Integrand size = 162, antiderivative size = 22 \[ \int \frac {e^{\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}} \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+x)} \, dx=e^{\frac {\log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2}} \]
Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}} \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+x)} \, dx=e^{\frac {\log ^2\left (2-\frac {x^2}{\log (-20+x)}\right )}{x^2}} \]
Integrate[(E^(Log[(-x^2 + 2*Log[-20 + x])/Log[-20 + x]]^2/x^2)*((2*x^3 + ( 80*x^2 - 4*x^3)*Log[-20 + x])*Log[(-x^2 + 2*Log[-20 + x])/Log[-20 + x]] + ((-40*x^2 + 2*x^3)*Log[-20 + x] + (80 - 4*x)*Log[-20 + x]^2)*Log[(-x^2 + 2 *Log[-20 + x])/Log[-20 + x]]^2))/((20*x^5 - x^6)*Log[-20 + x] + (-40*x^3 + 2*x^4)*Log[-20 + 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.
| \(\displaystyle \int \frac {e^{\frac {\log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )}{x^2}} \left (\left (\left (2 x^3-40 x^2\right ) \log (x-20)+(80-4 x) \log ^2(x-20)\right ) \log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )+\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (x-20)\right ) \log \left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )\right )}{\left (20 x^5-x^6\right ) \log (x-20)+\left (2 x^4-40 x^3\right ) \log ^2(x-20)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {\log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )}{x^2}} \left (\left (\left (2 x^3-40 x^2\right ) \log (x-20)+(80-4 x) \log ^2(x-20)\right ) \log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )+\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (x-20)\right ) \log \left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )\right )}{(20-x) x^3 \left (x^2-2 \log (x-20)\right ) \log (x-20)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 e^{\frac {\log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )}{x^2}} (-x+2 x \log (x-20)-40 \log (x-20)) \log \left (2-\frac {x^2}{\log (x-20)}\right )}{(x-20) x \left (x^2-2 \log (x-20)\right ) \log (x-20)}-\frac {2 e^{\frac {\log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )}{x^2}} \log ^2\left (2-\frac {x^2}{\log (x-20)}\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \int \frac {e^{\frac {\log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )}{x^2}} \log \left (2-\frac {x^2}{\log (x-20)}\right )}{x \left (x^2-2 \log (x-20)\right )}dx-2 \int \frac {e^{\frac {\log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )}{x^2}} \log \left (2-\frac {x^2}{\log (x-20)}\right )}{(x-20) \left (x^2-2 \log (x-20)\right ) \log (x-20)}dx-2 \int \frac {e^{\frac {\log ^2\left (\frac {2 \log (x-20)-x^2}{\log (x-20)}\right )}{x^2}} \log ^2\left (2-\frac {x^2}{\log (x-20)}\right )}{x^3}dx\) |
Int[(E^(Log[(-x^2 + 2*Log[-20 + x])/Log[-20 + x]]^2/x^2)*((2*x^3 + (80*x^2 - 4*x^3)*Log[-20 + x])*Log[(-x^2 + 2*Log[-20 + x])/Log[-20 + x]] + ((-40* x^2 + 2*x^3)*Log[-20 + x] + (80 - 4*x)*Log[-20 + x]^2)*Log[(-x^2 + 2*Log[- 20 + x])/Log[-20 + x]]^2))/((20*x^5 - x^6)*Log[-20 + x] + (-40*x^3 + 2*x^4 )*Log[-20 + x]^2),x]
3.12.57.3.1 Defintions of rubi rules used
Time = 146.02 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\ln \left (\frac {2 \ln \left (x -20\right )-x^{2}}{\ln \left (x -20\right )}\right )^{2}}{x^{2}}}\) | \(28\) |
| risch | \({\mathrm e}^{\frac {{\left (-i \pi {\operatorname {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )}^{3}-i \pi {\operatorname {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )}^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x -20\right )}\right )-i \pi {\operatorname {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )}^{2} \operatorname {csgn}\left (i \left (-2 \ln \left (x -20\right )+x^{2}\right )\right )+i \pi \,\operatorname {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x -20\right )}\right ) \operatorname {csgn}\left (i \left (-2 \ln \left (x -20\right )+x^{2}\right )\right )+2 i \pi {\operatorname {csgn}\left (\frac {i \left (-2 \ln \left (x -20\right )+x^{2}\right )}{\ln \left (x -20\right )}\right )}^{2}-2 i \pi +2 \ln \left (\ln \left (x -20\right )\right )-2 \ln \left (-2 \ln \left (x -20\right )+x^{2}\right )\right )}^{2}}{4 x^{2}}}\) | \(210\) |
int((((-4*x+80)*ln(x-20)^2+(2*x^3-40*x^2)*ln(x-20))*ln((2*ln(x-20)-x^2)/ln (x-20))^2+((-4*x^3+80*x^2)*ln(x-20)+2*x^3)*ln((2*ln(x-20)-x^2)/ln(x-20)))* exp(ln((2*ln(x-20)-x^2)/ln(x-20))^2/x^2)/((2*x^4-40*x^3)*ln(x-20)^2+(-x^6+ 20*x^5)*ln(x-20)),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}} \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+x)} \, dx=e^{\left (\frac {\log \left (-\frac {x^{2} - 2 \, \log \left (x - 20\right )}{\log \left (x - 20\right )}\right )^{2}}{x^{2}}\right )} \]
integrate((((-4*x+80)*log(x-20)^2+(2*x^3-40*x^2)*log(x-20))*log((2*log(x-2 0)-x^2)/log(x-20))^2+((-4*x^3+80*x^2)*log(x-20)+2*x^3)*log((2*log(x-20)-x^ 2)/log(x-20)))*exp(log((2*log(x-20)-x^2)/log(x-20))^2/x^2)/((2*x^4-40*x^3) *log(x-20)^2+(-x^6+20*x^5)*log(x-20)),x, algorithm=\
Time = 2.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}} \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+x)} \, dx=e^{\frac {\log {\left (\frac {- x^{2} + 2 \log {\left (x - 20 \right )}}{\log {\left (x - 20 \right )}} \right )}^{2}}{x^{2}}} \]
integrate((((-4*x+80)*ln(x-20)**2+(2*x**3-40*x**2)*ln(x-20))*ln((2*ln(x-20 )-x**2)/ln(x-20))**2+((-4*x**3+80*x**2)*ln(x-20)+2*x**3)*ln((2*ln(x-20)-x* *2)/ln(x-20)))*exp(ln((2*ln(x-20)-x**2)/ln(x-20))**2/x**2)/((2*x**4-40*x** 3)*ln(x-20)**2+(-x**6+20*x**5)*ln(x-20)),x)
Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (21) = 42\).
Time = 0.38 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.50 \[ \int \frac {e^{\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}} \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+x)} \, dx=e^{\left (\frac {\log \left (-x^{2} + 2 \, \log \left (x - 20\right )\right )^{2}}{x^{2}} - \frac {2 \, \log \left (-x^{2} + 2 \, \log \left (x - 20\right )\right ) \log \left (\log \left (x - 20\right )\right )}{x^{2}} + \frac {\log \left (\log \left (x - 20\right )\right )^{2}}{x^{2}}\right )} \]
integrate((((-4*x+80)*log(x-20)^2+(2*x^3-40*x^2)*log(x-20))*log((2*log(x-2 0)-x^2)/log(x-20))^2+((-4*x^3+80*x^2)*log(x-20)+2*x^3)*log((2*log(x-20)-x^ 2)/log(x-20)))*exp(log((2*log(x-20)-x^2)/log(x-20))^2/x^2)/((2*x^4-40*x^3) *log(x-20)^2+(-x^6+20*x^5)*log(x-20)),x, algorithm=\
e^(log(-x^2 + 2*log(x - 20))^2/x^2 - 2*log(-x^2 + 2*log(x - 20))*log(log(x - 20))/x^2 + log(log(x - 20))^2/x^2)
Time = 5.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}} \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+x)} \, dx=e^{\left (\frac {\log \left (-\frac {x^{2}}{\log \left (x - 20\right )} + 2\right )^{2}}{x^{2}}\right )} \]
integrate((((-4*x+80)*log(x-20)^2+(2*x^3-40*x^2)*log(x-20))*log((2*log(x-2 0)-x^2)/log(x-20))^2+((-4*x^3+80*x^2)*log(x-20)+2*x^3)*log((2*log(x-20)-x^ 2)/log(x-20)))*exp(log((2*log(x-20)-x^2)/log(x-20))^2/x^2)/((2*x^4-40*x^3) *log(x-20)^2+(-x^6+20*x^5)*log(x-20)),x, algorithm=\
Time = 8.42 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {\log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )}{x^2}} \left (\left (2 x^3+\left (80 x^2-4 x^3\right ) \log (-20+x)\right ) \log \left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )+\left (\left (-40 x^2+2 x^3\right ) \log (-20+x)+(80-4 x) \log ^2(-20+x)\right ) \log ^2\left (\frac {-x^2+2 \log (-20+x)}{\log (-20+x)}\right )\right )}{\left (20 x^5-x^6\right ) \log (-20+x)+\left (-40 x^3+2 x^4\right ) \log ^2(-20+x)} \, dx={\mathrm {e}}^{\frac {{\ln \left (\frac {2\,\ln \left (x-20\right )-x^2}{\ln \left (x-20\right )}\right )}^2}{x^2}} \]
int((exp(log((2*log(x - 20) - x^2)/log(x - 20))^2/x^2)*(log((2*log(x - 20) - x^2)/log(x - 20))*(log(x - 20)*(80*x^2 - 4*x^3) + 2*x^3) - log((2*log(x - 20) - x^2)/log(x - 20))^2*(log(x - 20)^2*(4*x - 80) + log(x - 20)*(40*x ^2 - 2*x^3))))/(log(x - 20)*(20*x^5 - x^6) - log(x - 20)^2*(40*x^3 - 2*x^4 )),x)