Integrand size = 122, antiderivative size = 22 \[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx=e^{-e^{\left (-1+e^{16}+x\right )^2}} x^2 (-2+\log (x)) \] Output:
(ln(x)-2)*x^2/exp(exp((exp(16)+x-1)^2))
Time = 5.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx=e^{-e^{\left (-1+e^{16}+x\right )^2}} x^2 (-2+\log (x)) \] Input:
Integrate[(-3*x + E^(1 + E^32 - 2*x + x^2 + E^16*(-2 + 2*x))*(-4*x^2 + 4*E ^16*x^2 + 4*x^3) + (2*x + E^(1 + E^32 - 2*x + x^2 + E^16*(-2 + 2*x))*(2*x^ 2 - 2*E^16*x^2 - 2*x^3))*Log[x])/E^E^(1 + E^32 - 2*x + x^2 + E^16*(-2 + 2* x)),x]
Output:
(x^2*(-2 + Log[x]))/E^E^(-1 + E^16 + x)^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.
| \(\displaystyle \int e^{-e^{x^2-2 x+e^{16} (2 x-2)+e^{32}+1}} \left (e^{x^2-2 x+e^{16} (2 x-2)+e^{32}+1} \left (4 x^3+4 e^{16} x^2-4 x^2\right )+\left (e^{x^2-2 x+e^{16} (2 x-2)+e^{32}+1} \left (-2 x^3-2 e^{16} x^2+2 x^2\right )+2 x\right ) \log (x)-3 x\right ) \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int e^{-e^{x^2-2 \left (1-e^{16}\right ) x+\left (e^{16}-1\right )^2}} \left (e^{x^2-2 x+e^{16} (2 x-2)+e^{32}+1} \left (4 x^3+4 e^{16} x^2-4 x^2\right )+\left (e^{x^2-2 x+e^{16} (2 x-2)+e^{32}+1} \left (-2 x^3-2 e^{16} x^2+2 x^2\right )+2 x\right ) \log (x)-3 x\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int e^{-e^{x^2-2 \left (1-e^{16}\right ) x+\left (e^{16}-1\right )^2}} x \left (4 e^{\left (x+e^{16}-1\right )^2} x \left (x+e^{16}-1\right )+2 \left (1-e^{\left (x+e^{16}-1\right )^2} x \left (x+e^{16}-1\right )\right ) \log (x)-3\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (2 \left (-x-e^{16}+1\right ) x^2 \exp \left (x^2-e^{x^2-2 \left (1-e^{16}\right ) x+\left (e^{16}-1\right )^2}-2 \left (1-e^{16}\right ) x+\left (e^{16}-1\right )^2\right ) (\log (x)-2)+e^{-e^{x^2-2 \left (1-e^{16}\right ) x+\left (e^{16}-1\right )^2}} x (2 \log (x)-3)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \int \frac {\int \exp \left ((x-1)^2+2 e^{16} (x-1)-e^{\left (x+e^{16}-1\right )^2}+e^{32}\right ) x^3dx}{x}dx-4 \left (1-e^{16}\right ) \int \exp \left (x^2-2 \left (1-e^{16}\right ) x-e^{x^2-2 \left (1-e^{16}\right ) x+\left (-1+e^{16}\right )^2}+\left (-1+e^{16}\right )^2\right ) x^2dx-2 \left (1-e^{16}\right ) \int \frac {\int \exp \left ((x-1)^2+2 e^{16} (x-1)-e^{\left (x+e^{16}-1\right )^2}+e^{32}\right ) x^2dx}{x}dx+2 \left (1-e^{16}\right ) \log (x) \int \exp \left (x^2-2 \left (1-e^{16}\right ) x-e^{x^2-2 \left (1-e^{16}\right ) x+\left (-1+e^{16}\right )^2}+\left (-1+e^{16}\right )^2\right ) x^2dx+4 \int \exp \left (x^2-2 \left (1-e^{16}\right ) x-e^{x^2-2 \left (1-e^{16}\right ) x+\left (-1+e^{16}\right )^2}+\left (-1+e^{16}\right )^2\right ) x^3dx-2 \log (x) \int \exp \left (x^2-2 \left (1-e^{16}\right ) x-e^{x^2-2 \left (1-e^{16}\right ) x+\left (-1+e^{16}\right )^2}+\left (-1+e^{16}\right )^2\right ) x^3dx-3 \int e^{-e^{x^2-2 \left (1-e^{16}\right ) x+\left (-1+e^{16}\right )^2}} xdx-2 \int \frac {\int e^{-e^{x^2-2 \left (1-e^{16}\right ) x+\left (-1+e^{16}\right )^2}} xdx}{x}dx+2 \log (x) \int e^{-e^{x^2-2 \left (1-e^{16}\right ) x+\left (-1+e^{16}\right )^2}} xdx\) |
Input:
Int[(-3*x + E^(1 + E^32 - 2*x + x^2 + E^16*(-2 + 2*x))*(-4*x^2 + 4*E^16*x^ 2 + 4*x^3) + (2*x + E^(1 + E^32 - 2*x + x^2 + E^16*(-2 + 2*x))*(2*x^2 - 2* E^16*x^2 - 2*x^3))*Log[x])/E^E^(1 + E^32 - 2*x + x^2 + E^16*(-2 + 2*x)),x]
Output:
$Aborted
Time = 2.94 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45
| method | result | size |
| risch | \(\left (\ln \left (x \right )-2\right ) x^{2} {\mathrm e}^{-{\mathrm e}^{2 x \,{\mathrm e}^{16}+x^{2}+{\mathrm e}^{32}-2 \,{\mathrm e}^{16}-2 x +1}}\) | \(32\) |
| parallelrisch | \(\left (x^{2} \ln \left (x \right )-2 x^{2}\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{32}+\left (-2+2 x \right ) {\mathrm e}^{16}+x^{2}-2 x +1}}\) | \(38\) |
Input:
int((((-2*x^2*exp(16)-2*x^3+2*x^2)*exp(exp(16)^2+(-2+2*x)*exp(16)+x^2-2*x+ 1)+2*x)*ln(x)+(4*x^2*exp(16)+4*x^3-4*x^2)*exp(exp(16)^2+(-2+2*x)*exp(16)+x ^2-2*x+1)-3*x)/exp(exp(exp(16)^2+(-2+2*x)*exp(16)+x^2-2*x+1)),x,method=_RE TURNVERBOSE)
Output:
(ln(x)-2)*x^2*exp(-exp(2*x*exp(16)+x^2+exp(32)-2*exp(16)-2*x+1))
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx={\left (x^{2} \log \left (x\right ) - 2 \, x^{2}\right )} e^{\left (-e^{\left (x^{2} + 2 \, {\left (x - 1\right )} e^{16} - 2 \, x + e^{32} + 1\right )}\right )} \] Input:
integrate((((-2*x^2*exp(16)-2*x^3+2*x^2)*exp(exp(16)^2+(2*x-2)*exp(16)+x^2 -2*x+1)+2*x)*log(x)+(4*x^2*exp(16)+4*x^3-4*x^2)*exp(exp(16)^2+(2*x-2)*exp( 16)+x^2-2*x+1)-3*x)/exp(exp(exp(16)^2+(2*x-2)*exp(16)+x^2-2*x+1)),x, algor ithm="fricas")
Output:
(x^2*log(x) - 2*x^2)*e^(-e^(x^2 + 2*(x - 1)*e^16 - 2*x + e^32 + 1))
Time = 25.60 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx=\left (x^{2} \log {\left (x \right )} - 2 x^{2}\right ) e^{- e^{x^{2} - 2 x + \left (2 x - 2\right ) e^{16} + 1 + e^{32}}} \] Input:
integrate((((-2*x**2*exp(16)-2*x**3+2*x**2)*exp(exp(16)**2+(2*x-2)*exp(16) +x**2-2*x+1)+2*x)*ln(x)+(4*x**2*exp(16)+4*x**3-4*x**2)*exp(exp(16)**2+(2*x -2)*exp(16)+x**2-2*x+1)-3*x)/exp(exp(exp(16)**2+(2*x-2)*exp(16)+x**2-2*x+1 )),x)
Output:
(x**2*log(x) - 2*x**2)*exp(-exp(x**2 - 2*x + (2*x - 2)*exp(16) + 1 + exp(3 2)))
Time = 0.68 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx={\left (x^{2} \log \left (x\right ) - 2 \, x^{2}\right )} e^{\left (-e^{\left (x^{2} + 2 \, x e^{16} - 2 \, x + e^{32} - 2 \, e^{16} + 1\right )}\right )} \] Input:
integrate((((-2*x^2*exp(16)-2*x^3+2*x^2)*exp(exp(16)^2+(2*x-2)*exp(16)+x^2 -2*x+1)+2*x)*log(x)+(4*x^2*exp(16)+4*x^3-4*x^2)*exp(exp(16)^2+(2*x-2)*exp( 16)+x^2-2*x+1)-3*x)/exp(exp(exp(16)^2+(2*x-2)*exp(16)+x^2-2*x+1)),x, algor ithm="maxima")
Output:
(x^2*log(x) - 2*x^2)*e^(-e^(x^2 + 2*x*e^16 - 2*x + e^32 - 2*e^16 + 1))
\[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx=\int { {\left (4 \, {\left (x^{3} + x^{2} e^{16} - x^{2}\right )} e^{\left (x^{2} + 2 \, {\left (x - 1\right )} e^{16} - 2 \, x + e^{32} + 1\right )} - 2 \, {\left ({\left (x^{3} + x^{2} e^{16} - x^{2}\right )} e^{\left (x^{2} + 2 \, {\left (x - 1\right )} e^{16} - 2 \, x + e^{32} + 1\right )} - x\right )} \log \left (x\right ) - 3 \, x\right )} e^{\left (-e^{\left (x^{2} + 2 \, {\left (x - 1\right )} e^{16} - 2 \, x + e^{32} + 1\right )}\right )} \,d x } \] Input:
integrate((((-2*x^2*exp(16)-2*x^3+2*x^2)*exp(exp(16)^2+(2*x-2)*exp(16)+x^2 -2*x+1)+2*x)*log(x)+(4*x^2*exp(16)+4*x^3-4*x^2)*exp(exp(16)^2+(2*x-2)*exp( 16)+x^2-2*x+1)-3*x)/exp(exp(exp(16)^2+(2*x-2)*exp(16)+x^2-2*x+1)),x, algor ithm="giac")
Output:
integrate((4*(x^3 + x^2*e^16 - x^2)*e^(x^2 + 2*(x - 1)*e^16 - 2*x + e^32 + 1) - 2*((x^3 + x^2*e^16 - x^2)*e^(x^2 + 2*(x - 1)*e^16 - 2*x + e^32 + 1) - x)*log(x) - 3*x)*e^(-e^(x^2 + 2*(x - 1)*e^16 - 2*x + e^32 + 1)), x)
Time = 7.83 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx=x^2\,{\mathrm {e}}^{-{\mathrm {e}}^{-2\,{\mathrm {e}}^{16}}\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{x^2}\,\mathrm {e}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{16}}\,{\mathrm {e}}^{{\mathrm {e}}^{32}}}\,\left (\ln \left (x\right )-2\right ) \] Input:
int(exp(-exp(exp(32) - 2*x + x^2 + exp(16)*(2*x - 2) + 1))*(log(x)*(2*x - exp(exp(32) - 2*x + x^2 + exp(16)*(2*x - 2) + 1)*(2*x^2*exp(16) - 2*x^2 + 2*x^3)) - 3*x + exp(exp(32) - 2*x + x^2 + exp(16)*(2*x - 2) + 1)*(4*x^2*ex p(16) - 4*x^2 + 4*x^3)),x)
Output:
x^2*exp(-exp(-2*exp(16))*exp(-2*x)*exp(x^2)*exp(1)*exp(2*x*exp(16))*exp(ex p(32)))*(log(x) - 2)
\[ \int e^{-e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)}} \left (-3 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (-4 x^2+4 e^{16} x^2+4 x^3\right )+\left (2 x+e^{1+e^{32}-2 x+x^2+e^{16} (-2+2 x)} \left (2 x^2-2 e^{16} x^2-2 x^3\right )\right ) \log (x)\right ) \, dx=4 e^{e^{32}} \left (\int \frac {e^{2 e^{16} x +x^{2}} x^{3}}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}+2 e^{2 e^{16}+2 x} x}{e^{2 e^{16}+2 x}}}}d x \right ) e +4 e^{e^{32}} \left (\int \frac {e^{2 e^{16} x +x^{2}} x^{2}}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}+2 e^{2 e^{16}+2 x} x}{e^{2 e^{16}+2 x}}}}d x \right ) e^{17}-4 e^{e^{32}} \left (\int \frac {e^{2 e^{16} x +x^{2}} x^{2}}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}+2 e^{2 e^{16}+2 x} x}{e^{2 e^{16}+2 x}}}}d x \right ) e -2 e^{e^{32}} \left (\int \frac {e^{2 e^{16} x +x^{2}} \mathrm {log}\left (x \right ) x^{3}}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}+2 e^{2 e^{16}+2 x} x}{e^{2 e^{16}+2 x}}}}d x \right ) e -2 e^{e^{32}} \left (\int \frac {e^{2 e^{16} x +x^{2}} \mathrm {log}\left (x \right ) x^{2}}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}+2 e^{2 e^{16}+2 x} x}{e^{2 e^{16}+2 x}}}}d x \right ) e^{17}+2 e^{e^{32}} \left (\int \frac {e^{2 e^{16} x +x^{2}} \mathrm {log}\left (x \right ) x^{2}}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}+2 e^{2 e^{16}+2 x} x}{e^{2 e^{16}+2 x}}}}d x \right ) e +2 e^{2 e^{16}} \left (\int \frac {\mathrm {log}\left (x \right ) x}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}}{e^{2 e^{16}+2 x}}}}d x \right )-3 e^{2 e^{16}} \left (\int \frac {x}{e^{\frac {e^{e^{32}+2 e^{16} x +x^{2}} e +2 e^{2 e^{16}+2 x} e^{16}}{e^{2 e^{16}+2 x}}}}d x \right ) \] Input:
int((((-2*x^2*exp(16)-2*x^3+2*x^2)*exp(exp(16)^2+(2*x-2)*exp(16)+x^2-2*x+1 )+2*x)*log(x)+(4*x^2*exp(16)+4*x^3-4*x^2)*exp(exp(16)^2+(2*x-2)*exp(16)+x^ 2-2*x+1)-3*x)/exp(exp(exp(16)^2+(2*x-2)*exp(16)+x^2-2*x+1)),x)
Output:
4*e**(e**32)*int((e**(2*e**16*x + x**2)*x**3)/e**((e**(e**32 + 2*e**16*x + x**2)*e + 2*e**(2*e**16 + 2*x)*e**16 + 2*e**(2*e**16 + 2*x)*x)/e**(2*e**1 6 + 2*x)),x)*e + 4*e**(e**32)*int((e**(2*e**16*x + x**2)*x**2)/e**((e**(e* *32 + 2*e**16*x + x**2)*e + 2*e**(2*e**16 + 2*x)*e**16 + 2*e**(2*e**16 + 2 *x)*x)/e**(2*e**16 + 2*x)),x)*e**17 - 4*e**(e**32)*int((e**(2*e**16*x + x* *2)*x**2)/e**((e**(e**32 + 2*e**16*x + x**2)*e + 2*e**(2*e**16 + 2*x)*e**1 6 + 2*e**(2*e**16 + 2*x)*x)/e**(2*e**16 + 2*x)),x)*e - 2*e**(e**32)*int((e **(2*e**16*x + x**2)*log(x)*x**3)/e**((e**(e**32 + 2*e**16*x + x**2)*e + 2 *e**(2*e**16 + 2*x)*e**16 + 2*e**(2*e**16 + 2*x)*x)/e**(2*e**16 + 2*x)),x) *e - 2*e**(e**32)*int((e**(2*e**16*x + x**2)*log(x)*x**2)/e**((e**(e**32 + 2*e**16*x + x**2)*e + 2*e**(2*e**16 + 2*x)*e**16 + 2*e**(2*e**16 + 2*x)*x )/e**(2*e**16 + 2*x)),x)*e**17 + 2*e**(e**32)*int((e**(2*e**16*x + x**2)*l og(x)*x**2)/e**((e**(e**32 + 2*e**16*x + x**2)*e + 2*e**(2*e**16 + 2*x)*e* *16 + 2*e**(2*e**16 + 2*x)*x)/e**(2*e**16 + 2*x)),x)*e + 2*e**(2*e**16)*in t((log(x)*x)/e**((e**(e**32 + 2*e**16*x + x**2)*e + 2*e**(2*e**16 + 2*x)*e **16)/e**(2*e**16 + 2*x)),x) - 3*e**(2*e**16)*int(x/e**((e**(e**32 + 2*e** 16*x + x**2)*e + 2*e**(2*e**16 + 2*x)*e**16)/e**(2*e**16 + 2*x)),x)