Integrand size = 95, antiderivative size = 25 \[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx=e^{e^{11-x \left (x-\left (-x+\frac {\log (x)}{x}\right )^2\right )}} \] Output:
exp(exp(11-x*(x-(ln(x)/x-x)^2)))
\[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx=\int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx \] Input:
Integrate[(E^(E^((11*x - x^3 + x^4 - 2*x^2*Log[x] + Log[x]^2)/x) + (11*x - x^3 + x^4 - 2*x^2*Log[x] + Log[x]^2)/x)*(-2*x^2 - 2*x^3 + 3*x^4 + (2 - 2* x^2)*Log[x] - Log[x]^2))/x^2,x]
Output:
Integrate[(E^(E^((11*x - x^3 + x^4 - 2*x^2*Log[x] + Log[x]^2)/x) + (11*x - x^3 + x^4 - 2*x^2*Log[x] + Log[x]^2)/x)*(-2*x^2 - 2*x^3 + 3*x^4 + (2 - 2* x^2)*Log[x] - Log[x]^2))/x^2, 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.
| \(\displaystyle \int \frac {\left (3 x^4-2 x^3-2 x^2+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right ) \exp \left (\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}}\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (3 x^2 \exp \left (\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}}\right )-2 x \exp \left (\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}}\right )-2 \exp \left (\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}}\right )-\frac {\log ^2(x) \exp \left (\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}}\right )}{x^2}-\frac {2 \left (x^2-1\right ) \log (x) \exp \left (\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 x^2 \log (x)+11 x+\log ^2(x)}{x}}\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \exp \left (\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}}\right )dx-2 \int \exp \left (\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}}\right ) xdx+3 \int \exp \left (\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}}\right ) x^2dx-2 \int \exp \left (\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}}\right ) \log (x)dx+2 \int \frac {\exp \left (\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}}\right ) \log (x)}{x^2}dx-\int \frac {\exp \left (\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}+e^{\frac {x^4-x^3-2 \log (x) x^2+11 x+\log ^2(x)}{x}}\right ) \log ^2(x)}{x^2}dx\) |
Input:
Int[(E^(E^((11*x - x^3 + x^4 - 2*x^2*Log[x] + Log[x]^2)/x) + (11*x - x^3 + x^4 - 2*x^2*Log[x] + Log[x]^2)/x)*(-2*x^2 - 2*x^3 + 3*x^4 + (2 - 2*x^2)*L og[x] - Log[x]^2))/x^2,x]
Output:
$Aborted
Time = 0.96 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \({\mathrm e}^{x^{-2 x} {\mathrm e}^{\frac {x^{4}-x^{3}+\ln \left (x \right )^{2}+11 x}{x}}}\) | \(29\) |
| parallelrisch | \({\mathrm e}^{{\mathrm e}^{\frac {\ln \left (x \right )^{2}-2 x^{2} \ln \left (x \right )+x^{4}-x^{3}+11 x}{x}}}\) | \(30\) |
Input:
int((-ln(x)^2+(-2*x^2+2)*ln(x)+3*x^4-2*x^3-2*x^2)*exp((ln(x)^2-2*x^2*ln(x) +x^4-x^3+11*x)/x)*exp(exp((ln(x)^2-2*x^2*ln(x)+x^4-x^3+11*x)/x))/x^2,x,met hod=_RETURNVERBOSE)
Output:
exp(x^(-2*x)*exp((x^4-x^3+ln(x)^2+11*x)/x))
Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (21) = 42\).
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.48 \[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx=e^{\left (\frac {x^{4} - x^{3} - 2 \, x^{2} \log \left (x\right ) + x e^{\left (\frac {x^{4} - x^{3} - 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2} + 11 \, x}{x}\right )} + \log \left (x\right )^{2} + 11 \, x}{x} - \frac {x^{4} - x^{3} - 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2} + 11 \, x}{x}\right )} \] Input:
integrate((-log(x)^2+(-2*x^2+2)*log(x)+3*x^4-2*x^3-2*x^2)*exp((log(x)^2-2* x^2*log(x)+x^4-x^3+11*x)/x)*exp(exp((log(x)^2-2*x^2*log(x)+x^4-x^3+11*x)/x ))/x^2,x, algorithm="fricas")
Output:
e^((x^4 - x^3 - 2*x^2*log(x) + x*e^((x^4 - x^3 - 2*x^2*log(x) + log(x)^2 + 11*x)/x) + log(x)^2 + 11*x)/x - (x^4 - x^3 - 2*x^2*log(x) + log(x)^2 + 11 *x)/x)
Time = 0.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx=e^{e^{\frac {x^{4} - x^{3} - 2 x^{2} \log {\left (x \right )} + 11 x + \log {\left (x \right )}^{2}}{x}}} \] Input:
integrate((-ln(x)**2+(-2*x**2+2)*ln(x)+3*x**4-2*x**3-2*x**2)*exp((ln(x)**2 -2*x**2*ln(x)+x**4-x**3+11*x)/x)*exp(exp((ln(x)**2-2*x**2*ln(x)+x**4-x**3+ 11*x)/x))/x**2,x)
Output:
exp(exp((x**4 - x**3 - 2*x**2*log(x) + 11*x + log(x)**2)/x))
Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx=e^{\left (e^{\left (x^{3} - x^{2} - 2 \, x \log \left (x\right ) + \frac {\log \left (x\right )^{2}}{x} + 11\right )}\right )} \] Input:
integrate((-log(x)^2+(-2*x^2+2)*log(x)+3*x^4-2*x^3-2*x^2)*exp((log(x)^2-2* x^2*log(x)+x^4-x^3+11*x)/x)*exp(exp((log(x)^2-2*x^2*log(x)+x^4-x^3+11*x)/x ))/x^2,x, algorithm="maxima")
Output:
e^(e^(x^3 - x^2 - 2*x*log(x) + log(x)^2/x + 11))
\[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx=\int { \frac {{\left (3 \, x^{4} - 2 \, x^{3} - 2 \, x^{2} - 2 \, {\left (x^{2} - 1\right )} \log \left (x\right ) - \log \left (x\right )^{2}\right )} e^{\left (\frac {x^{4} - x^{3} - 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2} + 11 \, x}{x} + e^{\left (\frac {x^{4} - x^{3} - 2 \, x^{2} \log \left (x\right ) + \log \left (x\right )^{2} + 11 \, x}{x}\right )}\right )}}{x^{2}} \,d x } \] Input:
integrate((-log(x)^2+(-2*x^2+2)*log(x)+3*x^4-2*x^3-2*x^2)*exp((log(x)^2-2* x^2*log(x)+x^4-x^3+11*x)/x)*exp(exp((log(x)^2-2*x^2*log(x)+x^4-x^3+11*x)/x ))/x^2,x, algorithm="giac")
Output:
integrate((3*x^4 - 2*x^3 - 2*x^2 - 2*(x^2 - 1)*log(x) - log(x)^2)*e^((x^4 - x^3 - 2*x^2*log(x) + log(x)^2 + 11*x)/x + e^((x^4 - x^3 - 2*x^2*log(x) + log(x)^2 + 11*x)/x))/x^2, x)
Time = 1.46 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{x^3}\,{\mathrm {e}}^{11}\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{\frac {{\ln \left (x\right )}^2}{x}}}{x^{2\,x}}} \] Input:
int(-(exp((11*x - 2*x^2*log(x) + log(x)^2 - x^3 + x^4)/x)*exp(exp((11*x - 2*x^2*log(x) + log(x)^2 - x^3 + x^4)/x))*(log(x)^2 + 2*x^2 + 2*x^3 - 3*x^4 + log(x)*(2*x^2 - 2)))/x^2,x)
Output:
exp((exp(x^3)*exp(11)*exp(-x^2)*exp(log(x)^2/x))/x^(2*x))
\[ \int \frac {e^{e^{\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}}+\frac {11 x-x^3+x^4-2 x^2 \log (x)+\log ^2(x)}{x}} \left (-2 x^2-2 x^3+3 x^4+\left (2-2 x^2\right ) \log (x)-\log ^2(x)\right )}{x^2} \, dx=e^{11} \left (-2 \left (\int \frac {e^{\frac {e^{\frac {\mathrm {log}\left (x \right )^{2}+x^{4}}{x}} e^{11} x +x^{2 x} e^{x^{2}} \mathrm {log}\left (x \right )^{2}+x^{2 x} e^{x^{2}} x^{4}}{x^{2 x} e^{x^{2}} x}}}{x^{2 x} e^{x^{2}}}d x \right )-\left (\int \frac {e^{\frac {e^{\frac {\mathrm {log}\left (x \right )^{2}+x^{4}}{x}} e^{11} x +x^{2 x} e^{x^{2}} \mathrm {log}\left (x \right )^{2}+x^{2 x} e^{x^{2}} x^{4}}{x^{2 x} e^{x^{2}} x}} \mathrm {log}\left (x \right )^{2}}{x^{2 x} e^{x^{2}} x^{2}}d x \right )+3 \left (\int \frac {e^{\frac {e^{\frac {\mathrm {log}\left (x \right )^{2}+x^{4}}{x}} e^{11} x +x^{2 x} e^{x^{2}} \mathrm {log}\left (x \right )^{2}+x^{2 x} e^{x^{2}} x^{4}}{x^{2 x} e^{x^{2}} x}} x^{2}}{x^{2 x} e^{x^{2}}}d x \right )+2 \left (\int \frac {e^{\frac {e^{\frac {\mathrm {log}\left (x \right )^{2}+x^{4}}{x}} e^{11} x +x^{2 x} e^{x^{2}} \mathrm {log}\left (x \right )^{2}+x^{2 x} e^{x^{2}} x^{4}}{x^{2 x} e^{x^{2}} x}} \mathrm {log}\left (x \right )}{x^{2 x} e^{x^{2}} x^{2}}d x \right )-2 \left (\int \frac {e^{\frac {e^{\frac {\mathrm {log}\left (x \right )^{2}+x^{4}}{x}} e^{11} x +x^{2 x} e^{x^{2}} \mathrm {log}\left (x \right )^{2}+x^{2 x} e^{x^{2}} x^{4}}{x^{2 x} e^{x^{2}} x}} \mathrm {log}\left (x \right )}{x^{2 x} e^{x^{2}}}d x \right )-2 \left (\int \frac {e^{\frac {e^{\frac {\mathrm {log}\left (x \right )^{2}+x^{4}}{x}} e^{11} x +x^{2 x} e^{x^{2}} \mathrm {log}\left (x \right )^{2}+x^{2 x} e^{x^{2}} x^{4}}{x^{2 x} e^{x^{2}} x}} x}{x^{2 x} e^{x^{2}}}d x \right )\right ) \] Input:
int((-log(x)^2+(-2*x^2+2)*log(x)+3*x^4-2*x^3-2*x^2)*exp((log(x)^2-2*x^2*lo g(x)+x^4-x^3+11*x)/x)*exp(exp((log(x)^2-2*x^2*log(x)+x^4-x^3+11*x)/x))/x^2 ,x)
Output:
e**11*( - 2*int(e**((e**((log(x)**2 + x**4)/x)*e**11*x + x**(2*x)*e**(x**2 )*log(x)**2 + x**(2*x)*e**(x**2)*x**4)/(x**(2*x)*e**(x**2)*x))/(x**(2*x)*e **(x**2)),x) - int((e**((e**((log(x)**2 + x**4)/x)*e**11*x + x**(2*x)*e**( x**2)*log(x)**2 + x**(2*x)*e**(x**2)*x**4)/(x**(2*x)*e**(x**2)*x))*log(x)* *2)/(x**(2*x)*e**(x**2)*x**2),x) + 3*int((e**((e**((log(x)**2 + x**4)/x)*e **11*x + x**(2*x)*e**(x**2)*log(x)**2 + x**(2*x)*e**(x**2)*x**4)/(x**(2*x) *e**(x**2)*x))*x**2)/(x**(2*x)*e**(x**2)),x) + 2*int((e**((e**((log(x)**2 + x**4)/x)*e**11*x + x**(2*x)*e**(x**2)*log(x)**2 + x**(2*x)*e**(x**2)*x** 4)/(x**(2*x)*e**(x**2)*x))*log(x))/(x**(2*x)*e**(x**2)*x**2),x) - 2*int((e **((e**((log(x)**2 + x**4)/x)*e**11*x + x**(2*x)*e**(x**2)*log(x)**2 + x** (2*x)*e**(x**2)*x**4)/(x**(2*x)*e**(x**2)*x))*log(x))/(x**(2*x)*e**(x**2)) ,x) - 2*int((e**((e**((log(x)**2 + x**4)/x)*e**11*x + x**(2*x)*e**(x**2)*l og(x)**2 + x**(2*x)*e**(x**2)*x**4)/(x**(2*x)*e**(x**2)*x))*x)/(x**(2*x)*e **(x**2)),x))