Integrand size = 93, antiderivative size = 27 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{e^{2 x \left (-\frac {x}{1+e}+\log (x)\right )} x \log ^2(\log (2))} \] Output:
-exp(x*exp((ln(x)-x/(1+exp(1)))*x)^2*ln(ln(2))^2)
Time = 5.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{e^{-\frac {2 x^2}{1+e}} x^{1+2 x} \log ^2(\log (2))} \] Input:
Integrate[(E^((2*(-x^2 + (x + E*x)*Log[x]))/(1 + E) + E^((2*(-x^2 + (x + E *x)*Log[x]))/(1 + E))*x*Log[Log[2]]^2)*(-1 + E*(-1 - 2*x) - 2*x + 4*x^2 + (-2*x - 2*E*x)*Log[x])*Log[Log[2]]^2)/(1 + E),x]
Output:
-E^((x^(1 + 2*x)*Log[Log[2]]^2)/E^((2*x^2)/(1 + E)))
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.
| \(\displaystyle \int \frac {\log ^2(\log (2)) \left (4 x^2-2 x+e (-2 x-1)+(-2 e x-2 x) \log (x)-1\right ) \exp \left (x \log ^2(\log (2)) e^{\frac {2 \left ((e x+x) \log (x)-x^2\right )}{1+e}}+\frac {2 \left ((e x+x) \log (x)-x^2\right )}{1+e}\right )}{1+e} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\log ^2(\log (2)) \int -\exp \left (e^{-\frac {2 x^2}{1+e}} x^{\frac {2 (e x+x)}{1+e}+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) \left (-4 x^2+2 (1+e) \log (x) x+2 x+e (2 x+1)+1\right )dx}{1+e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\log ^2(\log (2)) \int \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) \left (-4 x^2+2 (1+e) \log (x) x+2 x+e (2 x+1)+1\right )dx}{1+e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\log ^2(\log (2)) \int \left (-4 \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) x^2+2 \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) (1+e) \log (x) x+2 \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) (1+e) x+\exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) (1+e)\right )dx}{1+e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\log ^2(\log (2)) \left ((1+e) \int \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right )dx+2 (1+e) \int \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) xdx-4 \int \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) x^2dx+2 (1+e) \int \exp \left (e^{-\frac {2 x^2}{1+e}} x^{2 x+1} \log ^2(\log (2))-\frac {2 \left (x^2-(1+e) x \log (x)\right )}{1+e}\right ) x \log (x)dx\right )}{1+e}\) |
Input:
Int[(E^((2*(-x^2 + (x + E*x)*Log[x]))/(1 + E) + E^((2*(-x^2 + (x + E*x)*Lo g[x]))/(1 + E))*x*Log[Log[2]]^2)*(-1 + E*(-1 - 2*x) - 2*x + 4*x^2 + (-2*x - 2*E*x)*Log[x])*Log[Log[2]]^2)/(1 + E),x]
Output:
$Aborted
Time = 2.93 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19
| method | result | size |
| risch | \(-{\mathrm e}^{x \ln \left (\ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {2 x \left ({\mathrm e} \ln \left (x \right )+\ln \left (x \right )-x \right )}{1+{\mathrm e}}}}\) | \(32\) |
| parallelrisch | \(-\frac {{\mathrm e}^{x \ln \left (\ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {2 x \left ({\mathrm e} \ln \left (x \right )+\ln \left (x \right )-x \right )}{1+{\mathrm e}}}} {\mathrm e}+{\mathrm e}^{x \ln \left (\ln \left (2\right )\right )^{2} {\mathrm e}^{\frac {2 x \left ({\mathrm e} \ln \left (x \right )+\ln \left (x \right )-x \right )}{1+{\mathrm e}}}}}{1+{\mathrm e}}\) | \(73\) |
Input:
int(((-2*x*exp(1)-2*x)*ln(x)+(-1-2*x)*exp(1)+4*x^2-2*x-1)*ln(ln(2))^2*exp( ((x*exp(1)+x)*ln(x)-x^2)/(1+exp(1)))^2*exp(x*ln(ln(2))^2*exp(((x*exp(1)+x) *ln(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x,method=_RETURNVERBOSE)
Output:
-exp(x*ln(ln(2))^2*exp(2*x*(exp(1)*ln(x)+ln(x)-x)/(1+exp(1))))
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (27) = 54\).
Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.11 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{\left (\frac {{\left (x e + x\right )} e^{\left (-\frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2} - 2 \, x^{2} + 2 \, {\left (x e + x\right )} \log \left (x\right )}{e + 1} + \frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \] Input:
integrate(((-2*exp(1)*x-2*x)*log(x)+(-1-2*x)*exp(1)+4*x^2-2*x-1)*log(log(2 ))^2*exp(((exp(1)*x+x)*log(x)-x^2)/(1+exp(1)))^2*exp(x*log(log(2))^2*exp(( (exp(1)*x+x)*log(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x, algorithm="fricas")
Output:
-e^(((x*e + x)*e^(-2*(x^2 - (x*e + x)*log(x))/(e + 1))*log(log(2))^2 - 2*x ^2 + 2*(x*e + x)*log(x))/(e + 1) + 2*(x^2 - (x*e + x)*log(x))/(e + 1))
Time = 1.86 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=- e^{x e^{\frac {2 \left (- x^{2} + \left (x + e x\right ) \log {\left (x \right )}\right )}{1 + e}} \log {\left (\log {\left (2 \right )} \right )}^{2}} \] Input:
integrate(((-2*exp(1)*x-2*x)*ln(x)+(-2*x-1)*exp(1)+4*x**2-2*x-1)*ln(ln(2)) **2*exp(((exp(1)*x+x)*ln(x)-x**2)/(1+exp(1)))**2*exp(x*ln(ln(2))**2*exp((( exp(1)*x+x)*ln(x)-x**2)/(1+exp(1)))**2)/(1+exp(1)),x)
Output:
-exp(x*exp(2*(-x**2 + (x + E*x)*log(x))/(1 + E))*log(log(2))**2)
Time = 0.43 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-e^{\left (x e^{\left (\frac {2 \, x e \log \left (x\right )}{e + 1} - \frac {2 \, x^{2}}{e + 1} + \frac {2 \, x \log \left (x\right )}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2}\right )} \] Input:
integrate(((-2*exp(1)*x-2*x)*log(x)+(-1-2*x)*exp(1)+4*x^2-2*x-1)*log(log(2 ))^2*exp(((exp(1)*x+x)*log(x)-x^2)/(1+exp(1)))^2*exp(x*log(log(2))^2*exp(( (exp(1)*x+x)*log(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x, algorithm="maxima")
Output:
-e^(x*e^(2*x*e*log(x)/(e + 1) - 2*x^2/(e + 1) + 2*x*log(x)/(e + 1))*log(lo g(2))^2)
\[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=\int { \frac {{\left (4 \, x^{2} - {\left (2 \, x + 1\right )} e - 2 \, {\left (x e + x\right )} \log \left (x\right ) - 2 \, x - 1\right )} e^{\left (x e^{\left (-\frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2} - \frac {2 \, {\left (x^{2} - {\left (x e + x\right )} \log \left (x\right )\right )}}{e + 1}\right )} \log \left (\log \left (2\right )\right )^{2}}{e + 1} \,d x } \] Input:
integrate(((-2*exp(1)*x-2*x)*log(x)+(-1-2*x)*exp(1)+4*x^2-2*x-1)*log(log(2 ))^2*exp(((exp(1)*x+x)*log(x)-x^2)/(1+exp(1)))^2*exp(x*log(log(2))^2*exp(( (exp(1)*x+x)*log(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x, algorithm="giac")
Output:
integrate((4*x^2 - (2*x + 1)*e - 2*(x*e + x)*log(x) - 2*x - 1)*e^(x*e^(-2* (x^2 - (x*e + x)*log(x))/(e + 1))*log(log(2))^2 - 2*(x^2 - (x*e + x)*log(x ))/(e + 1))*log(log(2))^2/(e + 1), x)
Time = 0.90 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=-{\mathrm {e}}^{x\,x^{\frac {2\,x\,\mathrm {e}}{\mathrm {e}+1}}\,x^{\frac {2\,x}{\mathrm {e}+1}}\,{\mathrm {e}}^{-\frac {2\,x^2}{\mathrm {e}+1}}\,{\ln \left (\ln \left (2\right )\right )}^2} \] Input:
int(-(exp(x*exp((2*(log(x)*(x + x*exp(1)) - x^2))/(exp(1) + 1))*log(log(2) )^2)*exp((2*(log(x)*(x + x*exp(1)) - x^2))/(exp(1) + 1))*log(log(2))^2*(2* x + log(x)*(2*x + 2*x*exp(1)) - 4*x^2 + exp(1)*(2*x + 1) + 1))/(exp(1) + 1 ),x)
Output:
-exp(x*x^((2*x*exp(1))/(exp(1) + 1))*x^((2*x)/(exp(1) + 1))*exp(-(2*x^2)/( exp(1) + 1))*log(log(2))^2)
\[ \int \frac {e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}+e^{\frac {2 \left (-x^2+(x+e x) \log (x)\right )}{1+e}} x \log ^2(\log (2))} \left (-1+e (-1-2 x)-2 x+4 x^2+(-2 x-2 e x) \log (x)\right ) \log ^2(\log (2))}{1+e} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} \left (4 \left (\int \frac {x^{2 x} e^{\frac {x^{2 x} \mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} x}{e^{\frac {2 x^{2}}{e +1}}}} x^{2}}{e^{\frac {2 x^{2}}{e +1}}}d x \right )-2 \left (\int \frac {x^{2 x} e^{\frac {x^{2 x} \mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} x}{e^{\frac {2 x^{2}}{e +1}}}} \mathrm {log}\left (x \right ) x}{e^{\frac {2 x^{2}}{e +1}}}d x \right ) e -2 \left (\int \frac {x^{2 x} e^{\frac {x^{2 x} \mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} x}{e^{\frac {2 x^{2}}{e +1}}}} \mathrm {log}\left (x \right ) x}{e^{\frac {2 x^{2}}{e +1}}}d x \right )-2 \left (\int \frac {x^{2 x} e^{\frac {x^{2 x} \mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} x}{e^{\frac {2 x^{2}}{e +1}}}} x}{e^{\frac {2 x^{2}}{e +1}}}d x \right ) e -2 \left (\int \frac {x^{2 x} e^{\frac {x^{2 x} \mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} x}{e^{\frac {2 x^{2}}{e +1}}}} x}{e^{\frac {2 x^{2}}{e +1}}}d x \right )-\left (\int \frac {x^{2 x} e^{\frac {x^{2 x} \mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} x}{e^{\frac {2 x^{2}}{e +1}}}}}{e^{\frac {2 x^{2}}{e +1}}}d x \right ) e -\left (\int \frac {x^{2 x} e^{\frac {x^{2 x} \mathrm {log}\left (\mathrm {log}\left (2\right )\right )^{2} x}{e^{\frac {2 x^{2}}{e +1}}}}}{e^{\frac {2 x^{2}}{e +1}}}d x \right )\right )}{e +1} \] Input:
int(((-2*exp(1)*x-2*x)*log(x)+(-2*x-1)*exp(1)+4*x^2-2*x-1)*log(log(2))^2*e xp(((exp(1)*x+x)*log(x)-x^2)/(1+exp(1)))^2*exp(x*log(log(2))^2*exp(((exp(1 )*x+x)*log(x)-x^2)/(1+exp(1)))^2)/(1+exp(1)),x)
Output:
(log(log(2))**2*(4*int((x**(2*x)*e**((x**(2*x)*log(log(2))**2*x)/e**((2*x* *2)/(e + 1)))*x**2)/e**((2*x**2)/(e + 1)),x) - 2*int((x**(2*x)*e**((x**(2* x)*log(log(2))**2*x)/e**((2*x**2)/(e + 1)))*log(x)*x)/e**((2*x**2)/(e + 1) ),x)*e - 2*int((x**(2*x)*e**((x**(2*x)*log(log(2))**2*x)/e**((2*x**2)/(e + 1)))*log(x)*x)/e**((2*x**2)/(e + 1)),x) - 2*int((x**(2*x)*e**((x**(2*x)*l og(log(2))**2*x)/e**((2*x**2)/(e + 1)))*x)/e**((2*x**2)/(e + 1)),x)*e - 2* int((x**(2*x)*e**((x**(2*x)*log(log(2))**2*x)/e**((2*x**2)/(e + 1)))*x)/e* *((2*x**2)/(e + 1)),x) - int((x**(2*x)*e**((x**(2*x)*log(log(2))**2*x)/e** ((2*x**2)/(e + 1))))/e**((2*x**2)/(e + 1)),x)*e - int((x**(2*x)*e**((x**(2 *x)*log(log(2))**2*x)/e**((2*x**2)/(e + 1))))/e**((2*x**2)/(e + 1)),x)))/( e + 1)