Integrand size = 333, antiderivative size = 34 \[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\frac {e^{\left (3-3 \left (-e^{4 x}+x+x^2\right )\right )^2}}{1+\log \left (3-e^x+x\right )} \] Output:
exp((3-3*x-3*x^2+3*exp(4*x))^2)/(1+ln(-exp(x)+3+x))
Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\frac {e^{9 \left (-1-e^{4 x}+x+x^2\right )^2}}{1+\log \left (3-e^x+x\right )} \] Input:
Integrate[(E^(9 + 9*E^(8*x) - 18*x - 9*x^2 + 18*x^3 + 9*x^4 + E^(4*x)*(18 - 18*x - 18*x^2))*(55 + E^(8*x)*(-216 + 72*E^x - 72*x) + 72*x - 144*x^2 - 162*x^3 - 36*x^4 + E^x*(-19 - 18*x + 54*x^2 + 36*x^3) + E^(4*x)*(-162 + 27 0*x + 324*x^2 + 72*x^3 + E^x*(54 - 108*x - 72*x^2))) + E^(9 + 9*E^(8*x) - 18*x - 9*x^2 + 18*x^3 + 9*x^4 + E^(4*x)*(18 - 18*x - 18*x^2))*(54 + E^(8*x )*(-216 + 72*E^x - 72*x) + 72*x - 144*x^2 - 162*x^3 - 36*x^4 + E^x*(-18 - 18*x + 54*x^2 + 36*x^3) + E^(4*x)*(-162 + 270*x + 324*x^2 + 72*x^3 + E^x*( 54 - 108*x - 72*x^2)))*Log[3 - E^x + x])/(-3 + E^x - x + (-6 + 2*E^x - 2*x )*Log[3 - E^x + x] + (-3 + E^x - x)*Log[3 - E^x + x]^2),x]
Output:
E^(9*(-1 - E^(4*x) + x + x^2)^2)/(1 + Log[3 - E^x + x])
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-36 x^4-162 x^3-144 x^2+e^x \left (36 x^3+54 x^2-18 x-19\right )+e^{4 x} \left (72 x^3+324 x^2+e^x \left (-72 x^2-108 x+54\right )+270 x-162\right )+72 x+e^{8 x} \left (-72 x+72 e^x-216\right )+55\right ) \exp \left (9 x^4+18 x^3-9 x^2+e^{4 x} \left (-18 x^2-18 x+18\right )-18 x+9 e^{8 x}+9\right )+\left (-36 x^4-162 x^3-144 x^2+e^x \left (36 x^3+54 x^2-18 x-18\right )+e^{4 x} \left (72 x^3+324 x^2+e^x \left (-72 x^2-108 x+54\right )+270 x-162\right )+72 x+e^{8 x} \left (-72 x+72 e^x-216\right )+54\right ) \exp \left (9 x^4+18 x^3-9 x^2+e^{4 x} \left (-18 x^2-18 x+18\right )-18 x+9 e^{8 x}+9\right ) \log \left (x-e^x+3\right )}{e^x-x+\left (-x+e^x-3\right ) \log ^2\left (x-e^x+3\right )+\left (-2 x+2 e^x-6\right ) \log \left (x-e^x+3\right )-3} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-36 x^4-162 x^3-144 x^2+e^x \left (36 x^3+54 x^2-18 x-19\right )+e^{4 x} \left (72 x^3+324 x^2+e^x \left (-72 x^2-108 x+54\right )+270 x-162\right )+72 x+e^{8 x} \left (-72 x+72 e^x-216\right )+55\right ) \left (-\exp \left (9 x^4+18 x^3-9 x^2+e^{4 x} \left (-18 x^2-18 x+18\right )-18 x+9 e^{8 x}+9\right )\right )-\left (-36 x^4-162 x^3-144 x^2+e^x \left (36 x^3+54 x^2-18 x-18\right )+e^{4 x} \left (72 x^3+324 x^2+e^x \left (-72 x^2-108 x+54\right )+270 x-162\right )+72 x+e^{8 x} \left (-72 x+72 e^x-216\right )+54\right ) \exp \left (9 x^4+18 x^3-9 x^2+e^{4 x} \left (-18 x^2-18 x+18\right )-18 x+9 e^{8 x}+9\right ) \log \left (x-e^x+3\right )}{\left (x-e^x+3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {36 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \log \left (x-e^x+3\right ) x^4}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {36 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} x^4}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {162 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \log \left (x-e^x+3\right ) x^3}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {36 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x^3}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x^3}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {162 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} x^3}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {36 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x^3}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x^3}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {144 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \log \left (x-e^x+3\right ) x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {54 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {324 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {72 e^{9 x^4+18 x^3-9 x^2-13 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {144 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {54 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {324 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {72 e^{9 x^4+18 x^3-9 x^2-13 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x^2}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \log \left (x-e^x+3\right ) x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {18 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {270 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {108 e^{9 x^4+18 x^3-9 x^2-13 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {72 e^{9 x^4+18 x^3-9 x^2-10 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right ) x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {18 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {270 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {108 e^{9 x^4+18 x^3-9 x^2-13 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {72 e^{9 x^4+18 x^3-9 x^2-10 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} x}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {54 e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \log \left (x-e^x+3\right )}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {18 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right )}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {162 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right )}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {54 e^{9 x^4+18 x^3-9 x^2-13 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right )}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {216 e^{9 x^4+18 x^3-9 x^2-10 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right )}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 x^4+18 x^3-9 x^2-9 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9} \log \left (x-e^x+3\right )}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {55 e^{9 \left (x^2+x-e^{4 x}-1\right )^2}}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {19 e^{9 x^4+18 x^3-9 x^2-17 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9}}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {162 e^{9 x^4+18 x^3-9 x^2-14 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9}}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {54 e^{9 x^4+18 x^3-9 x^2-13 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9}}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}-\frac {216 e^{9 x^4+18 x^3-9 x^2-10 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9}}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 x^4+18 x^3-9 x^2-9 x+9 e^{8 x}-18 e^{4 x} \left (x^2+x-1\right )+9}}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \left (36 x^4+162 x^3+144 x^2+18 e^{5 x} \left (4 x^2+6 x-3\right )-18 e^{4 x} \left (4 x^3+18 x^2+15 x-9\right )-e^x \left (36 x^3+54 x^2-18 x-19\right )-18 \left (-2 x^4-9 x^3-8 x^2+e^{5 x} \left (-4 x^2-6 x+3\right )+e^x \left (2 x^3+3 x^2-x-1\right )+e^{4 x} \left (4 x^3+18 x^2+15 x-9\right )+4 x+4 e^{9 x}-4 e^{8 x} (x+3)+3\right ) \log \left (x-e^x+3\right )-72 x-72 e^{9 x}+72 e^{8 x} (x+3)-55\right )}{\left (x-e^x+3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{9 \left (x^2+x-e^{4 x}-1\right )^2} (x+2)}{\left (x-e^x+3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 \left (x^2+x-e^{4 x}-1\right )^2+8 x}}{\log \left (x-e^x+3\right )+1}-\frac {18 e^{9 \left (x^2+x-e^{4 x}-1\right )^2+4 x} \left (4 x^2+6 x-3\right )}{\log \left (x-e^x+3\right )+1}+\frac {e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \left (36 x^3+36 x^3 \log \left (x-e^x+3\right )+54 x^2+54 x^2 \log \left (x-e^x+3\right )-18 x-18 x \log \left (x-e^x+3\right )-18 \log \left (x-e^x+3\right )-19\right )}{\left (\log \left (x-e^x+3\right )+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^{9 \left (x^2+x-e^{4 x}-1\right )^2} (x+2)}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 \left (x^2+x-e^{4 x}-1\right )^2+8 x}}{\log \left (x-e^x+3\right )+1}-\frac {18 e^{9 \left (x^2+x-e^{4 x}-1\right )^2+4 x} \left (4 x^2+6 x-3\right )}{\log \left (x-e^x+3\right )+1}+\frac {e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \left (36 x^3+36 x^3 \log \left (x-e^x+3\right )+54 x^2+54 x^2 \log \left (x-e^x+3\right )-18 x-18 x \log \left (x-e^x+3\right )-18 \log \left (x-e^x+3\right )-19\right )}{\left (\log \left (x-e^x+3\right )+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (-\frac {e^{9 \left (x^2+x-e^{4 x}-1\right )^2} (x+2)}{\left (-x+e^x-3\right ) \left (\log \left (x-e^x+3\right )+1\right )^2}+\frac {72 e^{9 \left (x^2+x-e^{4 x}-1\right )^2+8 x}}{\log \left (x-e^x+3\right )+1}-\frac {18 e^{9 \left (x^2+x-e^{4 x}-1\right )^2+4 x} \left (4 x^2+6 x-3\right )}{\log \left (x-e^x+3\right )+1}+\frac {e^{9 \left (x^2+x-e^{4 x}-1\right )^2} \left (36 x^3+36 x^3 \log \left (x-e^x+3\right )+54 x^2+54 x^2 \log \left (x-e^x+3\right )-18 x-18 x \log \left (x-e^x+3\right )-18 \log \left (x-e^x+3\right )-19\right )}{\left (\log \left (x-e^x+3\right )+1\right )^2}\right )dx\) |
Input:
Int[(E^(9 + 9*E^(8*x) - 18*x - 9*x^2 + 18*x^3 + 9*x^4 + E^(4*x)*(18 - 18*x - 18*x^2))*(55 + E^(8*x)*(-216 + 72*E^x - 72*x) + 72*x - 144*x^2 - 162*x^ 3 - 36*x^4 + E^x*(-19 - 18*x + 54*x^2 + 36*x^3) + E^(4*x)*(-162 + 270*x + 324*x^2 + 72*x^3 + E^x*(54 - 108*x - 72*x^2))) + E^(9 + 9*E^(8*x) - 18*x - 9*x^2 + 18*x^3 + 9*x^4 + E^(4*x)*(18 - 18*x - 18*x^2))*(54 + E^(8*x)*(-21 6 + 72*E^x - 72*x) + 72*x - 144*x^2 - 162*x^3 - 36*x^4 + E^x*(-18 - 18*x + 54*x^2 + 36*x^3) + E^(4*x)*(-162 + 270*x + 324*x^2 + 72*x^3 + E^x*(54 - 1 08*x - 72*x^2)))*Log[3 - E^x + x])/(-3 + E^x - x + (-6 + 2*E^x - 2*x)*Log[ 3 - E^x + x] + (-3 + E^x - x)*Log[3 - E^x + x]^2),x]
Output:
$Aborted
Time = 28.96 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.71
| method | result | size |
| parallelrisch | \(\frac {{\mathrm e}^{9 \,{\mathrm e}^{8 x}+\left (-18 x^{2}-18 x +18\right ) {\mathrm e}^{4 x}+9 x^{4}+18 x^{3}-9 x^{2}-18 x +9}}{1+\ln \left (-{\mathrm e}^{x}+3+x \right )}\) | \(58\) |
| risch | \(\frac {{\mathrm e}^{9 x^{4}-18 \,{\mathrm e}^{4 x} x^{2}+18 x^{3}-18 x \,{\mathrm e}^{4 x}-9 x^{2}+9 \,{\mathrm e}^{8 x}+18 \,{\mathrm e}^{4 x}-18 x +9}}{1+\ln \left (-{\mathrm e}^{x}+3+x \right )}\) | \(63\) |
Input:
int((((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-108*x+54)*exp(x)+72*x^3+32 4*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-18)*exp(x)-36*x^4-162*x^3-14 4*x^2+72*x+54)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*exp(4*x)+9*x^4+18*x^3-9* x^2-18*x+9)*ln(-exp(x)+3+x)+((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-108 *x+54)*exp(x)+72*x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-19)*e xp(x)-36*x^4-162*x^3-144*x^2+72*x+55)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*e xp(4*x)+9*x^4+18*x^3-9*x^2-18*x+9))/((exp(x)-3-x)*ln(-exp(x)+3+x)^2+(2*exp (x)-2*x-6)*ln(-exp(x)+3+x)+exp(x)-3-x),x,method=_RETURNVERBOSE)
Output:
exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*exp(4*x)+9*x^4+18*x^3-9*x^2-18*x+9)/(1+ ln(-exp(x)+3+x))
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\frac {e^{\left (9 \, x^{4} + 18 \, x^{3} - 9 \, x^{2} - 18 \, {\left (x^{2} + x - 1\right )} e^{\left (4 \, x\right )} - 18 \, x + 9 \, e^{\left (8 \, x\right )} + 9\right )}}{\log \left (x - e^{x} + 3\right ) + 1} \] Input:
integrate((((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-108*x+54)*exp(x)+72* x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-18)*exp(x)-36*x^4-162* x^3-144*x^2+72*x+54)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*exp(4*x)+9*x^4+18* x^3-9*x^2-18*x+9)*log(-exp(x)+3+x)+((72*exp(x)-72*x-216)*exp(4*x)^2+((-72* x^2-108*x+54)*exp(x)+72*x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18* x-19)*exp(x)-36*x^4-162*x^3-144*x^2+72*x+55)*exp(9*exp(4*x)^2+(-18*x^2-18* x+18)*exp(4*x)+9*x^4+18*x^3-9*x^2-18*x+9))/((exp(x)-3-x)*log(-exp(x)+3+x)^ 2+(2*exp(x)-2*x-6)*log(-exp(x)+3+x)+exp(x)-3-x),x, algorithm="fricas")
Output:
e^(9*x^4 + 18*x^3 - 9*x^2 - 18*(x^2 + x - 1)*e^(4*x) - 18*x + 9*e^(8*x) + 9)/(log(x - e^x + 3) + 1)
Time = 1.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.56 \[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\frac {e^{9 x^{4} + 18 x^{3} - 9 x^{2} - 18 x + \left (- 18 x^{2} - 18 x + 18\right ) e^{4 x} + 9 e^{8 x} + 9}}{\log {\left (x - e^{x} + 3 \right )} + 1} \] Input:
integrate((((72*exp(x)-72*x-216)*exp(4*x)**2+((-72*x**2-108*x+54)*exp(x)+7 2*x**3+324*x**2+270*x-162)*exp(4*x)+(36*x**3+54*x**2-18*x-18)*exp(x)-36*x* *4-162*x**3-144*x**2+72*x+54)*exp(9*exp(4*x)**2+(-18*x**2-18*x+18)*exp(4*x )+9*x**4+18*x**3-9*x**2-18*x+9)*ln(-exp(x)+3+x)+((72*exp(x)-72*x-216)*exp( 4*x)**2+((-72*x**2-108*x+54)*exp(x)+72*x**3+324*x**2+270*x-162)*exp(4*x)+( 36*x**3+54*x**2-18*x-19)*exp(x)-36*x**4-162*x**3-144*x**2+72*x+55)*exp(9*e xp(4*x)**2+(-18*x**2-18*x+18)*exp(4*x)+9*x**4+18*x**3-9*x**2-18*x+9))/((ex p(x)-3-x)*ln(-exp(x)+3+x)**2+(2*exp(x)-2*x-6)*ln(-exp(x)+3+x)+exp(x)-3-x), x)
Output:
exp(9*x**4 + 18*x**3 - 9*x**2 - 18*x + (-18*x**2 - 18*x + 18)*exp(4*x) + 9 *exp(8*x) + 9)/(log(x - exp(x) + 3) + 1)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (30) = 60\).
Time = 0.54 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.38 \[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\frac {e^{\left (9 \, x^{4} + 18 \, x^{3} - 18 \, x^{2} e^{\left (4 \, x\right )} + 9 \, e^{\left (8 \, x\right )} + 18 \, e^{\left (4 \, x\right )} + 9\right )}}{e^{\left (9 \, x^{2} + 18 \, x e^{\left (4 \, x\right )} + 18 \, x\right )} \log \left (x - e^{x} + 3\right ) + e^{\left (9 \, x^{2} + 18 \, x e^{\left (4 \, x\right )} + 18 \, x\right )}} \] Input:
integrate((((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-108*x+54)*exp(x)+72* x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-18)*exp(x)-36*x^4-162* x^3-144*x^2+72*x+54)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*exp(4*x)+9*x^4+18* x^3-9*x^2-18*x+9)*log(-exp(x)+3+x)+((72*exp(x)-72*x-216)*exp(4*x)^2+((-72* x^2-108*x+54)*exp(x)+72*x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18* x-19)*exp(x)-36*x^4-162*x^3-144*x^2+72*x+55)*exp(9*exp(4*x)^2+(-18*x^2-18* x+18)*exp(4*x)+9*x^4+18*x^3-9*x^2-18*x+9))/((exp(x)-3-x)*log(-exp(x)+3+x)^ 2+(2*exp(x)-2*x-6)*log(-exp(x)+3+x)+exp(x)-3-x),x, algorithm="maxima")
Output:
e^(9*x^4 + 18*x^3 - 18*x^2*e^(4*x) + 9*e^(8*x) + 18*e^(4*x) + 9)/(e^(9*x^2 + 18*x*e^(4*x) + 18*x)*log(x - e^x + 3) + e^(9*x^2 + 18*x*e^(4*x) + 18*x) )
\[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\int { \frac {18 \, {\left (2 \, x^{4} + 9 \, x^{3} + 8 \, x^{2} + 4 \, {\left (x - e^{x} + 3\right )} e^{\left (8 \, x\right )} - {\left (4 \, x^{3} + 18 \, x^{2} - {\left (4 \, x^{2} + 6 \, x - 3\right )} e^{x} + 15 \, x - 9\right )} e^{\left (4 \, x\right )} - {\left (2 \, x^{3} + 3 \, x^{2} - x - 1\right )} e^{x} - 4 \, x - 3\right )} e^{\left (9 \, x^{4} + 18 \, x^{3} - 9 \, x^{2} - 18 \, {\left (x^{2} + x - 1\right )} e^{\left (4 \, x\right )} - 18 \, x + 9 \, e^{\left (8 \, x\right )} + 9\right )} \log \left (x - e^{x} + 3\right ) + {\left (36 \, x^{4} + 162 \, x^{3} + 144 \, x^{2} + 72 \, {\left (x - e^{x} + 3\right )} e^{\left (8 \, x\right )} - 18 \, {\left (4 \, x^{3} + 18 \, x^{2} - {\left (4 \, x^{2} + 6 \, x - 3\right )} e^{x} + 15 \, x - 9\right )} e^{\left (4 \, x\right )} - {\left (36 \, x^{3} + 54 \, x^{2} - 18 \, x - 19\right )} e^{x} - 72 \, x - 55\right )} e^{\left (9 \, x^{4} + 18 \, x^{3} - 9 \, x^{2} - 18 \, {\left (x^{2} + x - 1\right )} e^{\left (4 \, x\right )} - 18 \, x + 9 \, e^{\left (8 \, x\right )} + 9\right )}}{{\left (x - e^{x} + 3\right )} \log \left (x - e^{x} + 3\right )^{2} + 2 \, {\left (x - e^{x} + 3\right )} \log \left (x - e^{x} + 3\right ) + x - e^{x} + 3} \,d x } \] Input:
integrate((((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-108*x+54)*exp(x)+72* x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-18)*exp(x)-36*x^4-162* x^3-144*x^2+72*x+54)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*exp(4*x)+9*x^4+18* x^3-9*x^2-18*x+9)*log(-exp(x)+3+x)+((72*exp(x)-72*x-216)*exp(4*x)^2+((-72* x^2-108*x+54)*exp(x)+72*x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18* x-19)*exp(x)-36*x^4-162*x^3-144*x^2+72*x+55)*exp(9*exp(4*x)^2+(-18*x^2-18* x+18)*exp(4*x)+9*x^4+18*x^3-9*x^2-18*x+9))/((exp(x)-3-x)*log(-exp(x)+3+x)^ 2+(2*exp(x)-2*x-6)*log(-exp(x)+3+x)+exp(x)-3-x),x, algorithm="giac")
Output:
integrate((18*(2*x^4 + 9*x^3 + 8*x^2 + 4*(x - e^x + 3)*e^(8*x) - (4*x^3 + 18*x^2 - (4*x^2 + 6*x - 3)*e^x + 15*x - 9)*e^(4*x) - (2*x^3 + 3*x^2 - x - 1)*e^x - 4*x - 3)*e^(9*x^4 + 18*x^3 - 9*x^2 - 18*(x^2 + x - 1)*e^(4*x) - 1 8*x + 9*e^(8*x) + 9)*log(x - e^x + 3) + (36*x^4 + 162*x^3 + 144*x^2 + 72*( x - e^x + 3)*e^(8*x) - 18*(4*x^3 + 18*x^2 - (4*x^2 + 6*x - 3)*e^x + 15*x - 9)*e^(4*x) - (36*x^3 + 54*x^2 - 18*x - 19)*e^x - 72*x - 55)*e^(9*x^4 + 18 *x^3 - 9*x^2 - 18*(x^2 + x - 1)*e^(4*x) - 18*x + 9*e^(8*x) + 9))/((x - e^x + 3)*log(x - e^x + 3)^2 + 2*(x - e^x + 3)*log(x - e^x + 3) + x - e^x + 3) , x)
Time = 0.97 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.03 \[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\frac {{\mathrm {e}}^{9\,{\mathrm {e}}^{8\,x}}\,{\mathrm {e}}^{18\,{\mathrm {e}}^{4\,x}}\,{\mathrm {e}}^{-18\,x}\,{\mathrm {e}}^9\,{\mathrm {e}}^{-18\,x\,{\mathrm {e}}^{4\,x}}\,{\mathrm {e}}^{-9\,x^2}\,{\mathrm {e}}^{9\,x^4}\,{\mathrm {e}}^{18\,x^3}\,{\mathrm {e}}^{-18\,x^2\,{\mathrm {e}}^{4\,x}}}{\ln \left (x-{\mathrm {e}}^x+3\right )+1} \] Input:
int((exp(9*exp(8*x) - 18*x - exp(4*x)*(18*x + 18*x^2 - 18) - 9*x^2 + 18*x^ 3 + 9*x^4 + 9)*(exp(8*x)*(72*x - 72*exp(x) + 216) - 72*x + 144*x^2 + 162*x ^3 + 36*x^4 + exp(x)*(18*x - 54*x^2 - 36*x^3 + 19) - exp(4*x)*(270*x - exp (x)*(108*x + 72*x^2 - 54) + 324*x^2 + 72*x^3 - 162) - 55) + exp(9*exp(8*x) - 18*x - exp(4*x)*(18*x + 18*x^2 - 18) - 9*x^2 + 18*x^3 + 9*x^4 + 9)*log( x - exp(x) + 3)*(exp(8*x)*(72*x - 72*exp(x) + 216) - 72*x + 144*x^2 + 162* x^3 + 36*x^4 + exp(x)*(18*x - 54*x^2 - 36*x^3 + 18) - exp(4*x)*(270*x - ex p(x)*(108*x + 72*x^2 - 54) + 324*x^2 + 72*x^3 - 162) - 54))/(x - exp(x) + log(x - exp(x) + 3)*(2*x - 2*exp(x) + 6) + log(x - exp(x) + 3)^2*(x - exp( x) + 3) + 3),x)
Output:
(exp(9*exp(8*x))*exp(18*exp(4*x))*exp(-18*x)*exp(9)*exp(-18*x*exp(4*x))*ex p(-9*x^2)*exp(9*x^4)*exp(18*x^3)*exp(-18*x^2*exp(4*x)))/(log(x - exp(x) + 3) + 1)
\[ \int \frac {e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (55+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-19-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right )+e^{9+9 e^{8 x}-18 x-9 x^2+18 x^3+9 x^4+e^{4 x} \left (18-18 x-18 x^2\right )} \left (54+e^{8 x} \left (-216+72 e^x-72 x\right )+72 x-144 x^2-162 x^3-36 x^4+e^x \left (-18-18 x+54 x^2+36 x^3\right )+e^{4 x} \left (-162+270 x+324 x^2+72 x^3+e^x \left (54-108 x-72 x^2\right )\right )\right ) \log \left (3-e^x+x\right )}{-3+e^x-x+\left (-6+2 e^x-2 x\right ) \log \left (3-e^x+x\right )+\left (-3+e^x-x\right ) \log ^2\left (3-e^x+x\right )} \, dx=\int \frac {\left (\left (72 \,{\mathrm e}^{x}-72 x -216\right ) \left ({\mathrm e}^{4 x}\right )^{2}+\left (\left (-72 x^{2}-108 x +54\right ) {\mathrm e}^{x}+72 x^{3}+324 x^{2}+270 x -162\right ) {\mathrm e}^{4 x}+\left (36 x^{3}+54 x^{2}-18 x -18\right ) {\mathrm e}^{x}-36 x^{4}-162 x^{3}-144 x^{2}+72 x +54\right ) {\mathrm e}^{9 \left ({\mathrm e}^{4 x}\right )^{2}+\left (-18 x^{2}-18 x +18\right ) {\mathrm e}^{4 x}+9 x^{4}+18 x^{3}-9 x^{2}-18 x +9} \mathrm {log}\left (-{\mathrm e}^{x}+3+x \right )+\left (\left (72 \,{\mathrm e}^{x}-72 x -216\right ) \left ({\mathrm e}^{4 x}\right )^{2}+\left (\left (-72 x^{2}-108 x +54\right ) {\mathrm e}^{x}+72 x^{3}+324 x^{2}+270 x -162\right ) {\mathrm e}^{4 x}+\left (36 x^{3}+54 x^{2}-18 x -19\right ) {\mathrm e}^{x}-36 x^{4}-162 x^{3}-144 x^{2}+72 x +55\right ) {\mathrm e}^{9 \left ({\mathrm e}^{4 x}\right )^{2}+\left (-18 x^{2}-18 x +18\right ) {\mathrm e}^{4 x}+9 x^{4}+18 x^{3}-9 x^{2}-18 x +9}}{\left ({\mathrm e}^{x}-3-x \right ) \mathrm {log}\left (-{\mathrm e}^{x}+3+x \right )^{2}+\left (2 \,{\mathrm e}^{x}-2 x -6\right ) \mathrm {log}\left (-{\mathrm e}^{x}+3+x \right )+{\mathrm e}^{x}-3-x}d x \] Input:
int((((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-108*x+54)*exp(x)+72*x^3+32 4*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-18)*exp(x)-36*x^4-162*x^3-14 4*x^2+72*x+54)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*exp(4*x)+9*x^4+18*x^3-9* x^2-18*x+9)*log(-exp(x)+3+x)+((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-10 8*x+54)*exp(x)+72*x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-19)* exp(x)-36*x^4-162*x^3-144*x^2+72*x+55)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)* exp(4*x)+9*x^4+18*x^3-9*x^2-18*x+9))/((exp(x)-3-x)*log(-exp(x)+3+x)^2+(2*e xp(x)-2*x-6)*log(-exp(x)+3+x)+exp(x)-3-x),x)
Output:
int((((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-108*x+54)*exp(x)+72*x^3+32 4*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-18)*exp(x)-36*x^4-162*x^3-14 4*x^2+72*x+54)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)*exp(4*x)+9*x^4+18*x^3-9* x^2-18*x+9)*log(-exp(x)+3+x)+((72*exp(x)-72*x-216)*exp(4*x)^2+((-72*x^2-10 8*x+54)*exp(x)+72*x^3+324*x^2+270*x-162)*exp(4*x)+(36*x^3+54*x^2-18*x-19)* exp(x)-36*x^4-162*x^3-144*x^2+72*x+55)*exp(9*exp(4*x)^2+(-18*x^2-18*x+18)* exp(4*x)+9*x^4+18*x^3-9*x^2-18*x+9))/((exp(x)-3-x)*log(-exp(x)+3+x)^2+(2*e xp(x)-2*x-6)*log(-exp(x)+3+x)+exp(x)-3-x),x)