Integrand size = 230, antiderivative size = 30 \[ \int \frac {-5 e^{21+x} x+e^{42+2 x} \left (3 e^4-9 x+x^2\right )+\left (e^{42+2 x} \left (-6 e^4+6 x\right )+e^{21+x} \left (e^4 (-5-5 x)+5 x+5 x^2\right )\right ) \log \left (-e^4+x\right )}{e^{42+2 x} \left (9 e^4-9 x\right )+\left (e^{21+x} \left (-30 e^4+30 x\right )+e^{42+2 x} \left (36 x-6 x^2+e^4 (-36+6 x)\right )\right ) \log \left (-e^4+x\right )+\left (25 e^4-25 x+e^{21+x} \left (e^4 (60-10 x)-60 x+10 x^2\right )+e^{42+2 x} \left (-36 x+12 x^2-x^3+e^4 \left (36-12 x+x^2\right )\right )\right ) \log ^2\left (-e^4+x\right )} \, dx=\frac {x}{3-\left (6+5 e^{-21-x}-x\right ) \log \left (-e^4+x\right )} \]
Time = 0.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-5 e^{21+x} x+e^{42+2 x} \left (3 e^4-9 x+x^2\right )+\left (e^{42+2 x} \left (-6 e^4+6 x\right )+e^{21+x} \left (e^4 (-5-5 x)+5 x+5 x^2\right )\right ) \log \left (-e^4+x\right )}{e^{42+2 x} \left (9 e^4-9 x\right )+\left (e^{21+x} \left (-30 e^4+30 x\right )+e^{42+2 x} \left (36 x-6 x^2+e^4 (-36+6 x)\right )\right ) \log \left (-e^4+x\right )+\left (25 e^4-25 x+e^{21+x} \left (e^4 (60-10 x)-60 x+10 x^2\right )+e^{42+2 x} \left (-36 x+12 x^2-x^3+e^4 \left (36-12 x+x^2\right )\right )\right ) \log ^2\left (-e^4+x\right )} \, dx=\frac {e^{21+x} x}{3 e^{21+x}+\left (-5+e^{21+x} (-6+x)\right ) \log \left (-e^4+x\right )} \]
Integrate[(-5*E^(21 + x)*x + E^(42 + 2*x)*(3*E^4 - 9*x + x^2) + (E^(42 + 2 *x)*(-6*E^4 + 6*x) + E^(21 + x)*(E^4*(-5 - 5*x) + 5*x + 5*x^2))*Log[-E^4 + x])/(E^(42 + 2*x)*(9*E^4 - 9*x) + (E^(21 + x)*(-30*E^4 + 30*x) + E^(42 + 2*x)*(36*x - 6*x^2 + E^4*(-36 + 6*x)))*Log[-E^4 + x] + (25*E^4 - 25*x + E^ (21 + x)*(E^4*(60 - 10*x) - 60*x + 10*x^2) + E^(42 + 2*x)*(-36*x + 12*x^2 - x^3 + E^4*(36 - 12*x + x^2)))*Log[-E^4 + 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.
| \(\displaystyle \int \frac {e^{2 x+42} \left (x^2-9 x+3 e^4\right )+\left (e^{x+21} \left (5 x^2+5 x+e^4 (-5 x-5)\right )+e^{2 x+42} \left (6 x-6 e^4\right )\right ) \log \left (x-e^4\right )-5 e^{x+21} x}{\left (e^{2 x+42} \left (-6 x^2+36 x+e^4 (6 x-36)\right )+e^{x+21} \left (30 x-30 e^4\right )\right ) \log \left (x-e^4\right )+\left (e^{x+21} \left (10 x^2-60 x+e^4 (60-10 x)\right )+e^{2 x+42} \left (-x^3+12 x^2+e^4 \left (x^2-12 x+36\right )-36 x\right )-25 x+25 e^4\right ) \log ^2\left (x-e^4\right )+e^{2 x+42} \left (9 e^4-9 x\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (-5 x+e^{x+21} \left ((x-9) x+3 e^4\right )-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{x+21} \left (e^{x+21} (x-9) x-5 x+3 e^{x+25}-\left (e^4-x\right ) \left (5 x+6 e^{x+21}+5\right ) \log \left (x-e^4\right )\right )}{\left (e^4-x\right ) \left (3 e^{x+21}+\left (e^{x+21} (x-6)-5\right ) \log \left (x-e^4\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 e^{x+21} x \left (x^2 \log ^2\left (x-e^4\right )-5 \left (1+\frac {e^4}{5}\right ) x \log ^2\left (x-e^4\right )+5 e^4 \log ^2\left (x-e^4\right )+3 x \log \left (x-e^4\right )-3 e^4 \log \left (x-e^4\right )-3\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )^2}+\frac {e^{x+21} \left (x^2-9 x+6 x \log \left (x-e^4\right )-6 e^4 \log \left (x-e^4\right )+3 e^4\right )}{\left (e^4-x\right ) \left (x \log \left (x-e^4\right )-6 \log \left (x-e^4\right )+3\right ) \left (3 e^{x+21}-6 e^{x+21} \log \left (x-e^4\right )+e^{x+21} x \log \left (x-e^4\right )-5 \log \left (x-e^4\right )\right )}\right )dx\) |
Int[(-5*E^(21 + x)*x + E^(42 + 2*x)*(3*E^4 - 9*x + x^2) + (E^(42 + 2*x)*(- 6*E^4 + 6*x) + E^(21 + x)*(E^4*(-5 - 5*x) + 5*x + 5*x^2))*Log[-E^4 + x])/( E^(42 + 2*x)*(9*E^4 - 9*x) + (E^(21 + x)*(-30*E^4 + 30*x) + E^(42 + 2*x)*( 36*x - 6*x^2 + E^4*(-36 + 6*x)))*Log[-E^4 + x] + (25*E^4 - 25*x + E^(21 + x)*(E^4*(60 - 10*x) - 60*x + 10*x^2) + E^(42 + 2*x)*(-36*x + 12*x^2 - x^3 + E^4*(36 - 12*x + x^2)))*Log[-E^4 + x]^2),x]
3.12.2.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.83 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.70
| method | result | size |
| risch | \(\frac {x \,{\mathrm e}^{x +21}}{{\mathrm e}^{x +21} \ln \left (x -{\mathrm e}^{4}\right ) x -6 \,{\mathrm e}^{x +21} \ln \left (x -{\mathrm e}^{4}\right )+3 \,{\mathrm e}^{x +21}-5 \ln \left (x -{\mathrm e}^{4}\right )}\) | \(51\) |
| parallelrisch | \(\frac {x \,{\mathrm e}^{x +21}}{{\mathrm e}^{x +21} \ln \left (x -{\mathrm e}^{4}\right ) x -6 \,{\mathrm e}^{x +21} \ln \left (x -{\mathrm e}^{4}\right )+3 \,{\mathrm e}^{x +21}-5 \ln \left (x -{\mathrm e}^{4}\right )}\) | \(51\) |
int((((-6*exp(4)+6*x)*exp(x+21)^2+((-5*x-5)*exp(4)+5*x^2+5*x)*exp(x+21))*l n(x-exp(4))+(3*exp(4)+x^2-9*x)*exp(x+21)^2-5*x*exp(x+21))/((((x^2-12*x+36) *exp(4)-x^3+12*x^2-36*x)*exp(x+21)^2+((-10*x+60)*exp(4)+10*x^2-60*x)*exp(x +21)+25*exp(4)-25*x)*ln(x-exp(4))^2+(((6*x-36)*exp(4)-6*x^2+36*x)*exp(x+21 )^2+(-30*exp(4)+30*x)*exp(x+21))*ln(x-exp(4))+(9*exp(4)-9*x)*exp(x+21)^2), x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {-5 e^{21+x} x+e^{42+2 x} \left (3 e^4-9 x+x^2\right )+\left (e^{42+2 x} \left (-6 e^4+6 x\right )+e^{21+x} \left (e^4 (-5-5 x)+5 x+5 x^2\right )\right ) \log \left (-e^4+x\right )}{e^{42+2 x} \left (9 e^4-9 x\right )+\left (e^{21+x} \left (-30 e^4+30 x\right )+e^{42+2 x} \left (36 x-6 x^2+e^4 (-36+6 x)\right )\right ) \log \left (-e^4+x\right )+\left (25 e^4-25 x+e^{21+x} \left (e^4 (60-10 x)-60 x+10 x^2\right )+e^{42+2 x} \left (-36 x+12 x^2-x^3+e^4 \left (36-12 x+x^2\right )\right )\right ) \log ^2\left (-e^4+x\right )} \, dx=\frac {x e^{\left (x + 21\right )}}{{\left ({\left (x - 6\right )} e^{\left (x + 21\right )} - 5\right )} \log \left (x - e^{4}\right ) + 3 \, e^{\left (x + 21\right )}} \]
integrate((((-6*exp(4)+6*x)*exp(x+21)^2+((-5*x-5)*exp(4)+5*x^2+5*x)*exp(x+ 21))*log(x-exp(4))+(3*exp(4)+x^2-9*x)*exp(x+21)^2-5*x*exp(x+21))/((((x^2-1 2*x+36)*exp(4)-x^3+12*x^2-36*x)*exp(x+21)^2+((-10*x+60)*exp(4)+10*x^2-60*x )*exp(x+21)+25*exp(4)-25*x)*log(x-exp(4))^2+(((6*x-36)*exp(4)-6*x^2+36*x)* exp(x+21)^2+(-30*exp(4)+30*x)*exp(x+21))*log(x-exp(4))+(9*exp(4)-9*x)*exp( x+21)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).
Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.73 \[ \int \frac {-5 e^{21+x} x+e^{42+2 x} \left (3 e^4-9 x+x^2\right )+\left (e^{42+2 x} \left (-6 e^4+6 x\right )+e^{21+x} \left (e^4 (-5-5 x)+5 x+5 x^2\right )\right ) \log \left (-e^4+x\right )}{e^{42+2 x} \left (9 e^4-9 x\right )+\left (e^{21+x} \left (-30 e^4+30 x\right )+e^{42+2 x} \left (36 x-6 x^2+e^4 (-36+6 x)\right )\right ) \log \left (-e^4+x\right )+\left (25 e^4-25 x+e^{21+x} \left (e^4 (60-10 x)-60 x+10 x^2\right )+e^{42+2 x} \left (-36 x+12 x^2-x^3+e^4 \left (36-12 x+x^2\right )\right )\right ) \log ^2\left (-e^4+x\right )} \, dx=\frac {5 x \log {\left (x - e^{4} \right )}}{- 5 x \log {\left (x - e^{4} \right )}^{2} + \left (x^{2} \log {\left (x - e^{4} \right )}^{2} - 12 x \log {\left (x - e^{4} \right )}^{2} + 6 x \log {\left (x - e^{4} \right )} + 36 \log {\left (x - e^{4} \right )}^{2} - 36 \log {\left (x - e^{4} \right )} + 9\right ) e^{x + 21} + 30 \log {\left (x - e^{4} \right )}^{2} - 15 \log {\left (x - e^{4} \right )}} + \frac {x}{\left (x - 6\right ) \log {\left (x - e^{4} \right )} + 3} \]
integrate((((-6*exp(4)+6*x)*exp(x+21)**2+((-5*x-5)*exp(4)+5*x**2+5*x)*exp( x+21))*ln(x-exp(4))+(3*exp(4)+x**2-9*x)*exp(x+21)**2-5*x*exp(x+21))/((((x* *2-12*x+36)*exp(4)-x**3+12*x**2-36*x)*exp(x+21)**2+((-10*x+60)*exp(4)+10*x **2-60*x)*exp(x+21)+25*exp(4)-25*x)*ln(x-exp(4))**2+(((6*x-36)*exp(4)-6*x* *2+36*x)*exp(x+21)**2+(-30*exp(4)+30*x)*exp(x+21))*ln(x-exp(4))+(9*exp(4)- 9*x)*exp(x+21)**2),x)
5*x*log(x - exp(4))/(-5*x*log(x - exp(4))**2 + (x**2*log(x - exp(4))**2 - 12*x*log(x - exp(4))**2 + 6*x*log(x - exp(4)) + 36*log(x - exp(4))**2 - 36 *log(x - exp(4)) + 9)*exp(x + 21) + 30*log(x - exp(4))**2 - 15*log(x - exp (4))) + x/((x - 6)*log(x - exp(4)) + 3)
Time = 0.33 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-5 e^{21+x} x+e^{42+2 x} \left (3 e^4-9 x+x^2\right )+\left (e^{42+2 x} \left (-6 e^4+6 x\right )+e^{21+x} \left (e^4 (-5-5 x)+5 x+5 x^2\right )\right ) \log \left (-e^4+x\right )}{e^{42+2 x} \left (9 e^4-9 x\right )+\left (e^{21+x} \left (-30 e^4+30 x\right )+e^{42+2 x} \left (36 x-6 x^2+e^4 (-36+6 x)\right )\right ) \log \left (-e^4+x\right )+\left (25 e^4-25 x+e^{21+x} \left (e^4 (60-10 x)-60 x+10 x^2\right )+e^{42+2 x} \left (-36 x+12 x^2-x^3+e^4 \left (36-12 x+x^2\right )\right )\right ) \log ^2\left (-e^4+x\right )} \, dx=\frac {x e^{\left (x + 21\right )}}{{\left ({\left (x e^{21} - 6 \, e^{21}\right )} e^{x} - 5\right )} \log \left (x - e^{4}\right ) + 3 \, e^{\left (x + 21\right )}} \]
integrate((((-6*exp(4)+6*x)*exp(x+21)^2+((-5*x-5)*exp(4)+5*x^2+5*x)*exp(x+ 21))*log(x-exp(4))+(3*exp(4)+x^2-9*x)*exp(x+21)^2-5*x*exp(x+21))/((((x^2-1 2*x+36)*exp(4)-x^3+12*x^2-36*x)*exp(x+21)^2+((-10*x+60)*exp(4)+10*x^2-60*x )*exp(x+21)+25*exp(4)-25*x)*log(x-exp(4))^2+(((6*x-36)*exp(4)-6*x^2+36*x)* exp(x+21)^2+(-30*exp(4)+30*x)*exp(x+21))*log(x-exp(4))+(9*exp(4)-9*x)*exp( x+21)^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (25) = 50\).
Time = 0.50 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.07 \[ \int \frac {-5 e^{21+x} x+e^{42+2 x} \left (3 e^4-9 x+x^2\right )+\left (e^{42+2 x} \left (-6 e^4+6 x\right )+e^{21+x} \left (e^4 (-5-5 x)+5 x+5 x^2\right )\right ) \log \left (-e^4+x\right )}{e^{42+2 x} \left (9 e^4-9 x\right )+\left (e^{21+x} \left (-30 e^4+30 x\right )+e^{42+2 x} \left (36 x-6 x^2+e^4 (-36+6 x)\right )\right ) \log \left (-e^4+x\right )+\left (25 e^4-25 x+e^{21+x} \left (e^4 (60-10 x)-60 x+10 x^2\right )+e^{42+2 x} \left (-36 x+12 x^2-x^3+e^4 \left (36-12 x+x^2\right )\right )\right ) \log ^2\left (-e^4+x\right )} \, dx=\frac {{\left (x + 21\right )} e^{\left (x + 21\right )} - 21 \, e^{\left (x + 21\right )}}{{\left (x + 21\right )} e^{\left (x + 21\right )} \log \left (x - e^{4}\right ) - 27 \, e^{\left (x + 21\right )} \log \left (x - e^{4}\right ) + 3 \, e^{\left (x + 21\right )} - 5 \, \log \left (x - e^{4}\right )} \]
integrate((((-6*exp(4)+6*x)*exp(x+21)^2+((-5*x-5)*exp(4)+5*x^2+5*x)*exp(x+ 21))*log(x-exp(4))+(3*exp(4)+x^2-9*x)*exp(x+21)^2-5*x*exp(x+21))/((((x^2-1 2*x+36)*exp(4)-x^3+12*x^2-36*x)*exp(x+21)^2+((-10*x+60)*exp(4)+10*x^2-60*x )*exp(x+21)+25*exp(4)-25*x)*log(x-exp(4))^2+(((6*x-36)*exp(4)-6*x^2+36*x)* exp(x+21)^2+(-30*exp(4)+30*x)*exp(x+21))*log(x-exp(4))+(9*exp(4)-9*x)*exp( x+21)^2),x, algorithm=\
((x + 21)*e^(x + 21) - 21*e^(x + 21))/((x + 21)*e^(x + 21)*log(x - e^4) - 27*e^(x + 21)*log(x - e^4) + 3*e^(x + 21) - 5*log(x - e^4))
Time = 12.63 (sec) , antiderivative size = 285, normalized size of antiderivative = 9.50 \[ \int \frac {-5 e^{21+x} x+e^{42+2 x} \left (3 e^4-9 x+x^2\right )+\left (e^{42+2 x} \left (-6 e^4+6 x\right )+e^{21+x} \left (e^4 (-5-5 x)+5 x+5 x^2\right )\right ) \log \left (-e^4+x\right )}{e^{42+2 x} \left (9 e^4-9 x\right )+\left (e^{21+x} \left (-30 e^4+30 x\right )+e^{42+2 x} \left (36 x-6 x^2+e^4 (-36+6 x)\right )\right ) \log \left (-e^4+x\right )+\left (25 e^4-25 x+e^{21+x} \left (e^4 (60-10 x)-60 x+10 x^2\right )+e^{42+2 x} \left (-36 x+12 x^2-x^3+e^4 \left (36-12 x+x^2\right )\right )\right ) \log ^2\left (-e^4+x\right )} \, dx=\frac {25\,x\,{\mathrm {e}}^{x+25}+60\,x\,{\mathrm {e}}^{2\,x+46}+15\,x\,{\mathrm {e}}^{2\,x+50}+36\,x\,{\mathrm {e}}^{3\,x+67}+3\,x\,{\mathrm {e}}^{3\,x+71}-25\,x^2\,{\mathrm {e}}^{x+21}-60\,x^2\,{\mathrm {e}}^{2\,x+42}+25\,x^3\,{\mathrm {e}}^{2\,x+42}-40\,x^2\,{\mathrm {e}}^{2\,x+46}-36\,x^2\,{\mathrm {e}}^{3\,x+63}+15\,x^3\,{\mathrm {e}}^{3\,x+63}-x^4\,{\mathrm {e}}^{3\,x+63}-18\,x^2\,{\mathrm {e}}^{3\,x+67}+x^3\,{\mathrm {e}}^{3\,x+67}}{\left (3\,{\mathrm {e}}^{x+21}-\ln \left (x-{\mathrm {e}}^4\right )\,\left (6\,{\mathrm {e}}^{x+21}-x\,{\mathrm {e}}^{x+21}+5\right )\right )\,\left (60\,{\mathrm {e}}^{x+25}-25\,x+15\,{\mathrm {e}}^{x+29}+25\,{\mathrm {e}}^4+36\,{\mathrm {e}}^{2\,x+46}+3\,{\mathrm {e}}^{2\,x+50}-60\,x\,{\mathrm {e}}^{x+21}-40\,x\,{\mathrm {e}}^{x+25}-36\,x\,{\mathrm {e}}^{2\,x+42}-18\,x\,{\mathrm {e}}^{2\,x+46}+25\,x^2\,{\mathrm {e}}^{x+21}+15\,x^2\,{\mathrm {e}}^{2\,x+42}-x^3\,{\mathrm {e}}^{2\,x+42}+x^2\,{\mathrm {e}}^{2\,x+46}\right )} \]
int(-(log(x - exp(4))*(exp(x + 21)*(5*x + 5*x^2 - exp(4)*(5*x + 5)) + exp( 2*x + 42)*(6*x - 6*exp(4))) - 5*x*exp(x + 21) + exp(2*x + 42)*(3*exp(4) - 9*x + x^2))/(log(x - exp(4))^2*(25*x - 25*exp(4) + exp(x + 21)*(60*x - 10* x^2 + exp(4)*(10*x - 60)) + exp(2*x + 42)*(36*x - exp(4)*(x^2 - 12*x + 36) - 12*x^2 + x^3)) - log(x - exp(4))*(exp(2*x + 42)*(36*x - 6*x^2 + exp(4)* (6*x - 36)) + exp(x + 21)*(30*x - 30*exp(4))) + exp(2*x + 42)*(9*x - 9*exp (4))),x)
(25*x*exp(x + 25) + 60*x*exp(2*x + 46) + 15*x*exp(2*x + 50) + 36*x*exp(3*x + 67) + 3*x*exp(3*x + 71) - 25*x^2*exp(x + 21) - 60*x^2*exp(2*x + 42) + 2 5*x^3*exp(2*x + 42) - 40*x^2*exp(2*x + 46) - 36*x^2*exp(3*x + 63) + 15*x^3 *exp(3*x + 63) - x^4*exp(3*x + 63) - 18*x^2*exp(3*x + 67) + x^3*exp(3*x + 67))/((3*exp(x + 21) - log(x - exp(4))*(6*exp(x + 21) - x*exp(x + 21) + 5) )*(60*exp(x + 25) - 25*x + 15*exp(x + 29) + 25*exp(4) + 36*exp(2*x + 46) + 3*exp(2*x + 50) - 60*x*exp(x + 21) - 40*x*exp(x + 25) - 36*x*exp(2*x + 42 ) - 18*x*exp(2*x + 46) + 25*x^2*exp(x + 21) + 15*x^2*exp(2*x + 42) - x^3*e xp(2*x + 42) + x^2*exp(2*x + 46)))