Integrand size = 362, antiderivative size = 30 \[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\left (1+3 e^{5-\left (4+e^x+\frac {5}{x}\right )^2 x \log ^2(x)}+x\right )^2 \] Output:
(1+x+3/exp(ln(x)^2*(5/x+exp(x)+4)^2*x-5))^2
Leaf count is larger than twice the leaf count of optimal. \(61\) vs. \(2(30)=60\).
Time = 0.40 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.03 \[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx=9 e^{10-\frac {2 \left (5+\left (4+e^x\right ) x\right )^2 \log ^2(x)}{x}}+6 e^{5-\frac {\left (5+\left (4+e^x\right ) x\right )^2 \log ^2(x)}{x}} (1+x)+x (2+x) \] Input:
Integrate[(E^((2*(-5*x + (25 + 40*x + 16*x^2 + E^(2*x)*x^2 + E^x*(10*x + 8 *x^2))*Log[x]^2))/x)*(2*x^2 + 2*x^3) + (-900 - 1440*x - 576*x^2 - 36*E^(2* x)*x^2 + E^x*(-360*x - 288*x^2))*Log[x] + (450 - 288*x^2 + E^x*(-324*x^2 - 144*x^3) + E^(2*x)*(-18*x^2 - 36*x^3))*Log[x]^2 + E^((-5*x + (25 + 40*x + 16*x^2 + E^(2*x)*x^2 + E^x*(10*x + 8*x^2))*Log[x]^2)/x)*(6*x^2 + (-300 - 780*x - 672*x^2 - 192*x^3 + E^x*(-120*x - 216*x^2 - 96*x^3) + E^(2*x)*(-12 *x^2 - 12*x^3))*Log[x] + (150 + 150*x - 96*x^2 - 96*x^3 + E^x*(-108*x^2 - 156*x^3 - 48*x^4) + E^(2*x)*(-6*x^2 - 18*x^3 - 12*x^4))*Log[x]^2))/(E^((2* (-5*x + (25 + 40*x + 16*x^2 + E^(2*x)*x^2 + E^x*(10*x + 8*x^2))*Log[x]^2)) /x)*x^2),x]
Output:
9*E^(10 - (2*(5 + (4 + E^x)*x)^2*Log[x]^2)/x) + 6*E^(5 - ((5 + (4 + E^x)*x )^2*Log[x]^2)/x)*(1 + x) + 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.
| \(\displaystyle \int \frac {\exp \left (-\frac {2 \left (\left (e^{2 x} x^2+16 x^2+e^x \left (8 x^2+10 x\right )+40 x+25\right ) \log ^2(x)-5 x\right )}{x}\right ) \left (\left (2 x^3+2 x^2\right ) \exp \left (\frac {2 \left (\left (e^{2 x} x^2+16 x^2+e^x \left (8 x^2+10 x\right )+40 x+25\right ) \log ^2(x)-5 x\right )}{x}\right )+\left (6 x^2+\left (-192 x^3-672 x^2+e^x \left (-96 x^3-216 x^2-120 x\right )+e^{2 x} \left (-12 x^3-12 x^2\right )-780 x-300\right ) \log (x)+\left (-96 x^3-96 x^2+e^x \left (-48 x^4-156 x^3-108 x^2\right )+e^{2 x} \left (-12 x^4-18 x^3-6 x^2\right )+150 x+150\right ) \log ^2(x)\right ) \exp \left (\frac {\left (e^{2 x} x^2+16 x^2+e^x \left (8 x^2+10 x\right )+40 x+25\right ) \log ^2(x)-5 x}{x}\right )+\left (-36 e^{2 x} x^2-576 x^2+e^x \left (-288 x^2-360 x\right )-1440 x-900\right ) \log (x)+\left (-288 x^2+e^x \left (-144 x^3-324 x^2\right )+e^{2 x} \left (-36 x^3-18 x^2\right )+450\right ) \log ^2(x)\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 e^{-\frac {2 \left (\left (e^x+4\right ) x+5\right )^2 \log ^2(x)}{x}} \left ((x+1) e^{\frac {\left (\left (e^x+4\right ) x+5\right )^2 \log ^2(x)}{x}}+3 e^5\right ) \left (x^2 e^{\frac {\left (\left (e^x+4\right ) x+5\right )^2 \log ^2(x)}{x}}-3 e^5 \left (2 e^x \left (e^x+4\right ) x^3+\left (18 e^x+e^{2 x}+16\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (e^x+4\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2+3 e^5 \left (-2 e^x \left (4+e^x\right ) x^3-\left (16+18 e^x+e^{2 x}\right ) x^2+25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (x+1-\frac {9 e^{10-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^x x+4 x+5\right ) \log (x) \left (2 e^x \log (x) x^2+2 e^x x+e^x \log (x) x+4 \log (x) x+8 x-5 \log (x)+10\right )}{x^2}-\frac {3 e^{5-\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (8 e^x \log ^2(x) x^4+2 e^{2 x} \log ^2(x) x^4+26 e^x \log ^2(x) x^3+3 e^{2 x} \log ^2(x) x^3+16 \log ^2(x) x^3+16 e^x \log (x) x^3+2 e^{2 x} \log (x) x^3+32 \log (x) x^3+18 e^x \log ^2(x) x^2+e^{2 x} \log ^2(x) x^2+16 \log ^2(x) x^2+36 e^x \log (x) x^2+2 e^{2 x} \log (x) x^2+112 \log (x) x^2-x^2-25 \log ^2(x) x+20 e^x \log (x) x+130 \log (x) x-25 \log ^2(x)+50 \log (x)\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {e^{-\frac {2 \left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} (x+1)+3 e^5\right ) \left (e^{\frac {\left (\left (4+e^x\right ) x+5\right )^2 \log ^2(x)}{x}} x^2-3 e^5 \left (2 e^x \left (4+e^x\right ) x^3+\left (16+18 e^x+e^{2 x}\right ) x^2-25\right ) \log ^2(x)-6 e^5 \left (\left (4+e^x\right ) x+5\right )^2 \log (x)\right )}{x^2}dx\) |
Input:
Int[(E^((2*(-5*x + (25 + 40*x + 16*x^2 + E^(2*x)*x^2 + E^x*(10*x + 8*x^2)) *Log[x]^2))/x)*(2*x^2 + 2*x^3) + (-900 - 1440*x - 576*x^2 - 36*E^(2*x)*x^2 + E^x*(-360*x - 288*x^2))*Log[x] + (450 - 288*x^2 + E^x*(-324*x^2 - 144*x ^3) + E^(2*x)*(-18*x^2 - 36*x^3))*Log[x]^2 + E^((-5*x + (25 + 40*x + 16*x^ 2 + E^(2*x)*x^2 + E^x*(10*x + 8*x^2))*Log[x]^2)/x)*(6*x^2 + (-300 - 780*x - 672*x^2 - 192*x^3 + E^x*(-120*x - 216*x^2 - 96*x^3) + E^(2*x)*(-12*x^2 - 12*x^3))*Log[x] + (150 + 150*x - 96*x^2 - 96*x^3 + E^x*(-108*x^2 - 156*x^ 3 - 48*x^4) + E^(2*x)*(-6*x^2 - 18*x^3 - 12*x^4))*Log[x]^2))/(E^((2*(-5*x + (25 + 40*x + 16*x^2 + E^(2*x)*x^2 + E^x*(10*x + 8*x^2))*Log[x]^2))/x)*x^ 2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(143\) vs. \(2(29)=58\).
Time = 2.61 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.80
| method | result | size |
| risch | \(x^{2}+2 x +\left (6+6 x \right ) {\mathrm e}^{-\frac {8 \,{\mathrm e}^{x} \ln \left (x \right )^{2} x^{2}+{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{2}+10 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}+16 x^{2} \ln \left (x \right )^{2}+40 x \ln \left (x \right )^{2}+25 \ln \left (x \right )^{2}-5 x}{x}}+9 \,{\mathrm e}^{-\frac {2 \left (8 \,{\mathrm e}^{x} \ln \left (x \right )^{2} x^{2}+{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{2}+10 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}+16 x^{2} \ln \left (x \right )^{2}+40 x \ln \left (x \right )^{2}+25 \ln \left (x \right )^{2}-5 x \right )}{x}}\) | \(144\) |
| parallelrisch | \(-\frac {\left (-{\mathrm e}^{\frac {2 \left ({\mathrm e}^{2 x} x^{2}+\left (8 x^{2}+10 x \right ) {\mathrm e}^{x}+16 x^{2}+40 x +25\right ) \ln \left (x \right )^{2}-10 x}{x}} x^{3}-2 x^{2} {\mathrm e}^{\frac {2 \left ({\mathrm e}^{2 x} x^{2}+\left (8 x^{2}+10 x \right ) {\mathrm e}^{x}+16 x^{2}+40 x +25\right ) \ln \left (x \right )^{2}-10 x}{x}}-6 \,{\mathrm e}^{\frac {\left ({\mathrm e}^{2 x} x^{2}+\left (8 x^{2}+10 x \right ) {\mathrm e}^{x}+16 x^{2}+40 x +25\right ) \ln \left (x \right )^{2}-5 x}{x}} x^{2}-6 \,{\mathrm e}^{\frac {\left ({\mathrm e}^{2 x} x^{2}+\left (8 x^{2}+10 x \right ) {\mathrm e}^{x}+16 x^{2}+40 x +25\right ) \ln \left (x \right )^{2}-5 x}{x}} x -9 x \right ) {\mathrm e}^{-\frac {2 \left (\left ({\mathrm e}^{2 x} x^{2}+\left (8 x^{2}+10 x \right ) {\mathrm e}^{x}+16 x^{2}+40 x +25\right ) \ln \left (x \right )^{2}-5 x \right )}{x}}}{x}\) | \(254\) |
Input:
int(((2*x^3+2*x^2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)* ln(x)^2-5*x)/x)^2+(((-12*x^4-18*x^3-6*x^2)*exp(x)^2+(-48*x^4-156*x^3-108*x ^2)*exp(x)-96*x^3-96*x^2+150*x+150)*ln(x)^2+((-12*x^3-12*x^2)*exp(x)^2+(-9 6*x^3-216*x^2-120*x)*exp(x)-192*x^3-672*x^2-780*x-300)*ln(x)+6*x^2)*exp((( exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*ln(x)^2-5*x)/x)+((-36*x^3 -18*x^2)*exp(x)^2+(-144*x^3-324*x^2)*exp(x)-288*x^2+450)*ln(x)^2+(-36*exp( x)^2*x^2+(-288*x^2-360*x)*exp(x)-576*x^2-1440*x-900)*ln(x))/x^2/exp(((exp( x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*ln(x)^2-5*x)/x)^2,x,method=_R ETURNVERBOSE)
Output:
x^2+2*x+(6+6*x)*exp(-(8*exp(x)*ln(x)^2*x^2+exp(2*x)*ln(x)^2*x^2+10*x*exp(x )*ln(x)^2+16*x^2*ln(x)^2+40*x*ln(x)^2+25*ln(x)^2-5*x)/x)+9*exp(-2*(8*exp(x )*ln(x)^2*x^2+exp(2*x)*ln(x)^2*x^2+10*x*exp(x)*ln(x)^2+16*x^2*ln(x)^2+40*x *ln(x)^2+25*ln(x)^2-5*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (28) = 56\).
Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 5.10 \[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx={\left ({\left (x^{2} + 2 \, x\right )} e^{\left (\frac {2 \, {\left ({\left (x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 2 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{x} + 40 \, x + 25\right )} \log \left (x\right )^{2} - 5 \, x\right )}}{x}\right )} + 6 \, {\left (x + 1\right )} e^{\left (\frac {{\left (x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 2 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{x} + 40 \, x + 25\right )} \log \left (x\right )^{2} - 5 \, x}{x}\right )} + 9\right )} e^{\left (-\frac {2 \, {\left ({\left (x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 2 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{x} + 40 \, x + 25\right )} \log \left (x\right )^{2} - 5 \, x\right )}}{x}\right )} \] Input:
integrate(((2*x^3+2*x^2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40* x+25)*log(x)^2-5*x)/x)^2+(((-12*x^4-18*x^3-6*x^2)*exp(x)^2+(-48*x^4-156*x^ 3-108*x^2)*exp(x)-96*x^3-96*x^2+150*x+150)*log(x)^2+((-12*x^3-12*x^2)*exp( x)^2+(-96*x^3-216*x^2-120*x)*exp(x)-192*x^3-672*x^2-780*x-300)*log(x)+6*x^ 2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x) +((-36*x^3-18*x^2)*exp(x)^2+(-144*x^3-324*x^2)*exp(x)-288*x^2+450)*log(x)^ 2+(-36*exp(x)^2*x^2+(-288*x^2-360*x)*exp(x)-576*x^2-1440*x-900)*log(x))/x^ 2/exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x)^ 2,x, algorithm="fricas")
Output:
((x^2 + 2*x)*e^(2*((x^2*e^(2*x) + 16*x^2 + 2*(4*x^2 + 5*x)*e^x + 40*x + 25 )*log(x)^2 - 5*x)/x) + 6*(x + 1)*e^(((x^2*e^(2*x) + 16*x^2 + 2*(4*x^2 + 5* x)*e^x + 40*x + 25)*log(x)^2 - 5*x)/x) + 9)*e^(-2*((x^2*e^(2*x) + 16*x^2 + 2*(4*x^2 + 5*x)*e^x + 40*x + 25)*log(x)^2 - 5*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (26) = 52\).
Time = 68.70 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.30 \[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx=x^{2} + 2 x + \left (6 x + 6\right ) e^{- \frac {- 5 x + \left (x^{2} e^{2 x} + 16 x^{2} + 40 x + \left (8 x^{2} + 10 x\right ) e^{x} + 25\right ) \log {\left (x \right )}^{2}}{x}} + 9 e^{- \frac {2 \left (- 5 x + \left (x^{2} e^{2 x} + 16 x^{2} + 40 x + \left (8 x^{2} + 10 x\right ) e^{x} + 25\right ) \log {\left (x \right )}^{2}\right )}{x}} \] Input:
integrate(((2*x**3+2*x**2)*exp(((exp(x)**2*x**2+(8*x**2+10*x)*exp(x)+16*x* *2+40*x+25)*ln(x)**2-5*x)/x)**2+(((-12*x**4-18*x**3-6*x**2)*exp(x)**2+(-48 *x**4-156*x**3-108*x**2)*exp(x)-96*x**3-96*x**2+150*x+150)*ln(x)**2+((-12* x**3-12*x**2)*exp(x)**2+(-96*x**3-216*x**2-120*x)*exp(x)-192*x**3-672*x**2 -780*x-300)*ln(x)+6*x**2)*exp(((exp(x)**2*x**2+(8*x**2+10*x)*exp(x)+16*x** 2+40*x+25)*ln(x)**2-5*x)/x)+((-36*x**3-18*x**2)*exp(x)**2+(-144*x**3-324*x **2)*exp(x)-288*x**2+450)*ln(x)**2+(-36*exp(x)**2*x**2+(-288*x**2-360*x)*e xp(x)-576*x**2-1440*x-900)*ln(x))/x**2/exp(((exp(x)**2*x**2+(8*x**2+10*x)* exp(x)+16*x**2+40*x+25)*ln(x)**2-5*x)/x)**2,x)
Output:
x**2 + 2*x + (6*x + 6)*exp(-(-5*x + (x**2*exp(2*x) + 16*x**2 + 40*x + (8*x **2 + 10*x)*exp(x) + 25)*log(x)**2)/x) + 9*exp(-2*(-5*x + (x**2*exp(2*x) + 16*x**2 + 40*x + (8*x**2 + 10*x)*exp(x) + 25)*log(x)**2)/x)
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.27 \[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx=x^{2} + 3 \, {\left (2 \, {\left (x e^{5} + e^{5}\right )} e^{\left (-x e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 8 \, x e^{x} \log \left (x\right )^{2} + 16 \, x \log \left (x\right )^{2} + 10 \, e^{x} \log \left (x\right )^{2} + 40 \, \log \left (x\right )^{2} + \frac {25 \, \log \left (x\right )^{2}}{x}\right )} + 3 \, e^{\left (-2 \, x e^{\left (2 \, x\right )} \log \left (x\right )^{2} + 10\right )}\right )} e^{\left (-16 \, x e^{x} \log \left (x\right )^{2} - 32 \, x \log \left (x\right )^{2} - 20 \, e^{x} \log \left (x\right )^{2} - 80 \, \log \left (x\right )^{2} - \frac {50 \, \log \left (x\right )^{2}}{x}\right )} + 2 \, x \] Input:
integrate(((2*x^3+2*x^2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40* x+25)*log(x)^2-5*x)/x)^2+(((-12*x^4-18*x^3-6*x^2)*exp(x)^2+(-48*x^4-156*x^ 3-108*x^2)*exp(x)-96*x^3-96*x^2+150*x+150)*log(x)^2+((-12*x^3-12*x^2)*exp( x)^2+(-96*x^3-216*x^2-120*x)*exp(x)-192*x^3-672*x^2-780*x-300)*log(x)+6*x^ 2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x) +((-36*x^3-18*x^2)*exp(x)^2+(-144*x^3-324*x^2)*exp(x)-288*x^2+450)*log(x)^ 2+(-36*exp(x)^2*x^2+(-288*x^2-360*x)*exp(x)-576*x^2-1440*x-900)*log(x))/x^ 2/exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x)^ 2,x, algorithm="maxima")
Output:
x^2 + 3*(2*(x*e^5 + e^5)*e^(-x*e^(2*x)*log(x)^2 + 8*x*e^x*log(x)^2 + 16*x* log(x)^2 + 10*e^x*log(x)^2 + 40*log(x)^2 + 25*log(x)^2/x) + 3*e^(-2*x*e^(2 *x)*log(x)^2 + 10))*e^(-16*x*e^x*log(x)^2 - 32*x*log(x)^2 - 20*e^x*log(x)^ 2 - 80*log(x)^2 - 50*log(x)^2/x) + 2*x
\[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\int { -\frac {2 \, {\left (9 \, {\left (16 \, x^{2} + {\left (2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (4 \, x^{3} + 9 \, x^{2}\right )} e^{x} - 25\right )} \log \left (x\right )^{2} - {\left (x^{3} + x^{2}\right )} e^{\left (\frac {2 \, {\left ({\left (x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 2 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{x} + 40 \, x + 25\right )} \log \left (x\right )^{2} - 5 \, x\right )}}{x}\right )} + 3 \, {\left ({\left (16 \, x^{3} + 16 \, x^{2} + {\left (2 \, x^{4} + 3 \, x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (4 \, x^{4} + 13 \, x^{3} + 9 \, x^{2}\right )} e^{x} - 25 \, x - 25\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (16 \, x^{3} + 56 \, x^{2} + {\left (x^{3} + x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (4 \, x^{3} + 9 \, x^{2} + 5 \, x\right )} e^{x} + 65 \, x + 25\right )} \log \left (x\right )\right )} e^{\left (\frac {{\left (x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 2 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{x} + 40 \, x + 25\right )} \log \left (x\right )^{2} - 5 \, x}{x}\right )} + 18 \, {\left (x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 2 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{x} + 40 \, x + 25\right )} \log \left (x\right )\right )} e^{\left (-\frac {2 \, {\left ({\left (x^{2} e^{\left (2 \, x\right )} + 16 \, x^{2} + 2 \, {\left (4 \, x^{2} + 5 \, x\right )} e^{x} + 40 \, x + 25\right )} \log \left (x\right )^{2} - 5 \, x\right )}}{x}\right )}}{x^{2}} \,d x } \] Input:
integrate(((2*x^3+2*x^2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40* x+25)*log(x)^2-5*x)/x)^2+(((-12*x^4-18*x^3-6*x^2)*exp(x)^2+(-48*x^4-156*x^ 3-108*x^2)*exp(x)-96*x^3-96*x^2+150*x+150)*log(x)^2+((-12*x^3-12*x^2)*exp( x)^2+(-96*x^3-216*x^2-120*x)*exp(x)-192*x^3-672*x^2-780*x-300)*log(x)+6*x^ 2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x) +((-36*x^3-18*x^2)*exp(x)^2+(-144*x^3-324*x^2)*exp(x)-288*x^2+450)*log(x)^ 2+(-36*exp(x)^2*x^2+(-288*x^2-360*x)*exp(x)-576*x^2-1440*x-900)*log(x))/x^ 2/exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x)^ 2,x, algorithm="giac")
Output:
integrate(-2*(9*(16*x^2 + (2*x^3 + x^2)*e^(2*x) + 2*(4*x^3 + 9*x^2)*e^x - 25)*log(x)^2 - (x^3 + x^2)*e^(2*((x^2*e^(2*x) + 16*x^2 + 2*(4*x^2 + 5*x)*e ^x + 40*x + 25)*log(x)^2 - 5*x)/x) + 3*((16*x^3 + 16*x^2 + (2*x^4 + 3*x^3 + x^2)*e^(2*x) + 2*(4*x^4 + 13*x^3 + 9*x^2)*e^x - 25*x - 25)*log(x)^2 - x^ 2 + 2*(16*x^3 + 56*x^2 + (x^3 + x^2)*e^(2*x) + 2*(4*x^3 + 9*x^2 + 5*x)*e^x + 65*x + 25)*log(x))*e^(((x^2*e^(2*x) + 16*x^2 + 2*(4*x^2 + 5*x)*e^x + 40 *x + 25)*log(x)^2 - 5*x)/x) + 18*(x^2*e^(2*x) + 16*x^2 + 2*(4*x^2 + 5*x)*e ^x + 40*x + 25)*log(x))*e^(-2*((x^2*e^(2*x) + 16*x^2 + 2*(4*x^2 + 5*x)*e^x + 40*x + 25)*log(x)^2 - 5*x)/x)/x^2, x)
Time = 7.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.03 \[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx=2\,x+9\,{\mathrm {e}}^{10-80\,{\ln \left (x\right )}^2-\frac {50\,{\ln \left (x\right )}^2}{x}-20\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2-2\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2-16\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2-32\,x\,{\ln \left (x\right )}^2}+{\mathrm {e}}^{5-40\,{\ln \left (x\right )}^2-\frac {25\,{\ln \left (x\right )}^2}{x}-10\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2-x\,{\mathrm {e}}^{2\,x}\,{\ln \left (x\right )}^2-8\,x\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2-16\,x\,{\ln \left (x\right )}^2}\,\left (6\,x+6\right )+x^2 \] Input:
int(-(exp((2*(5*x - log(x)^2*(40*x + x^2*exp(2*x) + exp(x)*(10*x + 8*x^2) + 16*x^2 + 25)))/x)*(exp(-(5*x - log(x)^2*(40*x + x^2*exp(2*x) + exp(x)*(1 0*x + 8*x^2) + 16*x^2 + 25))/x)*(log(x)^2*(exp(x)*(108*x^2 + 156*x^3 + 48* x^4) - 150*x + exp(2*x)*(6*x^2 + 18*x^3 + 12*x^4) + 96*x^2 + 96*x^3 - 150) + log(x)*(780*x + exp(2*x)*(12*x^2 + 12*x^3) + 672*x^2 + 192*x^3 + exp(x) *(120*x + 216*x^2 + 96*x^3) + 300) - 6*x^2) - exp(-(2*(5*x - log(x)^2*(40* x + x^2*exp(2*x) + exp(x)*(10*x + 8*x^2) + 16*x^2 + 25)))/x)*(2*x^2 + 2*x^ 3) + log(x)^2*(exp(x)*(324*x^2 + 144*x^3) + exp(2*x)*(18*x^2 + 36*x^3) + 2 88*x^2 - 450) + log(x)*(1440*x + 36*x^2*exp(2*x) + exp(x)*(360*x + 288*x^2 ) + 576*x^2 + 900)))/x^2,x)
Output:
2*x + 9*exp(10 - 80*log(x)^2 - (50*log(x)^2)/x - 20*exp(x)*log(x)^2 - 2*x* exp(2*x)*log(x)^2 - 16*x*exp(x)*log(x)^2 - 32*x*log(x)^2) + exp(5 - 40*log (x)^2 - (25*log(x)^2)/x - 10*exp(x)*log(x)^2 - x*exp(2*x)*log(x)^2 - 8*x*e xp(x)*log(x)^2 - 16*x*log(x)^2)*(6*x + 6) + x^2
Time = 0.19 (sec) , antiderivative size = 350, normalized size of antiderivative = 11.67 \[ \int \frac {e^{-\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (e^{\frac {2 \left (-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)\right )}{x}} \left (2 x^2+2 x^3\right )+\left (-900-1440 x-576 x^2-36 e^{2 x} x^2+e^x \left (-360 x-288 x^2\right )\right ) \log (x)+\left (450-288 x^2+e^x \left (-324 x^2-144 x^3\right )+e^{2 x} \left (-18 x^2-36 x^3\right )\right ) \log ^2(x)+e^{\frac {-5 x+\left (25+40 x+16 x^2+e^{2 x} x^2+e^x \left (10 x+8 x^2\right )\right ) \log ^2(x)}{x}} \left (6 x^2+\left (-300-780 x-672 x^2-192 x^3+e^x \left (-120 x-216 x^2-96 x^3\right )+e^{2 x} \left (-12 x^2-12 x^3\right )\right ) \log (x)+\left (150+150 x-96 x^2-96 x^3+e^x \left (-108 x^2-156 x^3-48 x^4\right )+e^{2 x} \left (-6 x^2-18 x^3-12 x^4\right )\right ) \log ^2(x)\right )\right )}{x^2} \, dx=\frac {e^{\frac {2 e^{2 x} \mathrm {log}\left (x \right )^{2} x^{2}+16 e^{x} \mathrm {log}\left (x \right )^{2} x^{2}+20 e^{x} \mathrm {log}\left (x \right )^{2} x +32 \mathrm {log}\left (x \right )^{2} x^{2}+80 \mathrm {log}\left (x \right )^{2} x +50 \mathrm {log}\left (x \right )^{2}}{x}} x^{2}+2 e^{\frac {2 e^{2 x} \mathrm {log}\left (x \right )^{2} x^{2}+16 e^{x} \mathrm {log}\left (x \right )^{2} x^{2}+20 e^{x} \mathrm {log}\left (x \right )^{2} x +32 \mathrm {log}\left (x \right )^{2} x^{2}+80 \mathrm {log}\left (x \right )^{2} x +50 \mathrm {log}\left (x \right )^{2}}{x}} x +6 e^{\frac {e^{2 x} \mathrm {log}\left (x \right )^{2} x^{2}+8 e^{x} \mathrm {log}\left (x \right )^{2} x^{2}+10 e^{x} \mathrm {log}\left (x \right )^{2} x +16 \mathrm {log}\left (x \right )^{2} x^{2}+40 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}}{x}} e^{5} x +6 e^{\frac {e^{2 x} \mathrm {log}\left (x \right )^{2} x^{2}+8 e^{x} \mathrm {log}\left (x \right )^{2} x^{2}+10 e^{x} \mathrm {log}\left (x \right )^{2} x +16 \mathrm {log}\left (x \right )^{2} x^{2}+40 \mathrm {log}\left (x \right )^{2} x +25 \mathrm {log}\left (x \right )^{2}}{x}} e^{5}+9 e^{10}}{e^{\frac {2 e^{2 x} \mathrm {log}\left (x \right )^{2} x^{2}+16 e^{x} \mathrm {log}\left (x \right )^{2} x^{2}+20 e^{x} \mathrm {log}\left (x \right )^{2} x +32 \mathrm {log}\left (x \right )^{2} x^{2}+80 \mathrm {log}\left (x \right )^{2} x +50 \mathrm {log}\left (x \right )^{2}}{x}}} \] Input:
int(((2*x^3+2*x^2)*exp(((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)* log(x)^2-5*x)/x)^2+(((-12*x^4-18*x^3-6*x^2)*exp(x)^2+(-48*x^4-156*x^3-108* x^2)*exp(x)-96*x^3-96*x^2+150*x+150)*log(x)^2+((-12*x^3-12*x^2)*exp(x)^2+( -96*x^3-216*x^2-120*x)*exp(x)-192*x^3-672*x^2-780*x-300)*log(x)+6*x^2)*exp (((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x)+((-36 *x^3-18*x^2)*exp(x)^2+(-144*x^3-324*x^2)*exp(x)-288*x^2+450)*log(x)^2+(-36 *exp(x)^2*x^2+(-288*x^2-360*x)*exp(x)-576*x^2-1440*x-900)*log(x))/x^2/exp( ((exp(x)^2*x^2+(8*x^2+10*x)*exp(x)+16*x^2+40*x+25)*log(x)^2-5*x)/x)^2,x)
Output:
(e**((2*e**(2*x)*log(x)**2*x**2 + 16*e**x*log(x)**2*x**2 + 20*e**x*log(x)* *2*x + 32*log(x)**2*x**2 + 80*log(x)**2*x + 50*log(x)**2)/x)*x**2 + 2*e**( (2*e**(2*x)*log(x)**2*x**2 + 16*e**x*log(x)**2*x**2 + 20*e**x*log(x)**2*x + 32*log(x)**2*x**2 + 80*log(x)**2*x + 50*log(x)**2)/x)*x + 6*e**((e**(2*x )*log(x)**2*x**2 + 8*e**x*log(x)**2*x**2 + 10*e**x*log(x)**2*x + 16*log(x) **2*x**2 + 40*log(x)**2*x + 25*log(x)**2)/x)*e**5*x + 6*e**((e**(2*x)*log( x)**2*x**2 + 8*e**x*log(x)**2*x**2 + 10*e**x*log(x)**2*x + 16*log(x)**2*x* *2 + 40*log(x)**2*x + 25*log(x)**2)/x)*e**5 + 9*e**10)/e**((2*e**(2*x)*log (x)**2*x**2 + 16*e**x*log(x)**2*x**2 + 20*e**x*log(x)**2*x + 32*log(x)**2* x**2 + 80*log(x)**2*x + 50*log(x)**2)/x)