Integrand size = 178, antiderivative size = 33 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {-4 x+\log \left (e^{e^{x^2 (-5+\log (x))}}+\frac {6}{x}\right )}{x-x^2} \]
Time = 0.32 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 x-\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x) x} \]
Integrate[(-6 + 6*x - 24*x^2 + E^E^(-5*x^2 + x^2*Log[x])*(-4*x^3 + E^(-5*x ^2 + x^2*Log[x])*(-9*x^3 + 9*x^4 + (2*x^3 - 2*x^4)*Log[x])) + (-6 + 12*x + E^E^(-5*x^2 + x^2*Log[x])*(-x + 2*x^2))*Log[(6 + E^E^(-5*x^2 + x^2*Log[x] )*x)/x])/(6*x^2 - 12*x^3 + 6*x^4 + E^E^(-5*x^2 + x^2*Log[x])*(x^3 - 2*x^4 + x^5)),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.
\(\displaystyle \int \frac {-24 x^2+\left (\left (2 x^2-x\right ) e^{e^{x^2 \log (x)-5 x^2}}+12 x-6\right ) \log \left (\frac {x e^{e^{x^2 \log (x)-5 x^2}}+6}{x}\right )+e^{e^{x^2 \log (x)-5 x^2}} \left (e^{x^2 \log (x)-5 x^2} \left (9 x^4-9 x^3+\left (2 x^3-2 x^4\right ) \log (x)\right )-4 x^3\right )+6 x-6}{6 x^4-12 x^3+6 x^2+\left (x^5-2 x^4+x^3\right ) e^{e^{x^2 \log (x)-5 x^2}}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-24 x^2+\left (\left (2 x^2-x\right ) e^{e^{x^2 \log (x)-5 x^2}}+12 x-6\right ) \log \left (\frac {x e^{e^{x^2 \log (x)-5 x^2}}+6}{x}\right )+e^{e^{x^2 \log (x)-5 x^2}} \left (e^{x^2 \log (x)-5 x^2} \left (9 x^4-9 x^3+\left (2 x^3-2 x^4\right ) \log (x)\right )-4 x^3\right )+6 x-6}{(1-x)^2 x^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 e^{e^{-5 x^2} x^{x^2}} x}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}-\frac {24}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}+\frac {6}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right ) x}-\frac {6}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right ) x^2}+\frac {(2 x-1) \log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(x-1)^2 x^2}-\frac {e^{e^{-5 x^2} x^{x^2}-5 x^2} x^{x^2+1} (2 \log (x)-9)}{(x-1) \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -24 \int \frac {1}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(x-1)^2 \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(x-1) \left (e^{e^{-5 x^2} x^{x^2}} x+6\right )}dx+\frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}\) |
Int[(-6 + 6*x - 24*x^2 + E^E^(-5*x^2 + x^2*Log[x])*(-4*x^3 + E^(-5*x^2 + x ^2*Log[x])*(-9*x^3 + 9*x^4 + (2*x^3 - 2*x^4)*Log[x])) + (-6 + 12*x + E^E^( -5*x^2 + x^2*Log[x])*(-x + 2*x^2))*Log[(6 + E^E^(-5*x^2 + x^2*Log[x])*x)/x ])/(6*x^2 - 12*x^3 + 6*x^4 + E^E^(-5*x^2 + x^2*Log[x])*(x^3 - 2*x^4 + x^5) ),x]
3.13.32.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.73
\[-\frac {\ln \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x \left (-1+x \right )}+\frac {-i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )+i \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}} x +6\right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+8 x +2 \ln \left (x \right )}{2 x \left (-1+x \right )}\]
int((((2*x^2-x)*exp(exp(x^2*ln(x)-5*x^2))+12*x-6)*ln((x*exp(exp(x^2*ln(x)- 5*x^2))+6)/x)+(((-2*x^4+2*x^3)*ln(x)+9*x^4-9*x^3)*exp(x^2*ln(x)-5*x^2)-4*x ^3)*exp(exp(x^2*ln(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*exp(exp(x^2*l n(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x)
-1/x/(-1+x)*ln(exp(x^(x^2)*exp(-5*x^2))*x+6)+1/2*(-I*Pi*csgn(I*(exp(x^(x^2 )*exp(-5*x^2))*x+6))*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))^2+I*Pi*csgn( I*(exp(x^(x^2)*exp(-5*x^2))*x+6))*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6)) *csgn(I/x)+I*Pi*csgn(I/x*(exp(x^(x^2)*exp(-5*x^2))*x+6))^3-I*Pi*csgn(I/x*( exp(x^(x^2)*exp(-5*x^2))*x+6))^2*csgn(I/x)+8*x+2*ln(x))/x/(-1+x)
Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left (\frac {x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6}{x}\right )}{x^{2} - x} \]
integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^ 2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x) -5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*e xp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm=\
Time = 12.50 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=- \frac {\log {\left (\frac {x e^{e^{x^{2} \log {\left (x \right )} - 5 x^{2}}} + 6}{x} \right )}}{x^{2} - x} + \frac {4}{x - 1} \]
integrate((((2*x**2-x)*exp(exp(x**2*ln(x)-5*x**2))+12*x-6)*ln((x*exp(exp(x **2*ln(x)-5*x**2))+6)/x)+(((-2*x**4+2*x**3)*ln(x)+9*x**4-9*x**3)*exp(x**2* ln(x)-5*x**2)-4*x**3)*exp(exp(x**2*ln(x)-5*x**2))-24*x**2+6*x-6)/((x**5-2* x**4+x**3)*exp(exp(x**2*ln(x)-5*x**2))+6*x**4-12*x**3+6*x**2),x)
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left (x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6\right ) + \log \left (x\right )}{x^{2} - x} \]
integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^ 2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x) -5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*e xp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (33) = 66\).
Time = 1.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.36 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4 \, x - \log \left ({\left (x e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2} + e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6 \, e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} e^{\left (-x^{2} \log \left (x\right ) + 5 \, x^{2}\right )}\right ) + \log \left (x\right )}{x^{2} - x} \]
integrate((((2*x^2-x)*exp(exp(x^2*log(x)-5*x^2))+12*x-6)*log((x*exp(exp(x^ 2*log(x)-5*x^2))+6)/x)+(((-2*x^4+2*x^3)*log(x)+9*x^4-9*x^3)*exp(x^2*log(x) -5*x^2)-4*x^3)*exp(exp(x^2*log(x)-5*x^2))-24*x^2+6*x-6)/((x^5-2*x^4+x^3)*e xp(exp(x^2*log(x)-5*x^2))+6*x^4-12*x^3+6*x^2),x, algorithm=\
(4*x - log((x*e^(x^2*log(x) - 5*x^2 + e^(x^2*log(x) - 5*x^2)) + 6*e^(x^2*l og(x) - 5*x^2))*e^(-x^2*log(x) + 5*x^2)) + log(x))/(x^2 - x)
Time = 13.81 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+x^5\right )} \, dx=\frac {4\,x-\ln \left (\frac {x\,{\mathrm {e}}^{x^{x^2}\,{\mathrm {e}}^{-5\,x^2}}+6}{x}\right )}{x\,\left (x-1\right )} \]
int(-(log((x*exp(exp(x^2*log(x) - 5*x^2)) + 6)/x)*(exp(exp(x^2*log(x) - 5* x^2))*(x - 2*x^2) - 12*x + 6) - exp(exp(x^2*log(x) - 5*x^2))*(exp(x^2*log( x) - 5*x^2)*(log(x)*(2*x^3 - 2*x^4) - 9*x^3 + 9*x^4) - 4*x^3) - 6*x + 24*x ^2 + 6)/(exp(exp(x^2*log(x) - 5*x^2))*(x^3 - 2*x^4 + x^5) + 6*x^2 - 12*x^3 + 6*x^4),x)