Integrand size = 71, antiderivative size = 25 \[ \int \frac {e^{-2 e^{3-x^2}} \left (-54 x-104 e^{3-x^2} x^3+\left (2 x+4 e^{3-x^2} x^3\right ) \log (x)\right )}{-17576+2028 \log (x)-78 \log ^2(x)+\log ^3(x)} \, dx=\frac {e^{-2 e^{3-x^2}} x^2}{(26-\log (x))^2} \]
Time = 0.88 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-2 e^{3-x^2}} \left (-54 x-104 e^{3-x^2} x^3+\left (2 x+4 e^{3-x^2} x^3\right ) \log (x)\right )}{-17576+2028 \log (x)-78 \log ^2(x)+\log ^3(x)} \, dx=\frac {e^{-2 e^{3-x^2}} x^2}{(-26+\log (x))^2} \]
Integrate[(-54*x - 104*E^(3 - x^2)*x^3 + (2*x + 4*E^(3 - x^2)*x^3)*Log[x]) /(E^(2*E^(3 - x^2))*(-17576 + 2028*Log[x] - 78*Log[x]^2 + Log[x]^3)),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.
\(\displaystyle \int \frac {e^{-2 e^{3-x^2}} \left (-104 e^{3-x^2} x^3+\left (4 e^{3-x^2} x^3+2 x\right ) \log (x)-54 x\right )}{\log ^3(x)-78 \log ^2(x)+2028 \log (x)-17576} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{-2 e^{3-x^2}} \left (104 e^{3-x^2} x^3-\left (4 e^{3-x^2} x^3+2 x\right ) \log (x)+54 x\right )}{(26-\log (x))^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {2 e^{-2 e^{3-x^2}} x (\log (x)-27)}{(\log (x)-26)^3}+\frac {4 e^{-x^2-2 e^{3-x^2}+3} x^3}{(\log (x)-26)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \int \frac {e^{-2 e^{3-x^2}} x}{(\log (x)-26)^3}dx+2 \int \frac {e^{-2 e^{3-x^2}} x}{(\log (x)-26)^2}dx+4 \int \frac {e^{-x^2-2 e^{3-x^2}+3} x^3}{(\log (x)-26)^2}dx\) |
Int[(-54*x - 104*E^(3 - x^2)*x^3 + (2*x + 4*E^(3 - x^2)*x^3)*Log[x])/(E^(2 *E^(3 - x^2))*(-17576 + 2028*Log[x] - 78*Log[x]^2 + Log[x]^3)),x]
3.13.11.3.1 Defintions of rubi rules used
Time = 0.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88
method | result | size |
risch | \(\frac {x^{2} {\mathrm e}^{-2 \,{\mathrm e}^{-x^{2}+3}}}{\left (-26+\ln \left (x \right )\right )^{2}}\) | \(22\) |
parallelrisch | \(\frac {x^{2} {\mathrm e}^{-2 \,{\mathrm e}^{-x^{2}+3}}}{\ln \left (x \right )^{2}-52 \ln \left (x \right )+676}\) | \(28\) |
int(((4*x^3*exp(-x^2+3)+2*x)*ln(x)-104*x^3*exp(-x^2+3)-54*x)/(ln(x)^3-78*l n(x)^2+2028*ln(x)-17576)/exp(exp(-x^2+3))^2,x,method=_RETURNVERBOSE)
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-2 e^{3-x^2}} \left (-54 x-104 e^{3-x^2} x^3+\left (2 x+4 e^{3-x^2} x^3\right ) \log (x)\right )}{-17576+2028 \log (x)-78 \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{2} e^{\left (-2 \, e^{\left (-x^{2} + 3\right )}\right )}}{\log \left (x\right )^{2} - 52 \, \log \left (x\right ) + 676} \]
integrate(((4*x^3*exp(-x^2+3)+2*x)*log(x)-104*x^3*exp(-x^2+3)-54*x)/(log(x )^3-78*log(x)^2+2028*log(x)-17576)/exp(exp(-x^2+3))^2,x, algorithm=\
Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-2 e^{3-x^2}} \left (-54 x-104 e^{3-x^2} x^3+\left (2 x+4 e^{3-x^2} x^3\right ) \log (x)\right )}{-17576+2028 \log (x)-78 \log ^2(x)+\log ^3(x)} \, dx=\frac {x^{2} e^{- 2 e^{3 - x^{2}}}}{\log {\left (x \right )}^{2} - 52 \log {\left (x \right )} + 676} \]
integrate(((4*x**3*exp(-x**2+3)+2*x)*ln(x)-104*x**3*exp(-x**2+3)-54*x)/(ln (x)**3-78*ln(x)**2+2028*ln(x)-17576)/exp(exp(-x**2+3))**2,x)
\[ \int \frac {e^{-2 e^{3-x^2}} \left (-54 x-104 e^{3-x^2} x^3+\left (2 x+4 e^{3-x^2} x^3\right ) \log (x)\right )}{-17576+2028 \log (x)-78 \log ^2(x)+\log ^3(x)} \, dx=\int { -\frac {2 \, {\left (52 \, x^{3} e^{\left (-x^{2} + 3\right )} - {\left (2 \, x^{3} e^{\left (-x^{2} + 3\right )} + x\right )} \log \left (x\right ) + 27 \, x\right )} e^{\left (-2 \, e^{\left (-x^{2} + 3\right )}\right )}}{\log \left (x\right )^{3} - 78 \, \log \left (x\right )^{2} + 2028 \, \log \left (x\right ) - 17576} \,d x } \]
integrate(((4*x^3*exp(-x^2+3)+2*x)*log(x)-104*x^3*exp(-x^2+3)-54*x)/(log(x )^3-78*log(x)^2+2028*log(x)-17576)/exp(exp(-x^2+3))^2,x, algorithm=\
-2*integrate((52*x^3*e^(-x^2 + 3) - (2*x^3*e^(-x^2 + 3) + x)*log(x) + 27*x )*e^(-2*e^(-x^2 + 3))/(log(x)^3 - 78*log(x)^2 + 2028*log(x) - 17576), x)
\[ \int \frac {e^{-2 e^{3-x^2}} \left (-54 x-104 e^{3-x^2} x^3+\left (2 x+4 e^{3-x^2} x^3\right ) \log (x)\right )}{-17576+2028 \log (x)-78 \log ^2(x)+\log ^3(x)} \, dx=\int { -\frac {2 \, {\left (52 \, x^{3} e^{\left (-x^{2} + 3\right )} - {\left (2 \, x^{3} e^{\left (-x^{2} + 3\right )} + x\right )} \log \left (x\right ) + 27 \, x\right )} e^{\left (-2 \, e^{\left (-x^{2} + 3\right )}\right )}}{\log \left (x\right )^{3} - 78 \, \log \left (x\right )^{2} + 2028 \, \log \left (x\right ) - 17576} \,d x } \]
integrate(((4*x^3*exp(-x^2+3)+2*x)*log(x)-104*x^3*exp(-x^2+3)-54*x)/(log(x )^3-78*log(x)^2+2028*log(x)-17576)/exp(exp(-x^2+3))^2,x, algorithm=\
integrate(-2*(52*x^3*e^(-x^2 + 3) - (2*x^3*e^(-x^2 + 3) + x)*log(x) + 27*x )*e^(-2*e^(-x^2 + 3))/(log(x)^3 - 78*log(x)^2 + 2028*log(x) - 17576), x)
Time = 12.74 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {e^{-2 e^{3-x^2}} \left (-54 x-104 e^{3-x^2} x^3+\left (2 x+4 e^{3-x^2} x^3\right ) \log (x)\right )}{-17576+2028 \log (x)-78 \log ^2(x)+\log ^3(x)} \, dx=\frac {x^2\,{\mathrm {e}}^{-2\,{\mathrm {e}}^3\,{\mathrm {e}}^{-x^2}}}{{\left (\ln \left (x\right )-26\right )}^2} \]