Integrand size = 123, antiderivative size = 24 \[ \int \frac {e^{e^{-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )}-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )} \left (-x \log \left (e^3 x\right )+\log ^{18-12 x+2 x^2}\left (e^3 x\right ) \left (450-300 x+50 x^2+\left (-300 x+100 x^2\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )\right )\right )}{x \log \left (e^3 x\right )} \, dx=e^{e^{-x+25 \log ^{2 (-3+x)^2}\left (e^3 x\right )}} \]
Time = 0.74 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{e^{-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )}-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )} \left (-x \log \left (e^3 x\right )+\log ^{18-12 x+2 x^2}\left (e^3 x\right ) \left (450-300 x+50 x^2+\left (-300 x+100 x^2\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )\right )\right )}{x \log \left (e^3 x\right )} \, dx=e^{e^{-x+25 (3+\log (x))^{2 (-3+x)^2}}} \]
Integrate[(E^(E^(-x + 25*Log[E^3*x]^(18 - 12*x + 2*x^2)) - x + 25*Log[E^3* x]^(18 - 12*x + 2*x^2))*(-(x*Log[E^3*x]) + Log[E^3*x]^(18 - 12*x + 2*x^2)* (450 - 300*x + 50*x^2 + (-300*x + 100*x^2)*Log[E^3*x]*Log[Log[E^3*x]])))/( x*Log[E^3*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.
| \(\displaystyle \int \frac {\left (\log ^{2 x^2-12 x+18}\left (e^3 x\right ) \left (50 x^2+\left (100 x^2-300 x\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )-300 x+450\right )-x \log \left (e^3 x\right )\right ) \exp \left (25 \log ^{2 x^2-12 x+18}\left (e^3 x\right )+e^{25 \log ^{2 x^2-12 x+18}\left (e^3 x\right )-x}-x\right )}{x \log \left (e^3 x\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {50 (x-3) (\log (x)+3)^{2 x^2-12 x+17} (x+2 x \log (x) \log (\log (x)+3)+6 x \log (\log (x)+3)-3) \exp \left (25 \log ^{2 x^2-12 x+18}\left (e^3 x\right )+e^{25 \log ^{2 x^2-12 x+18}\left (e^3 x\right )-x}-x\right )}{x}-\exp \left (25 \log ^{2 x^2-12 x+18}\left (e^3 x\right )+e^{25 \log ^{2 x^2-12 x+18}\left (e^3 x\right )-x}-x\right )\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (\frac {50 (x-3) (\log (x)+3)^{2 x^2-12 x+17} (x+2 x (\log (x)+3) \log (\log (x)+3)-3)}{x}-1\right ) \exp \left (-x+25 (\log (x)+3)^{2 (x-3)^2}+e^{25 (\log (x)+3)^{2 (x-3)^2}-x}\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {50 (x-3) (\log (x)+3)^{2 x^2-12 x+17} (x+2 x \log (x) \log (\log (x)+3)+6 x \log (\log (x)+3)-3) \exp \left (-x+25 (\log (x)+3)^{2 (x-3)^2}+e^{25 (\log (x)+3)^{2 (x-3)^2}-x}\right )}{x}-\exp \left (-x+25 (\log (x)+3)^{2 (x-3)^2}+e^{25 (\log (x)+3)^{2 (x-3)^2}-x}\right )\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {50 (x-3) (\log (x)+3)^{2 x^2-12 x+17} (x+2 x \log (x) \log (\log (x)+3)+6 x \log (\log (x)+3)-3) \exp \left (-x+25 (\log (x)+3)^{2 (x-3)^2}+e^{25 (\log (x)+3)^{2 (x-3)^2}-x}\right )}{x}-\exp \left (-x+25 (\log (x)+3)^{2 (x-3)^2}+e^{25 (\log (x)+3)^{2 (x-3)^2}-x}\right )\right )dx\) |
Int[(E^(E^(-x + 25*Log[E^3*x]^(18 - 12*x + 2*x^2)) - x + 25*Log[E^3*x]^(18 - 12*x + 2*x^2))*(-(x*Log[E^3*x]) + Log[E^3*x]^(18 - 12*x + 2*x^2)*(450 - 300*x + 50*x^2 + (-300*x + 100*x^2)*Log[E^3*x]*Log[Log[E^3*x]])))/(x*Log[ E^3*x]),x]
3.3.91.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 101.93 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \({\mathrm e}^{{\mathrm e}^{25 \ln \left (x \,{\mathrm e}^{3}\right )^{2 \left (-3+x \right )^{2}}-x}}\) | \(22\) |
| parallelrisch | \({\mathrm e}^{{\mathrm e}^{25 \,{\mathrm e}^{\left (2 x^{2}-12 x +18\right ) \ln \left (\ln \left (x \,{\mathrm e}^{3}\right )\right )}-x}}\) | \(27\) |
int((((100*x^2-300*x)*ln(x*exp(3))*ln(ln(x*exp(3)))+50*x^2-300*x+450)*exp( (x^2-6*x+9)*ln(ln(x*exp(3))))^2-x*ln(x*exp(3)))*exp(25*exp((x^2-6*x+9)*ln( ln(x*exp(3))))^2-x)*exp(exp(25*exp((x^2-6*x+9)*ln(ln(x*exp(3))))^2-x))/x/l n(x*exp(3)),x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )}-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )} \left (-x \log \left (e^3 x\right )+\log ^{18-12 x+2 x^2}\left (e^3 x\right ) \left (450-300 x+50 x^2+\left (-300 x+100 x^2\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )\right )\right )}{x \log \left (e^3 x\right )} \, dx=e^{\left (e^{\left (-x + 25 \, \log \left (x e^{3}\right )^{2 \, x^{2} - 12 \, x + 18}\right )}\right )} \]
integrate((((100*x^2-300*x)*log(x*exp(3))*log(log(x*exp(3)))+50*x^2-300*x+ 450)*exp((x^2-6*x+9)*log(log(x*exp(3))))^2-x*log(x*exp(3)))*exp(25*exp((x^ 2-6*x+9)*log(log(x*exp(3))))^2-x)*exp(exp(25*exp((x^2-6*x+9)*log(log(x*exp (3))))^2-x))/x/log(x*exp(3)),x, algorithm=\
Timed out. \[ \int \frac {e^{e^{-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )}-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )} \left (-x \log \left (e^3 x\right )+\log ^{18-12 x+2 x^2}\left (e^3 x\right ) \left (450-300 x+50 x^2+\left (-300 x+100 x^2\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )\right )\right )}{x \log \left (e^3 x\right )} \, dx=\text {Timed out} \]
integrate((((100*x**2-300*x)*ln(x*exp(3))*ln(ln(x*exp(3)))+50*x**2-300*x+4 50)*exp((x**2-6*x+9)*ln(ln(x*exp(3))))**2-x*ln(x*exp(3)))*exp(25*exp((x**2 -6*x+9)*ln(ln(x*exp(3))))**2-x)*exp(exp(25*exp((x**2-6*x+9)*ln(ln(x*exp(3) )))**2-x))/x/ln(x*exp(3)),x)
Leaf count of result is larger than twice the leaf count of optimal. 494 vs. \(2 (21) = 42\).
Time = 31.79 (sec) , antiderivative size = 494, normalized size of antiderivative = 20.58 \[ \int \frac {e^{e^{-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )}-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )} \left (-x \log \left (e^3 x\right )+\log ^{18-12 x+2 x^2}\left (e^3 x\right ) \left (450-300 x+50 x^2+\left (-300 x+100 x^2\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )\right )\right )}{x \log \left (e^3 x\right )} \, dx =\text {Too large to display} \]
integrate((((100*x^2-300*x)*log(x*exp(3))*log(log(x*exp(3)))+50*x^2-300*x+ 450)*exp((x^2-6*x+9)*log(log(x*exp(3))))^2-x*log(x*exp(3)))*exp(25*exp((x^ 2-6*x+9)*log(log(x*exp(3))))^2-x)*exp(exp(25*exp((x^2-6*x+9)*log(log(x*exp (3))))^2-x))/x/log(x*exp(3)),x, algorithm=\
e^(e^(25*e^(2*x^2*log(log(x) + 3) - 12*x*log(log(x) + 3))*log(x)^18 + 1350 *e^(2*x^2*log(log(x) + 3) - 12*x*log(log(x) + 3))*log(x)^17 + 34425*e^(2*x ^2*log(log(x) + 3) - 12*x*log(log(x) + 3))*log(x)^16 + 550800*e^(2*x^2*log (log(x) + 3) - 12*x*log(log(x) + 3))*log(x)^15 + 6196500*e^(2*x^2*log(log( x) + 3) - 12*x*log(log(x) + 3))*log(x)^14 + 52050600*e^(2*x^2*log(log(x) + 3) - 12*x*log(log(x) + 3))*log(x)^13 + 338328900*e^(2*x^2*log(log(x) + 3) - 12*x*log(log(x) + 3))*log(x)^12 + 1739977200*e^(2*x^2*log(log(x) + 3) - 12*x*log(log(x) + 3))*log(x)^11 + 7177405950*e^(2*x^2*log(log(x) + 3) - 1 2*x*log(log(x) + 3))*log(x)^10 + 23924686500*e^(2*x^2*log(log(x) + 3) - 12 *x*log(log(x) + 3))*log(x)^9 + 64596653550*e^(2*x^2*log(log(x) + 3) - 12*x *log(log(x) + 3))*log(x)^8 + 140938153200*e^(2*x^2*log(log(x) + 3) - 12*x* log(log(x) + 3))*log(x)^7 + 246641768100*e^(2*x^2*log(log(x) + 3) - 12*x*l og(log(x) + 3))*log(x)^6 + 341503986600*e^(2*x^2*log(log(x) + 3) - 12*x*lo g(log(x) + 3))*log(x)^5 + 365897128500*e^(2*x^2*log(log(x) + 3) - 12*x*log (log(x) + 3))*log(x)^4 + 292717702800*e^(2*x^2*log(log(x) + 3) - 12*x*log( log(x) + 3))*log(x)^3 + 164653707825*e^(2*x^2*log(log(x) + 3) - 12*x*log(l og(x) + 3))*log(x)^2 + 58113073350*e^(2*x^2*log(log(x) + 3) - 12*x*log(log (x) + 3))*log(x) - x + 9685512225*e^(2*x^2*log(log(x) + 3) - 12*x*log(log( x) + 3))))
Timed out. \[ \int \frac {e^{e^{-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )}-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )} \left (-x \log \left (e^3 x\right )+\log ^{18-12 x+2 x^2}\left (e^3 x\right ) \left (450-300 x+50 x^2+\left (-300 x+100 x^2\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )\right )\right )}{x \log \left (e^3 x\right )} \, dx=\text {Timed out} \]
integrate((((100*x^2-300*x)*log(x*exp(3))*log(log(x*exp(3)))+50*x^2-300*x+ 450)*exp((x^2-6*x+9)*log(log(x*exp(3))))^2-x*log(x*exp(3)))*exp(25*exp((x^ 2-6*x+9)*log(log(x*exp(3))))^2-x)*exp(exp(25*exp((x^2-6*x+9)*log(log(x*exp (3))))^2-x))/x/log(x*exp(3)),x, algorithm=\
Time = 12.52 (sec) , antiderivative size = 399, normalized size of antiderivative = 16.62 \[ \int \frac {e^{e^{-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )}-x+25 \log ^{18-12 x+2 x^2}\left (e^3 x\right )} \left (-x \log \left (e^3 x\right )+\log ^{18-12 x+2 x^2}\left (e^3 x\right ) \left (450-300 x+50 x^2+\left (-300 x+100 x^2\right ) \log \left (e^3 x\right ) \log \left (\log \left (e^3 x\right )\right )\right )\right )}{x \log \left (e^3 x\right )} \, dx={\mathrm {e}}^{{\mathrm {e}}^{9685512225\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{25\,{\ln \left (x\right )}^{18}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{1350\,{\ln \left (x\right )}^{17}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{34425\,{\ln \left (x\right )}^{16}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{550800\,{\ln \left (x\right )}^{15}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{6196500\,{\ln \left (x\right )}^{14}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{52050600\,{\ln \left (x\right )}^{13}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{64596653550\,{\ln \left (x\right )}^8\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{338328900\,{\ln \left (x\right )}^{12}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{292717702800\,{\ln \left (x\right )}^3\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{58113073350\,\ln \left (x\right )\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{365897128500\,{\ln \left (x\right )}^4\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{164653707825\,{\ln \left (x\right )}^2\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{1739977200\,{\ln \left (x\right )}^{11}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{246641768100\,{\ln \left (x\right )}^6\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{341503986600\,{\ln \left (x\right )}^5\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{23924686500\,{\ln \left (x\right )}^9\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{7177405950\,{\ln \left (x\right )}^{10}\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}\,{\mathrm {e}}^{140938153200\,{\ln \left (x\right )}^7\,{\left (\ln \left (x\right )+3\right )}^{2\,x^2-12\,x}}} \]
int(-(exp(exp(25*exp(2*log(log(x*exp(3)))*(x^2 - 6*x + 9)) - x))*exp(25*ex p(2*log(log(x*exp(3)))*(x^2 - 6*x + 9)) - x)*(exp(2*log(log(x*exp(3)))*(x^ 2 - 6*x + 9))*(300*x - 50*x^2 + log(log(x*exp(3)))*log(x*exp(3))*(300*x - 100*x^2) - 450) + x*log(x*exp(3))))/(x*log(x*exp(3))),x)
exp(exp(9685512225*(log(x) + 3)^(2*x^2 - 12*x))*exp(25*log(x)^18*(log(x) + 3)^(2*x^2 - 12*x))*exp(1350*log(x)^17*(log(x) + 3)^(2*x^2 - 12*x))*exp(34 425*log(x)^16*(log(x) + 3)^(2*x^2 - 12*x))*exp(550800*log(x)^15*(log(x) + 3)^(2*x^2 - 12*x))*exp(6196500*log(x)^14*(log(x) + 3)^(2*x^2 - 12*x))*exp( 52050600*log(x)^13*(log(x) + 3)^(2*x^2 - 12*x))*exp(64596653550*log(x)^8*( log(x) + 3)^(2*x^2 - 12*x))*exp(338328900*log(x)^12*(log(x) + 3)^(2*x^2 - 12*x))*exp(-x)*exp(292717702800*log(x)^3*(log(x) + 3)^(2*x^2 - 12*x))*exp( 58113073350*log(x)*(log(x) + 3)^(2*x^2 - 12*x))*exp(365897128500*log(x)^4* (log(x) + 3)^(2*x^2 - 12*x))*exp(164653707825*log(x)^2*(log(x) + 3)^(2*x^2 - 12*x))*exp(1739977200*log(x)^11*(log(x) + 3)^(2*x^2 - 12*x))*exp(246641 768100*log(x)^6*(log(x) + 3)^(2*x^2 - 12*x))*exp(341503986600*log(x)^5*(lo g(x) + 3)^(2*x^2 - 12*x))*exp(23924686500*log(x)^9*(log(x) + 3)^(2*x^2 - 1 2*x))*exp(7177405950*log(x)^10*(log(x) + 3)^(2*x^2 - 12*x))*exp(1409381532 00*log(x)^7*(log(x) + 3)^(2*x^2 - 12*x)))