Integrand size = 159, antiderivative size = 25 \[ \int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx=\frac {1}{2 \left (3-e^{2 (5-x)^4}+\log (-2+x)\right )} \]
Leaf count is larger than twice the leaf count of optimal. \(60\) vs. \(2(25)=50\).
Time = 0.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx=\frac {e^{40 x \left (25+x^2\right )}}{2 \left (3 e^{40 x \left (25+x^2\right )}-e^{2 \left (625+150 x^2+x^4\right )}+e^{40 x \left (25+x^2\right )} \log (-2+x)\right )} \]
Integrate[(-1 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(2000 - 2200* x + 840*x^2 - 136*x^3 + 8*x^4))/(-36 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(24 - 12*x) + 18*x + E^(2500 - 2000*x + 600*x^2 - 80*x^3 + 4*x^4 )*(-4 + 2*x) + (-24 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(8 - 4* x) + 12*x)*Log[-2 + x] + (-4 + 2*x)*Log[-2 + x]^2),x]
E^(40*x*(25 + x^2))/(2*(3*E^(40*x*(25 + x^2)) - E^(2*(625 + 150*x^2 + x^4) ) + E^(40*x*(25 + x^2))*Log[-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.
| \(\displaystyle \int \frac {e^{2 x^4-40 x^3+300 x^2-1000 x+1250} \left (8 x^4-136 x^3+840 x^2-2200 x+2000\right )-1}{e^{2 x^4-40 x^3+300 x^2-1000 x+1250} (24-12 x)+e^{4 x^4-80 x^3+600 x^2-2000 x+2500} (2 x-4)+\left (e^{2 x^4-40 x^3+300 x^2-1000 x+1250} (8-4 x)+12 x-24\right ) \log (x-2)+18 x+(2 x-4) \log ^2(x-2)-36} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{2 (2-x) \left (3 e^{40 x \left (x^2+25\right )}+e^{40 x \left (x^2+25\right )} \log (x-2)-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (5-x)^4} (2-x) (5-x)^3\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {8 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^4}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {136 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^3}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {840 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x^2}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {2200 e^{2 (x-5)^4+80 x \left (x^2+25\right )} x}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}-\frac {e^{80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}+\frac {2000 e^{2 (x-5)^4+80 x \left (x^2+25\right )}}{(x-2) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 x^4+300 x^2+1250}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{2} \int \frac {e^{80 x \left (x^2+25\right )} \left (1-8 e^{2 (x-5)^4} (x-5)^3 (x-2)\right )}{(2-x) \left (e^{40 x \left (x^2+25\right )} \log (x-2)+3 e^{40 x \left (x^2+25\right )}-e^{2 \left (x^4+150 x^2+625\right )}\right )^2}dx\) |
Int[(-1 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(2000 - 2200*x + 84 0*x^2 - 136*x^3 + 8*x^4))/(-36 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x ^4)*(24 - 12*x) + 18*x + E^(2500 - 2000*x + 600*x^2 - 80*x^3 + 4*x^4)*(-4 + 2*x) + (-24 + E^(1250 - 1000*x + 300*x^2 - 40*x^3 + 2*x^4)*(8 - 4*x) + 1 2*x)*Log[-2 + x] + (-4 + 2*x)*Log[-2 + x]^2),x]
3.12.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.77 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {1}{2 \left ({\mathrm e}^{2 \left (-5+x \right )^{4}}-\ln \left (-2+x \right )-3\right )}\) | \(21\) |
| parallelrisch | \(\frac {1}{-2 \,{\mathrm e}^{2 x^{4}-40 x^{3}+300 x^{2}-1000 x +1250}+2 \ln \left (-2+x \right )+6}\) | \(34\) |
int(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500*x+625) ^2-1)/((2*x-4)*ln(-2+x)^2+((-4*x+8)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+12 *x-24)*ln(-2+x)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x+24)*exp (x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx=-\frac {1}{2 \, {\left (e^{\left (2 \, x^{4} - 40 \, x^{3} + 300 \, x^{2} - 1000 \, x + 1250\right )} - \log \left (x - 2\right ) - 3\right )}} \]
integrate(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500* x+625)^2-1)/((2*x-4)*log(-2+x)^2+((-4*x+8)*exp(x^4-20*x^3+150*x^2-500*x+62 5)^2+12*x-24)*log(-2+x)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x +24)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx=- \frac {1}{2 e^{2 x^{4} - 40 x^{3} + 300 x^{2} - 1000 x + 1250} - 2 \log {\left (x - 2 \right )} - 6} \]
integrate(((8*x**4-136*x**3+840*x**2-2200*x+2000)*exp(x**4-20*x**3+150*x** 2-500*x+625)**2-1)/((2*x-4)*ln(-2+x)**2+((-4*x+8)*exp(x**4-20*x**3+150*x** 2-500*x+625)**2+12*x-24)*ln(-2+x)+(2*x-4)*exp(x**4-20*x**3+150*x**2-500*x+ 625)**4+(-12*x+24)*exp(x**4-20*x**3+150*x**2-500*x+625)**2+18*x-36),x)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.12 \[ \int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx=\frac {e^{\left (40 \, x^{3} + 1000 \, x\right )}}{2 \, {\left ({\left (e^{\left (1000 \, x\right )} \log \left (x - 2\right ) + 3 \, e^{\left (1000 \, x\right )}\right )} e^{\left (40 \, x^{3}\right )} - e^{\left (2 \, x^{4} + 300 \, x^{2} + 1250\right )}\right )}} \]
integrate(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500* x+625)^2-1)/((2*x-4)*log(-2+x)^2+((-4*x+8)*exp(x^4-20*x^3+150*x^2-500*x+62 5)^2+12*x-24)*log(-2+x)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x +24)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x, algorithm=\
1/2*e^(40*x^3 + 1000*x)/((e^(1000*x)*log(x - 2) + 3*e^(1000*x))*e^(40*x^3) - e^(2*x^4 + 300*x^2 + 1250))
Time = 0.74 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.32 \[ \int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx=-\frac {1}{2 \, {\left (e^{\left (2 \, x^{4} - 40 \, x^{3} + 300 \, x^{2} - 1000 \, x + 1250\right )} - \log \left (x - 2\right ) - 3\right )}} \]
integrate(((8*x^4-136*x^3+840*x^2-2200*x+2000)*exp(x^4-20*x^3+150*x^2-500* x+625)^2-1)/((2*x-4)*log(-2+x)^2+((-4*x+8)*exp(x^4-20*x^3+150*x^2-500*x+62 5)^2+12*x-24)*log(-2+x)+(2*x-4)*exp(x^4-20*x^3+150*x^2-500*x+625)^4+(-12*x +24)*exp(x^4-20*x^3+150*x^2-500*x+625)^2+18*x-36),x, algorithm=\
Time = 9.80 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-1+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} \left (2000-2200 x+840 x^2-136 x^3+8 x^4\right )}{-36+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (24-12 x)+18 x+e^{2500-2000 x+600 x^2-80 x^3+4 x^4} (-4+2 x)+\left (-24+e^{1250-1000 x+300 x^2-40 x^3+2 x^4} (8-4 x)+12 x\right ) \log (-2+x)+(-4+2 x) \log ^2(-2+x)} \, dx=\frac {1}{2\,\left (\ln \left (x-2\right )-{\mathrm {e}}^{-1000\,x}\,{\mathrm {e}}^{1250}\,{\mathrm {e}}^{2\,x^4}\,{\mathrm {e}}^{-40\,x^3}\,{\mathrm {e}}^{300\,x^2}+3\right )} \]
int((exp(300*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250)*(840*x^2 - 2200*x - 136 *x^3 + 8*x^4 + 2000) - 1)/(18*x + log(x - 2)^2*(2*x - 4) - log(x - 2)*(exp (300*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250)*(4*x - 8) - 12*x + 24) - exp(30 0*x^2 - 1000*x - 40*x^3 + 2*x^4 + 1250)*(12*x - 24) + exp(600*x^2 - 2000*x - 80*x^3 + 4*x^4 + 2500)*(2*x - 4) - 36),x)