**2+40*x)*ln(x)+(2*x**2+10*x)*ln(3)-4*x**3-28*x**2- 40*x)*ln(5+x)**2+((-2*x-10)*ln(x)**2+(-2*x**2*ln(3)+4*x**3+8*x**2+2*x+10)* ln(x))*ln(5+x)-2*x*ln(x)**2)*exp(((x**2*ln(3)**2+(-4*x**3-8*x**2)*ln(3)+4* x**4+16*x**3+16*x**2)*ln(5+x)**2+(2*x*ln(3)-4*x**2-8*x)*ln(x)*ln(5+x)+ln(x )**2)/x**2/ln(5+x)**2)/(x**4+5*x**3)/ln(5+x)**3,x)
Output:
exp(((-4*x**2 - 8*x + 2*x*log(3))*log(x)*log(x + 5) + (4*x**4 + 16*x**3 + x**2*log(3)**2 + 16*x**2 + (-4*x**3 - 8*x**2)*log(3))*log(x + 5)**2 + log( x)**2)/(x**2*log(x + 5)**2))
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (25) = 50\).
Time = 0.57 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=\frac {1}{6561} \, e^{\left (4 \, x^{2} - 4 \, x \log \left (3\right ) + \log \left (3\right )^{2} + 16 \, x + \frac {2 \, \log \left (3\right ) \log \left (x\right )}{x \log \left (x + 5\right )} - \frac {4 \, \log \left (x\right )}{\log \left (x + 5\right )} - \frac {8 \, \log \left (x\right )}{x \log \left (x + 5\right )} + \frac {\log \left (x\right )^{2}}{x^{2} \log \left (x + 5\right )^{2}} + 16\right )} \] Input:
integrate((((-4*x^4-20*x^3)*log(3)+8*x^5+56*x^4+80*x^3)*log(5+x)^3+(((-2*x ^2-10*x)*log(3)+8*x^2+40*x)*log(x)+(2*x^2+10*x)*log(3)-4*x^3-28*x^2-40*x)* log(5+x)^2+((-2*x-10)*log(x)^2+(-2*x^2*log(3)+4*x^3+8*x^2+2*x+10)*log(x))* log(5+x)-2*x*log(x)^2)*exp(((x^2*log(3)^2+(-4*x^3-8*x^2)*log(3)+4*x^4+16*x ^3+16*x^2)*log(5+x)^2+(2*x*log(3)-4*x^2-8*x)*log(x)*log(5+x)+log(x)^2)/x^2 /log(5+x)^2)/(x^4+5*x^3)/log(5+x)^3,x, algorithm="maxima")
Output:
1/6561*e^(4*x^2 - 4*x*log(3) + log(3)^2 + 16*x + 2*log(3)*log(x)/(x*log(x + 5)) - 4*log(x)/log(x + 5) - 8*log(x)/(x*log(x + 5)) + log(x)^2/(x^2*log( x + 5)^2) + 16)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (25) = 50\).
Time = 0.51 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.92 \[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=e^{\left (4 \, x^{2} - 4 \, x \log \left (3\right ) + \log \left (3\right )^{2} + 16 \, x + \frac {2 \, \log \left (3\right ) \log \left (x\right )}{x \log \left (x + 5\right )} - \frac {4 \, \log \left (x\right )}{\log \left (x + 5\right )} - \frac {8 \, \log \left (x\right )}{x \log \left (x + 5\right )} + \frac {\log \left (x\right )^{2}}{x^{2} \log \left (x + 5\right )^{2}} - 8 \, \log \left (3\right ) + 16\right )} \] Input:
integrate((((-4*x^4-20*x^3)*log(3)+8*x^5+56*x^4+80*x^3)*log(5+x)^3+(((-2*x ^2-10*x)*log(3)+8*x^2+40*x)*log(x)+(2*x^2+10*x)*log(3)-4*x^3-28*x^2-40*x)* log(5+x)^2+((-2*x-10)*log(x)^2+(-2*x^2*log(3)+4*x^3+8*x^2+2*x+10)*log(x))* log(5+x)-2*x*log(x)^2)*exp(((x^2*log(3)^2+(-4*x^3-8*x^2)*log(3)+4*x^4+16*x ^3+16*x^2)*log(5+x)^2+(2*x*log(3)-4*x^2-8*x)*log(x)*log(5+x)+log(x)^2)/x^2 /log(5+x)^2)/(x^4+5*x^3)/log(5+x)^3,x, algorithm="giac")
Output:
e^(4*x^2 - 4*x*log(3) + log(3)^2 + 16*x + 2*log(3)*log(x)/(x*log(x + 5)) - 4*log(x)/log(x + 5) - 8*log(x)/(x*log(x + 5)) + log(x)^2/(x^2*log(x + 5)^ 2) - 8*log(3) + 16)
Time = 3.19 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.19 \[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=\frac {x^{\frac {2\,\ln \left (3\right )}{x\,\ln \left (x+5\right )}}\,{\mathrm {e}}^{{\ln \left (3\right )}^2}\,{\mathrm {e}}^{16\,x}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{\frac {{\ln \left (x\right )}^2}{x^2\,{\ln \left (x+5\right )}^2}}\,{\mathrm {e}}^{4\,x^2}}{6561\,3^{4\,x}\,x^{\frac {8}{x\,\ln \left (x+5\right )}}\,x^{\frac {4}{\ln \left (x+5\right )}}} \] Input:
int(-(exp((log(x)^2 + log(x + 5)^2*(x^2*log(3)^2 - log(3)*(8*x^2 + 4*x^3) + 16*x^2 + 16*x^3 + 4*x^4) - log(x + 5)*log(x)*(8*x - 2*x*log(3) + 4*x^2)) /(x^2*log(x + 5)^2))*(2*x*log(x)^2 - log(x + 5)*(log(x)*(2*x - 2*x^2*log(3 ) + 8*x^2 + 4*x^3 + 10) - log(x)^2*(2*x + 10)) - log(x + 5)^3*(80*x^3 - lo g(3)*(20*x^3 + 4*x^4) + 56*x^4 + 8*x^5) + log(x + 5)^2*(40*x - log(3)*(10* x + 2*x^2) + 28*x^2 + 4*x^3 - log(x)*(40*x - log(3)*(10*x + 2*x^2) + 8*x^2 ))))/(log(x + 5)^3*(5*x^3 + x^4)),x)
Output:
(x^((2*log(3))/(x*log(x + 5)))*exp(log(3)^2)*exp(16*x)*exp(16)*exp(log(x)^ 2/(x^2*log(x + 5)^2))*exp(4*x^2))/(6561*3^(4*x)*x^(8/(x*log(x + 5)))*x^(4/ log(x + 5)))
Time = 0.67 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.85 \[ \int \frac {e^{\frac {\log ^2(x)+\left (-8 x-4 x^2+2 x \log (3)\right ) \log (x) \log (5+x)+\left (16 x^2+16 x^3+4 x^4+\left (-8 x^2-4 x^3\right ) \log (3)+x^2 \log ^2(3)\right ) \log ^2(5+x)}{x^2 \log ^2(5+x)}} \left (-2 x \log ^2(x)+\left (\left (10+2 x+8 x^2+4 x^3-2 x^2 \log (3)\right ) \log (x)+(-10-2 x) \log ^2(x)\right ) \log (5+x)+\left (-40 x-28 x^2-4 x^3+\left (10 x+2 x^2\right ) \log (3)+\left (40 x+8 x^2+\left (-10 x-2 x^2\right ) \log (3)\right ) \log (x)\right ) \log ^2(5+x)+\left (80 x^3+56 x^4+8 x^5+\left (-20 x^3-4 x^4\right ) \log (3)\right ) \log ^3(5+x)\right )}{\left (5 x^3+x^4\right ) \log ^3(5+x)} \, dx=\frac {e^{\frac {\mathrm {log}\left (x +5\right )^{2} \mathrm {log}\left (3\right )^{2} x^{2}+4 \mathrm {log}\left (x +5\right )^{2} x^{4}+16 \mathrm {log}\left (x +5\right )^{2} x^{3}+2 \,\mathrm {log}\left (x +5\right ) \mathrm {log}\left (x \right ) \mathrm {log}\left (3\right ) x +\mathrm {log}\left (x \right )^{2}}{\mathrm {log}\left (x +5\right )^{2} x^{2}}} e^{16}}{6561 e^{\frac {4 \,\mathrm {log}\left (x \right ) x +8 \,\mathrm {log}\left (x \right )}{\mathrm {log}\left (x +5\right ) x}} 3^{4 x}} \] Input:
int((((-4*x^4-20*x^3)*log(3)+8*x^5+56*x^4+80*x^3)*log(5+x)^3+(((-2*x^2-10* x)*log(3)+8*x^2+40*x)*log(x)+(2*x^2+10*x)*log(3)-4*x^3-28*x^2-40*x)*log(5+ x)^2+((-2*x-10)*log(x)^2+(-2*x^2*log(3)+4*x^3+8*x^2+2*x+10)*log(x))*log(5+ x)-2*x*log(x)^2)*exp(((x^2*log(3)^2+(-4*x^3-8*x^2)*log(3)+4*x^4+16*x^3+16* x^2)*log(5+x)^2+(2*x*log(3)-4*x^2-8*x)*log(x)*log(5+x)+log(x)^2)/x^2/log(5 +x)^2)/(x^4+5*x^3)/log(5+x)^3,x)
Output:
(e**((log(x + 5)**2*log(3)**2*x**2 + 4*log(x + 5)**2*x**4 + 16*log(x + 5)* *2*x**3 + 2*log(x + 5)*log(x)*log(3)*x + log(x)**2)/(log(x + 5)**2*x**2))* e**16)/(6561*e**((4*log(x)*x + 8*log(x))/(log(x + 5)*x))*3**(4*x))