Integrand size = 115, antiderivative size = 26 \[ \int \frac {-1+e^{e^{e^{7+x^2}}}-\log (x)+2 e^{7+e^{7+x^2}+x^2} x^2 \log (x)+\left (-e^{e^{e^{7+x^2}}}+\log (x)\right ) \log \left (e^{-e^{e^{7+x^2}}} \left (e^{e^{e^{7+x^2}}} x-x \log (x)\right )\right )}{e^{e^{e^{7+x^2}}} x^2-x^2 \log (x)} \, dx=\frac {\log \left (x \left (1-e^{-e^{e^{7+x^2}}} \log (x)\right )\right )}{x} \]
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-1+e^{e^{e^{7+x^2}}}-\log (x)+2 e^{7+e^{7+x^2}+x^2} x^2 \log (x)+\left (-e^{e^{e^{7+x^2}}}+\log (x)\right ) \log \left (e^{-e^{e^{7+x^2}}} \left (e^{e^{e^{7+x^2}}} x-x \log (x)\right )\right )}{e^{e^{e^{7+x^2}}} x^2-x^2 \log (x)} \, dx=\frac {\log \left (x-e^{-e^{e^{7+x^2}}} x \log (x)\right )}{x} \]
Integrate[(-1 + E^E^E^(7 + x^2) - Log[x] + 2*E^(7 + E^(7 + x^2) + x^2)*x^2 *Log[x] + (-E^E^E^(7 + x^2) + Log[x])*Log[(E^E^E^(7 + x^2)*x - x*Log[x])/E ^E^E^(7 + x^2)])/(E^E^E^(7 + x^2)*x^2 - x^2*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.
| \(\displaystyle \int \frac {e^{e^{e^{x^2+7}}}+2 e^{x^2+e^{x^2+7}+7} x^2 \log (x)+\left (\log (x)-e^{e^{e^{x^2+7}}}\right ) \log \left (e^{-e^{e^{x^2+7}}} \left (e^{e^{e^{x^2+7}}} x-x \log (x)\right )\right )-\log (x)-1}{e^{e^{e^{x^2+7}}} x^2-x^2 \log (x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^{x^2+e^{x^2+7}+7} \log (x)}{e^{e^{e^{x^2+7}}}-\log (x)}-\frac {-e^{e^{e^{x^2+7}}}-\log \left (x-e^{-e^{e^{x^2+7}}} x \log (x)\right ) \log (x)+e^{e^{e^{x^2+7}}} \log \left (x-e^{-e^{e^{x^2+7}}} x \log (x)\right )+\log (x)+1}{x^2 \left (e^{e^{e^{x^2+7}}}-\log (x)\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\int \frac {1}{x^2 \left (e^{e^{e^{x^2+7}}}-\log (x)\right )}dx+2 \int \frac {e^{x^2+e^{x^2+7}+7} \log (x)}{e^{e^{e^{x^2+7}}}-\log (x)}dx-\int \frac {\log \left (x-e^{-e^{e^{x^2+7}}} x \log (x)\right )}{x^2}dx-\frac {1}{x}\) |
Int[(-1 + E^E^E^(7 + x^2) - Log[x] + 2*E^(7 + E^(7 + x^2) + x^2)*x^2*Log[x ] + (-E^E^E^(7 + x^2) + Log[x])*Log[(E^E^E^(7 + x^2)*x - x*Log[x])/E^E^E^( 7 + x^2)])/(E^E^E^(7 + x^2)*x^2 - x^2*Log[x]),x]
3.20.3.3.1 Defintions of rubi rules used
Time = 12.71 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
| method | result | size |
| parallelrisch | \(\frac {\ln \left (-x \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right )}{x}\) | \(32\) |
| risch | \(-\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right )}{x}+\frac {-2 i \pi {\operatorname {csgn}\left (i x \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}} \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i x \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right )+i \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i x \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}} \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right )+i \pi \,\operatorname {csgn}\left (i \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}} \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right )^{2}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}} \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right )^{2}-i \pi \operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}} \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right )^{3}+i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}} \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )\right ) {\operatorname {csgn}\left (i x \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right )}^{2}+i \pi {\operatorname {csgn}\left (i x \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}\right )}^{3}+2 i \pi +2 \ln \left (x \right )+2 \ln \left (-{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x^{2}+7}}}+\ln \left (x \right )\right )}{2 x}\) | \(470\) |
int(((-exp(exp(exp(x^2+7)))+ln(x))*ln((x*exp(exp(exp(x^2+7)))-x*ln(x))/exp (exp(exp(x^2+7))))+exp(exp(exp(x^2+7)))+2*x^2*exp(x^2+7)*ln(x)*exp(exp(x^2 +7))-ln(x)-1)/(x^2*exp(exp(exp(x^2+7)))-x^2*ln(x)),x,method=_RETURNVERBOSE )
Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {-1+e^{e^{e^{7+x^2}}}-\log (x)+2 e^{7+e^{7+x^2}+x^2} x^2 \log (x)+\left (-e^{e^{e^{7+x^2}}}+\log (x)\right ) \log \left (e^{-e^{e^{7+x^2}}} \left (e^{e^{e^{7+x^2}}} x-x \log (x)\right )\right )}{e^{e^{e^{7+x^2}}} x^2-x^2 \log (x)} \, dx=\frac {\log \left ({\left (x e^{\left (e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )} - x \log \left (x\right )\right )} e^{\left (-e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )}\right )}{x} \]
integrate(((-exp(exp(exp(x^2+7)))+log(x))*log((x*exp(exp(exp(x^2+7)))-x*lo g(x))/exp(exp(exp(x^2+7))))+exp(exp(exp(x^2+7)))+2*x^2*exp(x^2+7)*log(x)*e xp(exp(x^2+7))-log(x)-1)/(x^2*exp(exp(exp(x^2+7)))-x^2*log(x)),x, algorith m=\
Time = 29.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-1+e^{e^{e^{7+x^2}}}-\log (x)+2 e^{7+e^{7+x^2}+x^2} x^2 \log (x)+\left (-e^{e^{e^{7+x^2}}}+\log (x)\right ) \log \left (e^{-e^{e^{7+x^2}}} \left (e^{e^{e^{7+x^2}}} x-x \log (x)\right )\right )}{e^{e^{e^{7+x^2}}} x^2-x^2 \log (x)} \, dx=\frac {\log {\left (\left (x e^{e^{e^{x^{2} + 7}}} - x \log {\left (x \right )}\right ) e^{- e^{e^{x^{2} + 7}}} \right )}}{x} \]
integrate(((-exp(exp(exp(x**2+7)))+ln(x))*ln((x*exp(exp(exp(x**2+7)))-x*ln (x))/exp(exp(exp(x**2+7))))+exp(exp(exp(x**2+7)))+2*x**2*exp(x**2+7)*ln(x) *exp(exp(x**2+7))-ln(x)-1)/(x**2*exp(exp(exp(x**2+7)))-x**2*ln(x)),x)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-1+e^{e^{e^{7+x^2}}}-\log (x)+2 e^{7+e^{7+x^2}+x^2} x^2 \log (x)+\left (-e^{e^{e^{7+x^2}}}+\log (x)\right ) \log \left (e^{-e^{e^{7+x^2}}} \left (e^{e^{e^{7+x^2}}} x-x \log (x)\right )\right )}{e^{e^{e^{7+x^2}}} x^2-x^2 \log (x)} \, dx=-\frac {e^{\left (e^{\left (x^{2} + 7\right )}\right )} - \log \left (x\right ) - \log \left (e^{\left (e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )} - \log \left (x\right )\right )}{x} \]
integrate(((-exp(exp(exp(x^2+7)))+log(x))*log((x*exp(exp(exp(x^2+7)))-x*lo g(x))/exp(exp(exp(x^2+7))))+exp(exp(exp(x^2+7)))+2*x^2*exp(x^2+7)*log(x)*e xp(exp(x^2+7))-log(x)-1)/(x^2*exp(exp(exp(x^2+7)))-x^2*log(x)),x, algorith m=\
\[ \int \frac {-1+e^{e^{e^{7+x^2}}}-\log (x)+2 e^{7+e^{7+x^2}+x^2} x^2 \log (x)+\left (-e^{e^{e^{7+x^2}}}+\log (x)\right ) \log \left (e^{-e^{e^{7+x^2}}} \left (e^{e^{e^{7+x^2}}} x-x \log (x)\right )\right )}{e^{e^{e^{7+x^2}}} x^2-x^2 \log (x)} \, dx=\int { \frac {2 \, x^{2} e^{\left (x^{2} + e^{\left (x^{2} + 7\right )} + 7\right )} \log \left (x\right ) - {\left (e^{\left (e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )} - \log \left (x\right )\right )} \log \left ({\left (x e^{\left (e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )} - x \log \left (x\right )\right )} e^{\left (-e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )}\right ) + e^{\left (e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )} - \log \left (x\right ) - 1}{x^{2} e^{\left (e^{\left (e^{\left (x^{2} + 7\right )}\right )}\right )} - x^{2} \log \left (x\right )} \,d x } \]
integrate(((-exp(exp(exp(x^2+7)))+log(x))*log((x*exp(exp(exp(x^2+7)))-x*lo g(x))/exp(exp(exp(x^2+7))))+exp(exp(exp(x^2+7)))+2*x^2*exp(x^2+7)*log(x)*e xp(exp(x^2+7))-log(x)-1)/(x^2*exp(exp(exp(x^2+7)))-x^2*log(x)),x, algorith m=\
integrate((2*x^2*e^(x^2 + e^(x^2 + 7) + 7)*log(x) - (e^(e^(e^(x^2 + 7))) - log(x))*log((x*e^(e^(e^(x^2 + 7))) - x*log(x))*e^(-e^(e^(x^2 + 7)))) + e^ (e^(e^(x^2 + 7))) - log(x) - 1)/(x^2*e^(e^(e^(x^2 + 7))) - x^2*log(x)), x)
Time = 12.47 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-1+e^{e^{e^{7+x^2}}}-\log (x)+2 e^{7+e^{7+x^2}+x^2} x^2 \log (x)+\left (-e^{e^{e^{7+x^2}}}+\log (x)\right ) \log \left (e^{-e^{e^{7+x^2}}} \left (e^{e^{e^{7+x^2}}} x-x \log (x)\right )\right )}{e^{e^{e^{7+x^2}}} x^2-x^2 \log (x)} \, dx=\frac {\ln \left (x-x\,{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^{x^2}\,{\mathrm {e}}^7}}\,\ln \left (x\right )\right )}{x} \]