Integrand size = 154, antiderivative size = 27 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=e^5+\frac {e^5 (-4+x)}{-1+e^{\frac {25}{x^2+\log (4)}}} \] Output:
exp(5)+(-4+x)*exp(5)/(exp(1/(1/25*x^2+2/25*ln(2)))-1)
Time = 5.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=\frac {e^5 (-4+x)}{-1+e^{\frac {25}{x^2+\log (4)}}} \] Input:
Integrate[(-(E^5*x^4) - 2*E^5*x^2*Log[4] - E^5*Log[4]^2 + E^(25/(x^2 + Log [4]))*(E^5*(-200*x + 50*x^2 + x^4) + 2*E^5*x^2*Log[4] + E^5*Log[4]^2))/(x^ 4 + 2*x^2*Log[4] + Log[4]^2 + E^(25/(x^2 + Log[4]))*(-2*x^4 - 4*x^2*Log[4] - 2*Log[4]^2) + E^(50/(x^2 + Log[4]))*(x^4 + 2*x^2*Log[4] + Log[4]^2)),x]
Output:
(E^5*(-4 + x))/(-1 + E^(25/(x^2 + Log[4])))
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.
| \(\displaystyle \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)+e^{\frac {25}{x^2+\log (4)}} \left (2 e^5 x^2 \log (4)+e^5 \left (x^4+50 x^2-200 x\right )+e^5 \log ^2(4)\right )-e^5 \log ^2(4)}{x^4+2 x^2 \log (4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )+\log ^2(4)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^5 \left (e^{\frac {25}{x^2+\log (4)}} \left (x^4+2 x^2 (25+\log (4))-200 x+\log ^2(4)\right )-\left (x^2+\log (4)\right )^2\right )}{\left (1-e^{\frac {25}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^5 \int -\frac {\left (x^2+\log (4)\right )^2+e^{\frac {25}{x^2+\log (4)}} \left (-x^4-2 (25+\log (4)) x^2+200 x-\log ^2(4)\right )}{\left (1-e^{\frac {25}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -e^5 \int \frac {\left (x^2+\log (4)\right )^2+e^{\frac {25}{x^2+\log (4)}} \left (-x^4-2 (25+\log (4)) x^2+200 x-\log ^2(4)\right )}{\left (1-e^{\frac {25}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -e^5 \int \left (-\frac {2 (x-4) x}{25 \left (-1+e^{\frac {1}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {2 \left (2+3 e^{\frac {1}{x^2+\log (4)}}+3 e^{\frac {2}{x^2+\log (4)}}+2 e^{\frac {3}{x^2+\log (4)}}\right ) (x-4) x}{5 \left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {10 \left (2+3 e^{\frac {5}{x^2+\log (4)}}+3 e^{\frac {10}{x^2+\log (4)}}+2 e^{\frac {15}{x^2+\log (4)}}\right ) (x-4) x}{\left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}+\frac {3 e^{\frac {5}{x^2+\log (4)}} x^4+2 e^{\frac {10}{x^2+\log (4)}} x^4+e^{\frac {15}{x^2+\log (4)}} x^4+4 x^4+10 e^{\frac {15}{x^2+\log (4)}} \left (1+\frac {2 \log (2)}{5}\right ) x^2+30 e^{\frac {10}{x^2+\log (4)}} \left (1+\frac {4 \log (2)}{15}\right ) x^2+60 e^{\frac {5}{x^2+\log (4)}} \left (1+\frac {\log (2)}{5}\right ) x^2+100 \left (1+\frac {4 \log (2)}{25}\right ) x^2-240 e^{\frac {5}{x^2+\log (4)}} x-120 e^{\frac {10}{x^2+\log (4)}} x-40 e^{\frac {15}{x^2+\log (4)}} x-400 x+3 e^{\frac {5}{x^2+\log (4)}} \log ^2(4)+2 e^{\frac {10}{x^2+\log (4)}} \log ^2(4)+e^{\frac {15}{x^2+\log (4)}} \log ^2(4)+4 \log ^2(4)}{5 \left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {x^4+2 (1+\log (4)) x^2-8 x+\log ^2(4)}{25 \left (1-e^{\frac {1}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {2 e^{\frac {2}{x^2+\log (4)}} x^4+e^{\frac {3}{x^2+\log (4)}} x^4+3 e^{\frac {1}{x^2+\log (4)}} x^4+4 x^4+6 e^{\frac {2}{x^2+\log (4)}} \left (1+\frac {\log (16)}{3}\right ) x^2+2 e^{\frac {3}{x^2+\log (4)}} (1+\log (4)) x^2+12 e^{\frac {1}{x^2+\log (4)}} (1+\log (2)) x^2+20 \left (1+\frac {4 \log (2)}{5}\right ) x^2-24 e^{\frac {2}{x^2+\log (4)}} x-8 e^{\frac {3}{x^2+\log (4)}} x-48 e^{\frac {1}{x^2+\log (4)}} x-80 x+2 e^{\frac {2}{x^2+\log (4)}} \log ^2(4)+e^{\frac {3}{x^2+\log (4)}} \log ^2(4)+3 e^{\frac {1}{x^2+\log (4)}} \log ^2(4)+4 \log ^2(4)}{25 \left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -e^5 \int \frac {-25 e^{\frac {25}{x^2+\log (4)}} x^4+25 x^4+\log (1267650600228229401496703205376) x^2-1250 e^{\frac {25}{x^2+\log (4)}} \left (1+\frac {2 \log (2)}{25}\right ) x^2+5000 e^{\frac {25}{x^2+\log (4)}} x-25 e^{\frac {25}{x^2+\log (4)}} \log ^2(4)+25 \log ^2(4)}{25 \left (1-e^{\frac {25}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{25} e^5 \int \frac {-25 e^{\frac {25}{x^2+\log (4)}} x^4+25 x^4+\log (1267650600228229401496703205376) x^2-50 e^{\frac {25}{x^2+\log (4)}} (25+\log (4)) x^2+5000 e^{\frac {25}{x^2+\log (4)}} x-25 e^{\frac {25}{x^2+\log (4)}} \log ^2(4)+25 \log ^2(4)}{\left (1-e^{\frac {25}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{25} e^5 \int \frac {25 x^4+\log (1267650600228229401496703205376) x^2-25 e^{\frac {25}{x^2+\log (4)}} \left (x^4+2 (25+\log (4)) x^2-200 x+\log ^2(4)\right )+25 \log ^2(4)}{\left (1-e^{\frac {25}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{25} e^5 \int \left (-\frac {2 (x-4) x}{\left (-1+e^{\frac {1}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {10 \left (2+3 e^{\frac {1}{x^2+\log (4)}}+3 e^{\frac {2}{x^2+\log (4)}}+2 e^{\frac {3}{x^2+\log (4)}}\right ) (x-4) x}{\left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {250 \left (2+3 e^{\frac {5}{x^2+\log (4)}}+3 e^{\frac {10}{x^2+\log (4)}}+2 e^{\frac {15}{x^2+\log (4)}}\right ) (x-4) x}{\left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}+\frac {375 e^{\frac {5}{x^2+\log (4)}} x^4+250 e^{\frac {10}{x^2+\log (4)}} x^4+125 e^{\frac {15}{x^2+\log (4)}} x^4+500 x^4+1250 e^{\frac {15}{x^2+\log (4)}} \left (1+\frac {2 \log (2)}{5}\right ) x^2+3750 e^{\frac {10}{x^2+\log (4)}} \left (1+\frac {4 \log (2)}{15}\right ) x^2+7500 e^{\frac {5}{x^2+\log (4)}} \left (1+\frac {\log (2)}{5}\right ) x^2+12500 \left (1+\frac {4 \log (2)}{25}\right ) x^2-30000 e^{\frac {5}{x^2+\log (4)}} x-15000 e^{\frac {10}{x^2+\log (4)}} x-5000 e^{\frac {15}{x^2+\log (4)}} x-50000 x+375 e^{\frac {5}{x^2+\log (4)}} \log ^2(4)+250 e^{\frac {10}{x^2+\log (4)}} \log ^2(4)+125 e^{\frac {15}{x^2+\log (4)}} \log ^2(4)+500 \log ^2(4)}{25 \left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {1250 e^{\frac {2}{x^2+\log (4)}} x^4+625 e^{\frac {3}{x^2+\log (4)}} x^4+1875 e^{\frac {1}{x^2+\log (4)}} x^4+2500 x^4+1250 e^{\frac {3}{x^2+\log (4)}} (1+\log (4)) x^2+3750 e^{\frac {2}{x^2+\log (4)}} \left (1+\frac {4 \log (2)}{3}\right ) x^2+7500 e^{\frac {1}{x^2+\log (4)}} (1+\log (2)) x^2+12500 \left (1+\frac {4 \log (2)}{5}\right ) x^2-15000 e^{\frac {2}{x^2+\log (4)}} x-5000 e^{\frac {3}{x^2+\log (4)}} x-30000 e^{\frac {1}{x^2+\log (4)}} x-50000 x+1250 e^{\frac {2}{x^2+\log (4)}} \log ^2(4)+625 e^{\frac {3}{x^2+\log (4)}} \log ^2(4)+1875 e^{\frac {1}{x^2+\log (4)}} \log ^2(4)+2500 \log ^2(4)}{625 \left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {625 x^4+2 (625+25 \log (4)+12 \log (1267650600228229401496703205376)) x^2-5000 x+625 \log ^2(4)}{625 \left (1-e^{\frac {1}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{25} e^5 \int \left (-\frac {2 (x-4) x}{\left (-1+e^{\frac {1}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {10 \left (2+3 e^{\frac {1}{x^2+\log (4)}}+3 e^{\frac {2}{x^2+\log (4)}}+2 e^{\frac {3}{x^2+\log (4)}}\right ) (x-4) x}{\left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {250 \left (2+3 e^{\frac {5}{x^2+\log (4)}}+3 e^{\frac {10}{x^2+\log (4)}}+2 e^{\frac {15}{x^2+\log (4)}}\right ) (x-4) x}{\left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}+\frac {5 \left (3 e^{\frac {5}{x^2+\log (4)}} x^4+2 e^{\frac {10}{x^2+\log (4)}} x^4+e^{\frac {15}{x^2+\log (4)}} x^4+4 x^4+10 e^{\frac {15}{x^2+\log (4)}} \left (1+\frac {2 \log (2)}{5}\right ) x^2+30 e^{\frac {10}{x^2+\log (4)}} \left (1+\frac {4 \log (2)}{15}\right ) x^2+60 e^{\frac {5}{x^2+\log (4)}} \left (1+\frac {\log (2)}{5}\right ) x^2+100 \left (1+\frac {4 \log (2)}{25}\right ) x^2-240 e^{\frac {5}{x^2+\log (4)}} x-120 e^{\frac {10}{x^2+\log (4)}} x-40 e^{\frac {15}{x^2+\log (4)}} x-400 x+3 e^{\frac {5}{x^2+\log (4)}} \log ^2(4)+2 e^{\frac {10}{x^2+\log (4)}} \log ^2(4)+e^{\frac {15}{x^2+\log (4)}} \log ^2(4)+4 \log ^2(4)\right )}{\left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {2 e^{\frac {2}{x^2+\log (4)}} x^4+e^{\frac {3}{x^2+\log (4)}} x^4+3 e^{\frac {1}{x^2+\log (4)}} x^4+4 x^4+6 e^{\frac {2}{x^2+\log (4)}} \left (1+\frac {\log (16)}{3}\right ) x^2+2 e^{\frac {3}{x^2+\log (4)}} (1+\log (4)) x^2+12 e^{\frac {1}{x^2+\log (4)}} (1+\log (2)) x^2+20 \left (1+\frac {4 \log (2)}{5}\right ) x^2-24 e^{\frac {2}{x^2+\log (4)}} x-8 e^{\frac {3}{x^2+\log (4)}} x-48 e^{\frac {1}{x^2+\log (4)}} x-80 x+2 e^{\frac {2}{x^2+\log (4)}} \log ^2(4)+e^{\frac {3}{x^2+\log (4)}} \log ^2(4)+3 e^{\frac {1}{x^2+\log (4)}} \log ^2(4)+4 \log ^2(4)}{\left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {625 x^4+2 (625+25 \log (4)+12 \log (1267650600228229401496703205376)) x^2-5000 x+625 \log ^2(4)}{625 \left (1-e^{\frac {1}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{25} e^5 \int \left (-\frac {2 (x-4) x}{\left (-1+e^{\frac {1}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {10 \left (2+3 e^{\frac {1}{x^2+\log (4)}}+3 e^{\frac {2}{x^2+\log (4)}}+2 e^{\frac {3}{x^2+\log (4)}}\right ) (x-4) x}{\left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}-\frac {250 \left (2+3 e^{\frac {5}{x^2+\log (4)}}+3 e^{\frac {10}{x^2+\log (4)}}+2 e^{\frac {15}{x^2+\log (4)}}\right ) (x-4) x}{\left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right )^2 \left (x^2+\log (4)\right )^2}+\frac {5 \left (3 e^{\frac {5}{x^2+\log (4)}} x^4+2 e^{\frac {10}{x^2+\log (4)}} x^4+e^{\frac {15}{x^2+\log (4)}} x^4+4 x^4+10 e^{\frac {15}{x^2+\log (4)}} \left (1+\frac {2 \log (2)}{5}\right ) x^2+30 e^{\frac {10}{x^2+\log (4)}} \left (1+\frac {4 \log (2)}{15}\right ) x^2+60 e^{\frac {5}{x^2+\log (4)}} \left (1+\frac {\log (2)}{5}\right ) x^2+100 \left (1+\frac {4 \log (2)}{25}\right ) x^2-240 e^{\frac {5}{x^2+\log (4)}} x-120 e^{\frac {10}{x^2+\log (4)}} x-40 e^{\frac {15}{x^2+\log (4)}} x-400 x+3 e^{\frac {5}{x^2+\log (4)}} \log ^2(4)+2 e^{\frac {10}{x^2+\log (4)}} \log ^2(4)+e^{\frac {15}{x^2+\log (4)}} \log ^2(4)+4 \log ^2(4)\right )}{\left (1+e^{\frac {5}{x^2+\log (4)}}+e^{\frac {10}{x^2+\log (4)}}+e^{\frac {15}{x^2+\log (4)}}+e^{\frac {20}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {2 e^{\frac {2}{x^2+\log (4)}} x^4+e^{\frac {3}{x^2+\log (4)}} x^4+3 e^{\frac {1}{x^2+\log (4)}} x^4+4 x^4+6 e^{\frac {2}{x^2+\log (4)}} \left (1+\frac {\log (16)}{3}\right ) x^2+2 e^{\frac {3}{x^2+\log (4)}} (1+\log (4)) x^2+12 e^{\frac {1}{x^2+\log (4)}} (1+\log (2)) x^2+20 \left (1+\frac {4 \log (2)}{5}\right ) x^2-24 e^{\frac {2}{x^2+\log (4)}} x-8 e^{\frac {3}{x^2+\log (4)}} x-48 e^{\frac {1}{x^2+\log (4)}} x-80 x+2 e^{\frac {2}{x^2+\log (4)}} \log ^2(4)+e^{\frac {3}{x^2+\log (4)}} \log ^2(4)+3 e^{\frac {1}{x^2+\log (4)}} \log ^2(4)+4 \log ^2(4)}{\left (1+e^{\frac {1}{x^2+\log (4)}}+e^{\frac {2}{x^2+\log (4)}}+e^{\frac {3}{x^2+\log (4)}}+e^{\frac {4}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}+\frac {625 x^4+2 (625+25 \log (4)+12 \log (1267650600228229401496703205376)) x^2-5000 x+625 \log ^2(4)}{625 \left (1-e^{\frac {1}{x^2+\log (4)}}\right ) \left (x^2+\log (4)\right )^2}\right )dx\) |
Input:
Int[(-(E^5*x^4) - 2*E^5*x^2*Log[4] - E^5*Log[4]^2 + E^(25/(x^2 + Log[4]))* (E^5*(-200*x + 50*x^2 + x^4) + 2*E^5*x^2*Log[4] + E^5*Log[4]^2))/(x^4 + 2* x^2*Log[4] + Log[4]^2 + E^(25/(x^2 + Log[4]))*(-2*x^4 - 4*x^2*Log[4] - 2*L og[4]^2) + E^(50/(x^2 + Log[4]))*(x^4 + 2*x^2*Log[4] + Log[4]^2)),x]
Output:
$Aborted
Time = 0.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(\frac {\left (x -4\right ) {\mathrm e}^{5}}{{\mathrm e}^{\frac {25}{x^{2}+2 \ln \left (2\right )}}-1}\) | \(24\) |
| parallelrisch | \(\frac {x \,{\mathrm e}^{5}-4 \,{\mathrm e}^{5}}{{\mathrm e}^{\frac {25}{x^{2}+2 \ln \left (2\right )}}-1}\) | \(28\) |
| norman | \(\frac {x^{3} {\mathrm e}^{5}-4 x^{2} {\mathrm e}^{5}-8 \,{\mathrm e}^{5} \ln \left (2\right )+2 x \,{\mathrm e}^{5} \ln \left (2\right )}{\left ({\mathrm e}^{\frac {25}{x^{2}+2 \ln \left (2\right )}}-1\right ) \left (x^{2}+2 \ln \left (2\right )\right )}\) | \(56\) |
Input:
int(((4*exp(5)*ln(2)^2+4*x^2*exp(5)*ln(2)+(x^4+50*x^2-200*x)*exp(5))*exp(2 5/(x^2+2*ln(2)))-4*exp(5)*ln(2)^2-4*x^2*exp(5)*ln(2)-x^4*exp(5))/((4*ln(2) ^2+4*x^2*ln(2)+x^4)*exp(25/(x^2+2*ln(2)))^2+(-8*ln(2)^2-8*x^2*ln(2)-2*x^4) *exp(25/(x^2+2*ln(2)))+4*ln(2)^2+4*x^2*ln(2)+x^4),x,method=_RETURNVERBOSE)
Output:
(x-4)*exp(5)/(exp(25/(x^2+2*ln(2)))-1)
Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=\frac {{\left (x - 4\right )} e^{5}}{e^{\left (\frac {25}{x^{2} + 2 \, \log \left (2\right )}\right )} - 1} \] Input:
integrate(((4*exp(5)*log(2)^2+4*x^2*exp(5)*log(2)+(x^4+50*x^2-200*x)*exp(5 ))*exp(25/(x^2+2*log(2)))-4*exp(5)*log(2)^2-4*x^2*exp(5)*log(2)-x^4*exp(5) )/((4*log(2)^2+4*x^2*log(2)+x^4)*exp(25/(x^2+2*log(2)))^2+(-8*log(2)^2-8*x ^2*log(2)-2*x^4)*exp(25/(x^2+2*log(2)))+4*log(2)^2+4*x^2*log(2)+x^4),x, al gorithm="fricas")
Output:
(x - 4)*e^5/(e^(25/(x^2 + 2*log(2))) - 1)
Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=\frac {x e^{5} - 4 e^{5}}{e^{\frac {25}{x^{2} + 2 \log {\left (2 \right )}}} - 1} \] Input:
integrate(((4*exp(5)*ln(2)**2+4*x**2*exp(5)*ln(2)+(x**4+50*x**2-200*x)*exp (5))*exp(25/(x**2+2*ln(2)))-4*exp(5)*ln(2)**2-4*x**2*exp(5)*ln(2)-x**4*exp (5))/((4*ln(2)**2+4*x**2*ln(2)+x**4)*exp(25/(x**2+2*ln(2)))**2+(-8*ln(2)** 2-8*x**2*ln(2)-2*x**4)*exp(25/(x**2+2*ln(2)))+4*ln(2)**2+4*x**2*ln(2)+x**4 ),x)
Output:
(x*exp(5) - 4*exp(5))/(exp(25/(x**2 + 2*log(2))) - 1)
Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=\frac {x e^{5} - 4 \, e^{5}}{e^{\left (\frac {25}{x^{2} + 2 \, \log \left (2\right )}\right )} - 1} \] Input:
integrate(((4*exp(5)*log(2)^2+4*x^2*exp(5)*log(2)+(x^4+50*x^2-200*x)*exp(5 ))*exp(25/(x^2+2*log(2)))-4*exp(5)*log(2)^2-4*x^2*exp(5)*log(2)-x^4*exp(5) )/((4*log(2)^2+4*x^2*log(2)+x^4)*exp(25/(x^2+2*log(2)))^2+(-8*log(2)^2-8*x ^2*log(2)-2*x^4)*exp(25/(x^2+2*log(2)))+4*log(2)^2+4*x^2*log(2)+x^4),x, al gorithm="maxima")
Output:
(x*e^5 - 4*e^5)/(e^(25/(x^2 + 2*log(2))) - 1)
Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (26) = 52\).
Time = 0.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 6.70 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=\frac {x^{4} e^{\left (\frac {5 \, {\left (2 \, \log \left (2\right ) + 5\right )}}{2 \, \log \left (2\right )}\right )} - x^{4} e^{\left (\frac {25}{2 \, \log \left (2\right )} + 5\right )} + 4 \, x^{2} e^{\left (\frac {5 \, {\left (2 \, \log \left (2\right ) + 5\right )}}{2 \, \log \left (2\right )}\right )} \log \left (2\right ) - 4 \, x^{2} e^{\left (\frac {25}{2 \, \log \left (2\right )} + 5\right )} \log \left (2\right ) + 50 \, x^{2} e^{\left (\frac {5 \, {\left (2 \, \log \left (2\right ) + 5\right )}}{2 \, \log \left (2\right )}\right )} + 4 \, e^{\left (\frac {5 \, {\left (2 \, \log \left (2\right ) + 5\right )}}{2 \, \log \left (2\right )}\right )} \log \left (2\right )^{2} - 4 \, e^{\left (\frac {25}{2 \, \log \left (2\right )} + 5\right )} \log \left (2\right )^{2} - 200 \, x e^{\left (\frac {5 \, {\left (2 \, \log \left (2\right ) + 5\right )}}{2 \, \log \left (2\right )}\right )}}{50 \, {\left (x e^{\left (-\frac {25 \, x^{2}}{2 \, {\left (x^{2} \log \left (2\right ) + 2 \, \log \left (2\right )^{2}\right )}} + \frac {25}{\log \left (2\right )}\right )} - x e^{\left (\frac {25}{2 \, \log \left (2\right )}\right )}\right )}} \] Input:
integrate(((4*exp(5)*log(2)^2+4*x^2*exp(5)*log(2)+(x^4+50*x^2-200*x)*exp(5 ))*exp(25/(x^2+2*log(2)))-4*exp(5)*log(2)^2-4*x^2*exp(5)*log(2)-x^4*exp(5) )/((4*log(2)^2+4*x^2*log(2)+x^4)*exp(25/(x^2+2*log(2)))^2+(-8*log(2)^2-8*x ^2*log(2)-2*x^4)*exp(25/(x^2+2*log(2)))+4*log(2)^2+4*x^2*log(2)+x^4),x, al gorithm="giac")
Output:
1/50*(x^4*e^(5/2*(2*log(2) + 5)/log(2)) - x^4*e^(25/2/log(2) + 5) + 4*x^2* e^(5/2*(2*log(2) + 5)/log(2))*log(2) - 4*x^2*e^(25/2/log(2) + 5)*log(2) + 50*x^2*e^(5/2*(2*log(2) + 5)/log(2)) + 4*e^(5/2*(2*log(2) + 5)/log(2))*log (2)^2 - 4*e^(25/2/log(2) + 5)*log(2)^2 - 200*x*e^(5/2*(2*log(2) + 5)/log(2 )))/(x*e^(-25/2*x^2/(x^2*log(2) + 2*log(2)^2) + 25/log(2)) - x*e^(25/2/log (2)))
Time = 0.32 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.52 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=\frac {{\mathrm {e}}^5\,{\left (x^2+\ln \left (4\right )\right )}^2\,\left (x-4\right )}{{\left (x^2+2\,\ln \left (2\right )\right )}^2\,\left ({\mathrm {e}}^{\frac {25}{x^2+2\,\ln \left (2\right )}}-1\right )} \] Input:
int(-(4*exp(5)*log(2)^2 - exp(25/(2*log(2) + x^2))*(exp(5)*(50*x^2 - 200*x + x^4) + 4*exp(5)*log(2)^2 + 4*x^2*exp(5)*log(2)) + x^4*exp(5) + 4*x^2*ex p(5)*log(2))/(exp(50/(2*log(2) + x^2))*(4*x^2*log(2) + 4*log(2)^2 + x^4) + 4*x^2*log(2) + 4*log(2)^2 + x^4 - exp(25/(2*log(2) + x^2))*(8*x^2*log(2) + 8*log(2)^2 + 2*x^4)),x)
Output:
(exp(5)*(log(4) + x^2)^2*(x - 4))/((2*log(2) + x^2)^2*(exp(25/(2*log(2) + x^2)) - 1))
Time = 0.71 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48 \[ \int \frac {-e^5 x^4-2 e^5 x^2 \log (4)-e^5 \log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (e^5 \left (-200 x+50 x^2+x^4\right )+2 e^5 x^2 \log (4)+e^5 \log ^2(4)\right )}{x^4+2 x^2 \log (4)+\log ^2(4)+e^{\frac {25}{x^2+\log (4)}} \left (-2 x^4-4 x^2 \log (4)-2 \log ^2(4)\right )+e^{\frac {50}{x^2+\log (4)}} \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx=\frac {e^{5} \left (-4 e^{\frac {25}{2 \,\mathrm {log}\left (2\right )+x^{2}}}+x \right )}{e^{\frac {25}{2 \,\mathrm {log}\left (2\right )+x^{2}}}-1} \] Input:
int(((4*exp(5)*log(2)^2+4*x^2*exp(5)*log(2)+(x^4+50*x^2-200*x)*exp(5))*exp (25/(x^2+2*log(2)))-4*exp(5)*log(2)^2-4*x^2*exp(5)*log(2)-x^4*exp(5))/((4* log(2)^2+4*x^2*log(2)+x^4)*exp(25/(x^2+2*log(2)))^2+(-8*log(2)^2-8*x^2*log (2)-2*x^4)*exp(25/(x^2+2*log(2)))+4*log(2)^2+4*x^2*log(2)+x^4),x)
Output:
(e**5*( - 4*e**(25/(2*log(2) + x**2)) + x))/(e**(25/(2*log(2) + x**2)) - 1 )