Integrand size = 94, antiderivative size = 22 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=e^{\frac {e^x-x-3 x (x+\log (x))}{e^2}}+x \]
Time = 0.12 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=x+e^{\frac {e^x-x (1+3 x)}{e^2}} x^{-\frac {3 x}{e^2}} \]
Integrate[(-E^x + x + 3*x^2 + 3*x*Log[x] + E^(-2 + (E^x - x - 3*x^2 - 3*x* Log[x])/E^2)*(4 - E^x + 6*x + 3*Log[x])*(E^x - x - 3*x^2 - 3*x*Log[x]))/(- E^x + x + 3*x^2 + 3*x*Log[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.
\(\displaystyle \int \frac {3 x^2+e^{\frac {-3 x^2-x+e^x-3 x \log (x)}{e^2}-2} \left (6 x-e^x+3 \log (x)+4\right ) \left (-3 x^2-x+e^x-3 x \log (x)\right )+x-e^x+3 x \log (x)}{3 x^2+x-e^x+3 x \log (x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (e^{-\frac {x (3 x+1)}{e^2}+e^{x-2}-2} x^{-\frac {3 x}{e^2}} \left (-6 x+e^x-3 \log (x)-4\right )+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \int e^{-\frac {x (3 x+1)}{e^2}+e^{x-2}-2} x^{-\frac {3 x}{e^2}}dx+\int e^{-\frac {(3 x+1) x}{e^2}+x+e^{x-2}-2} x^{-\frac {3 x}{e^2}}dx-6 \int e^{-\frac {x (3 x+1)}{e^2}+e^{x-2}-2} x^{1-\frac {3 x}{e^2}}dx+3 \int \frac {\int e^{-\frac {x (3 x+1)}{e^2}+e^{x-2}-2} x^{-\frac {3 x}{e^2}}dx}{x}dx-3 \log (x) \int e^{-\frac {x (3 x+1)}{e^2}+e^{x-2}-2} x^{-\frac {3 x}{e^2}}dx+x\) |
Int[(-E^x + x + 3*x^2 + 3*x*Log[x] + E^(-2 + (E^x - x - 3*x^2 - 3*x*Log[x] )/E^2)*(4 - E^x + 6*x + 3*Log[x])*(E^x - x - 3*x^2 - 3*x*Log[x]))/(-E^x + x + 3*x^2 + 3*x*Log[x]),x]
3.18.16.3.1 Defintions of rubi rules used
Time = 14.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05
method | result | size |
risch | \(x +{\mathrm e}^{\left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right ) {\mathrm e}^{-2}}\) | \(23\) |
parallelrisch | \(\frac {18 x \,{\mathrm e}^{2 x}-72 \,{\mathrm e}^{x} x^{2}+324 x^{4} \ln \left (x \right )-108 \,{\mathrm e}^{x} x^{3}-12 \,{\mathrm e}^{x} x +162 x^{3} \ln \left (x \right )^{2}+54 x^{2} \ln \left (x \right )^{2}+6 \,{\mathrm e}^{2 x}-108 x^{2} {\mathrm e}^{x} \ln \left (x \right )-36 x \,{\mathrm e}^{x} \ln \left (x \right )+162 x^{5}+162 x^{4}+54 x^{3}+6 x^{2}+216 x^{3} \ln \left (x \right )+36 x^{2} \ln \left (x \right )-108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x \ln \left (x \right ) {\mathrm e}^{x}+18 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{2}+108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{3}+162 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{4}+18 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} {\mathrm e}^{2 x}+108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} x^{2} \ln \left (x \right )+324 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} \ln \left (x \right ) x^{3}+162 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} \ln \left (x \right )^{2} x^{2}-36 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} {\mathrm e}^{x} x -108 \,{\mathrm e}^{{\mathrm e}^{\ln \left (-3 x \ln \left (x \right )+{\mathrm e}^{x}-3 x^{2}-x \right )-2}} {\mathrm e}^{x} x^{2}}{18 \left (3 x \ln \left (x \right )-{\mathrm e}^{x}+3 x^{2}+x \right )^{2}}\) | \(402\) |
int(((3*ln(x)-exp(x)+6*x+4)*exp(ln(-3*x*ln(x)+exp(x)-3*x^2-x)-2)*exp(exp(l n(-3*x*ln(x)+exp(x)-3*x^2-x)-2))+3*x*ln(x)-exp(x)+3*x^2+x)/(3*x*ln(x)-exp( x)+3*x^2+x),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (20) = 40\).
Time = 0.31 (sec) , antiderivative size = 90, normalized size of antiderivative = 4.09 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=\frac {3 \, x^{3} + 3 \, x^{2} \log \left (x\right ) + x^{2} - x e^{x} - e^{\left (-{\left (3 \, x^{2} - e^{2} \log \left (-3 \, x^{2} - 3 \, x \log \left (x\right ) - x + e^{x}\right ) + 3 \, x \log \left (x\right ) + x + 2 \, e^{2} - e^{x}\right )} e^{\left (-2\right )} + 2\right )}}{3 \, x^{2} + 3 \, x \log \left (x\right ) + x - e^{x}} \]
integrate(((3*log(x)-exp(x)+6*x+4)*exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2)* exp(exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2))+3*x*log(x)-exp(x)+3*x^2+x)/(3* x*log(x)-exp(x)+3*x^2+x),x, algorithm=\
(3*x^3 + 3*x^2*log(x) + x^2 - x*e^x - e^(-(3*x^2 - e^2*log(-3*x^2 - 3*x*lo g(x) - x + e^x) + 3*x*log(x) + x + 2*e^2 - e^x)*e^(-2) + 2))/(3*x^2 + 3*x* log(x) + x - e^x)
Time = 0.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=x + e^{\frac {- 3 x^{2} - 3 x \log {\left (x \right )} - x + e^{x}}{e^{2}}} \]
integrate(((3*ln(x)-exp(x)+6*x+4)*exp(ln(-3*x*ln(x)+exp(x)-3*x**2-x)-2)*ex p(exp(ln(-3*x*ln(x)+exp(x)-3*x**2-x)-2))+3*x*ln(x)-exp(x)+3*x**2+x)/(3*x*l n(x)-exp(x)+3*x**2+x),x)
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.00 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx={\left (x e^{\left (3 \, x^{2} e^{\left (-2\right )} + x e^{\left (-2\right )}\right )} + e^{\left (-3 \, x e^{\left (-2\right )} \log \left (x\right ) + e^{\left (x - 2\right )}\right )}\right )} e^{\left (-3 \, x^{2} e^{\left (-2\right )} - x e^{\left (-2\right )}\right )} \]
integrate(((3*log(x)-exp(x)+6*x+4)*exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2)* exp(exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2))+3*x*log(x)-exp(x)+3*x^2+x)/(3* x*log(x)-exp(x)+3*x^2+x),x, algorithm=\
(x*e^(3*x^2*e^(-2) + x*e^(-2)) + e^(-3*x*e^(-2)*log(x) + e^(x - 2)))*e^(-3 *x^2*e^(-2) - x*e^(-2))
\[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=\int { \frac {3 \, x^{2} + {\left (6 \, x - e^{x} + 3 \, \log \left (x\right ) + 4\right )} e^{\left (e^{\left (\log \left (-3 \, x^{2} - 3 \, x \log \left (x\right ) - x + e^{x}\right ) - 2\right )} + \log \left (-3 \, x^{2} - 3 \, x \log \left (x\right ) - x + e^{x}\right ) - 2\right )} + 3 \, x \log \left (x\right ) + x - e^{x}}{3 \, x^{2} + 3 \, x \log \left (x\right ) + x - e^{x}} \,d x } \]
integrate(((3*log(x)-exp(x)+6*x+4)*exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2)* exp(exp(log(-3*x*log(x)+exp(x)-3*x^2-x)-2))+3*x*log(x)-exp(x)+3*x^2+x)/(3* x*log(x)-exp(x)+3*x^2+x),x, algorithm=\
integrate((3*x^2 + (6*x - e^x + 3*log(x) + 4)*e^(e^(log(-3*x^2 - 3*x*log(x ) - x + e^x) - 2) + log(-3*x^2 - 3*x*log(x) - x + e^x) - 2) + 3*x*log(x) + x - e^x)/(3*x^2 + 3*x*log(x) + x - e^x), x)
Time = 15.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {-e^x+x+3 x^2+3 x \log (x)+e^{-2+\frac {e^x-x-3 x^2-3 x \log (x)}{e^2}} \left (4-e^x+6 x+3 \log (x)\right ) \left (e^x-x-3 x^2-3 x \log (x)\right )}{-e^x+x+3 x^2+3 x \log (x)} \, dx=x+\frac {{\mathrm {e}}^{-3\,x^2\,{\mathrm {e}}^{-2}}\,{\mathrm {e}}^{{\mathrm {e}}^{-2}\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{-2}}}{x^{3\,x\,{\mathrm {e}}^{-2}}} \]