Integrand size = 125, antiderivative size = 27 \[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\frac {2 e^{\frac {4}{2+e^{2 \left (-\frac {2}{9}+\log (x)\right )^2}}} x}{\log (2)} \] Output:
2*x*exp(4/(2+exp((ln(x)-2/9)^2)^2))/ln(2)
Time = 0.22 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\frac {2 e^{\frac {4 x^{8/9}}{e^{\frac {8}{81}+2 \log ^2(x)}+2 x^{8/9}}} x}{\log (2)} \] Input:
Integrate[(E^(4/(2 + E^((2*(4 - 36*Log[x] + 81*Log[x]^2))/81)))*(72 + 18*E ^((4*(4 - 36*Log[x] + 81*Log[x]^2))/81) + E^((2*(4 - 36*Log[x] + 81*Log[x] ^2))/81)*(136 - 288*Log[x])))/(36*Log[2] + 36*E^((2*(4 - 36*Log[x] + 81*Lo g[x]^2))/81)*Log[2] + 9*E^((4*(4 - 36*Log[x] + 81*Log[x]^2))/81)*Log[2]),x ]
Output:
(2*E^((4*x^(8/9))/(E^(8/81 + 2*Log[x]^2) + 2*x^(8/9)))*x)/Log[2]
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.
| \(\displaystyle \int \frac {e^{\frac {4}{e^{\frac {2}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )}+2}} \left (e^{\frac {2}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )} (136-288 \log (x))+18 e^{\frac {4}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )}+72\right )}{36 \log (2) e^{\frac {2}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )}+9 \log (2) e^{\frac {4}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )}+36 \log (2)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x^{16/9} e^{\frac {4}{e^{\frac {2}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )}+2}} \left (e^{\frac {2}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )} (136-288 \log (x))+18 e^{\frac {4}{81} \left (81 \log ^2(x)-36 \log (x)+4\right )}+72\right )}{9 \log (2) \left (2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 e^{\frac {4}{2+\frac {e^{\frac {2}{81} \left (81 \log ^2(x)+4\right )}}{x^{8/9}}}} x^{16/9} \left (\frac {4 e^{\frac {2}{81} \left (81 \log ^2(x)+4\right )} (17-36 \log (x))}{x^{8/9}}+\frac {9 e^{\frac {4}{81} \left (81 \log ^2(x)+4\right )}}{x^{16/9}}+36\right )}{\left (2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}\right )^2}dx}{9 \log (2)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {e^{\frac {4}{2+\frac {e^{\frac {2}{81} \left (81 \log ^2(x)+4\right )}}{x^{8/9}}}} x^{16/9} \left (\frac {4 e^{\frac {2}{81} \left (81 \log ^2(x)+4\right )} (17-36 \log (x))}{x^{8/9}}+\frac {9 e^{\frac {4}{81} \left (81 \log ^2(x)+4\right )}}{x^{16/9}}+36\right )}{\left (2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}\right )^2}dx}{9 \log (2)}\) |
| \(\Big \downarrow \) 7267 |
| \(\displaystyle \frac {2 \int \frac {e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} x^{8/3} \left (\frac {4 e^{2 \log ^2(x)+\frac {8}{81}} (17-36 \log (x))}{x^{8/9}}+\frac {9 e^{4 \log ^2(x)+\frac {16}{81}}}{x^{16/9}}+36\right )}{\left (2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}\right )^2}d\sqrt [9]{x}}{\log (2)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 \int \left (\frac {32 e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} (9 \log (x)-2) x^{8/3}}{\left (2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}\right )^2}-\frac {16 e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} (9 \log (x)-2) x^{16/9}}{2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}}+9 e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} x^{8/9}\right )d\sqrt [9]{x}}{\log (2)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (9 \int e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} x^{8/9}d\sqrt [9]{x}+32 \int \frac {e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} x^{16/9}}{2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}}d\sqrt [9]{x}-64 \int \frac {e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} x^{8/3}}{\left (2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}\right )^2}d\sqrt [9]{x}-144 \int \frac {e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} x^{16/9} \log (x)}{2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}}d\sqrt [9]{x}+288 \int \frac {e^{\frac {4}{2+\frac {e^{2 \log ^2(x)+\frac {8}{81}}}{x^{8/9}}}} x^{8/3} \log (x)}{\left (2 x^{8/9}+e^{2 \log ^2(x)+\frac {8}{81}}\right )^2}d\sqrt [9]{x}\right )}{\log (2)}\) |
Input:
Int[(E^(4/(2 + E^((2*(4 - 36*Log[x] + 81*Log[x]^2))/81)))*(72 + 18*E^((4*( 4 - 36*Log[x] + 81*Log[x]^2))/81) + E^((2*(4 - 36*Log[x] + 81*Log[x]^2))/8 1)*(136 - 288*Log[x])))/(36*Log[2] + 36*E^((2*(4 - 36*Log[x] + 81*Log[x]^2 ))/81)*Log[2] + 9*E^((4*(4 - 36*Log[x] + 81*Log[x]^2))/81)*Log[2]),x]
Output:
$Aborted
Time = 41.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
| method | result | size |
| parallelrisch | \(\frac {2 x \,{\mathrm e}^{\frac {4}{{\mathrm e}^{2 \ln \left (x \right )^{2}+\ln \left (\frac {1}{x^{\frac {8}{9}}}\right )+\frac {8}{81}}+2}}}{\ln \left (2\right )}\) | \(28\) |
| risch | \(\frac {2 x \,{\mathrm e}^{\frac {4 x^{\frac {8}{9}}}{2 x^{\frac {8}{9}}+{\mathrm e}^{2 \ln \left (x \right )^{2}+\frac {8}{81}}}}}{\ln \left (2\right )}\) | \(31\) |
Input:
int((18*exp(ln(x)^2-4/9*ln(x)+4/81)^4+(-288*ln(x)+136)*exp(ln(x)^2-4/9*ln( x)+4/81)^2+72)*exp(4/(exp(ln(x)^2-4/9*ln(x)+4/81)^2+2))/(9*ln(2)*exp(ln(x) ^2-4/9*ln(x)+4/81)^4+36*ln(2)*exp(ln(x)^2-4/9*ln(x)+4/81)^2+36*ln(2)),x,me thod=_RETURNVERBOSE)
Output:
2*x/ln(2)*exp(4/(exp(ln(x)^2-4/9*ln(x)+4/81)^2+2))
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\frac {2 \, x e^{\left (\frac {4}{e^{\left (2 \, \log \left (x\right )^{2} - \frac {8}{9} \, \log \left (x\right ) + \frac {8}{81}\right )} + 2}\right )}}{\log \left (2\right )} \] Input:
integrate((18*exp(log(x)^2-4/9*log(x)+4/81)^4+(-288*log(x)+136)*exp(log(x) ^2-4/9*log(x)+4/81)^2+72)*exp(4/(exp(log(x)^2-4/9*log(x)+4/81)^2+2))/(9*lo g(2)*exp(log(x)^2-4/9*log(x)+4/81)^4+36*log(2)*exp(log(x)^2-4/9*log(x)+4/8 1)^2+36*log(2)),x, algorithm="fricas")
Output:
2*x*e^(4/(e^(2*log(x)^2 - 8/9*log(x) + 8/81) + 2))/log(2)
Timed out. \[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\text {Timed out} \] Input:
integrate((18*exp(ln(x)**2-4/9*ln(x)+4/81)**4+(-288*ln(x)+136)*exp(ln(x)** 2-4/9*ln(x)+4/81)**2+72)*exp(4/(exp(ln(x)**2-4/9*ln(x)+4/81)**2+2))/(9*ln( 2)*exp(ln(x)**2-4/9*ln(x)+4/81)**4+36*ln(2)*exp(ln(x)**2-4/9*ln(x)+4/81)** 2+36*ln(2)),x)
Output:
Timed out
Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\frac {2 \, x e^{\left (\frac {4 \, x^{\frac {8}{9}}}{2 \, x^{\frac {8}{9}} + e^{\left (2 \, \log \left (x\right )^{2} + \frac {8}{81}\right )}}\right )}}{\log \left (2\right )} \] Input:
integrate((18*exp(log(x)^2-4/9*log(x)+4/81)^4+(-288*log(x)+136)*exp(log(x) ^2-4/9*log(x)+4/81)^2+72)*exp(4/(exp(log(x)^2-4/9*log(x)+4/81)^2+2))/(9*lo g(2)*exp(log(x)^2-4/9*log(x)+4/81)^4+36*log(2)*exp(log(x)^2-4/9*log(x)+4/8 1)^2+36*log(2)),x, algorithm="maxima")
Output:
2*x*e^(4*x^(8/9)/(2*x^(8/9) + e^(2*log(x)^2 + 8/81)))/log(2)
Timed out. \[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\text {Timed out} \] Input:
integrate((18*exp(log(x)^2-4/9*log(x)+4/81)^4+(-288*log(x)+136)*exp(log(x) ^2-4/9*log(x)+4/81)^2+72)*exp(4/(exp(log(x)^2-4/9*log(x)+4/81)^2+2))/(9*lo g(2)*exp(log(x)^2-4/9*log(x)+4/81)^4+36*log(2)*exp(log(x)^2-4/9*log(x)+4/8 1)^2+36*log(2)),x, algorithm="giac")
Output:
Timed out
Time = 4.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\frac {2\,x\,{\mathrm {e}}^{\frac {4}{\frac {{\mathrm {e}}^{2\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{8/81}}{x^{8/9}}+2}}}{\ln \left (2\right )} \] Input:
int((exp(4/(exp(2*log(x)^2 - (8*log(x))/9 + 8/81) + 2))*(18*exp(4*log(x)^2 - (16*log(x))/9 + 16/81) - exp(2*log(x)^2 - (8*log(x))/9 + 8/81)*(288*log (x) - 136) + 72))/(36*log(2) + 36*exp(2*log(x)^2 - (8*log(x))/9 + 8/81)*lo g(2) + 9*exp(4*log(x)^2 - (16*log(x))/9 + 16/81)*log(2)),x)
Output:
(2*x*exp(4/((exp(2*log(x)^2)*exp(8/81))/x^(8/9) + 2)))/log(2)
\[ \int \frac {e^{\frac {4}{2+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}}} \left (72+18 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )}+e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} (136-288 \log (x))\right )}{36 \log (2)+36 e^{\frac {2}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)+9 e^{\frac {4}{81} \left (4-36 \log (x)+81 \log ^2(x)\right )} \log (2)} \, dx=\int \frac {\left (18 \left ({\mathrm e}^{\mathrm {log}\left (x \right )^{2}-\frac {4 \,\mathrm {log}\left (x \right )}{9}+\frac {4}{81}}\right )^{4}+\left (-288 \,\mathrm {log}\left (x \right )+136\right ) \left ({\mathrm e}^{\mathrm {log}\left (x \right )^{2}-\frac {4 \,\mathrm {log}\left (x \right )}{9}+\frac {4}{81}}\right )^{2}+72\right ) {\mathrm e}^{\frac {4}{\left ({\mathrm e}^{\mathrm {log}\left (x \right )^{2}-\frac {4 \,\mathrm {log}\left (x \right )}{9}+\frac {4}{81}}\right )^{2}+2}}}{9 \,\mathrm {log}\left (2\right ) \left ({\mathrm e}^{\mathrm {log}\left (x \right )^{2}-\frac {4 \,\mathrm {log}\left (x \right )}{9}+\frac {4}{81}}\right )^{4}+36 \,\mathrm {log}\left (2\right ) \left ({\mathrm e}^{\mathrm {log}\left (x \right )^{2}-\frac {4 \,\mathrm {log}\left (x \right )}{9}+\frac {4}{81}}\right )^{2}+36 \,\mathrm {log}\left (2\right )}d x \] Input:
int((18*exp(log(x)^2-4/9*log(x)+4/81)^4+(-288*log(x)+136)*exp(log(x)^2-4/9 *log(x)+4/81)^2+72)*exp(4/(exp(log(x)^2-4/9*log(x)+4/81)^2+2))/(9*log(2)*e xp(log(x)^2-4/9*log(x)+4/81)^4+36*log(2)*exp(log(x)^2-4/9*log(x)+4/81)^2+3 6*log(2)),x)
Output:
int((18*exp(log(x)^2-4/9*log(x)+4/81)^4+(-288*log(x)+136)*exp(log(x)^2-4/9 *log(x)+4/81)^2+72)*exp(4/(exp(log(x)^2-4/9*log(x)+4/81)^2+2))/(9*log(2)*e xp(log(x)^2-4/9*log(x)+4/81)^4+36*log(2)*exp(log(x)^2-4/9*log(x)+4/81)^2+3 6*log(2)),x)