Integrand size = 251, antiderivative size = 26 \[ \int \frac {e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right ) \left (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right )\right )} \, dx=\log \left (\log \left (20+\log \left (5-e^{-5+\left (x+\frac {5}{x^2 \log (x)}\right )^2}\right )\right )\right ) \]
Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right ) \left (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right )\right )} \, dx=\log \left (\log \left (20+\log \left (5-e^{-5+\frac {\left (5+x^3 \log (x)\right )^2}{x^4 \log ^2(x)}}\right )\right )\right ) \]
Integrate[(E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2 ))*(-50 + (-100 - 10*x^3)*Log[x] - 10*x^3*Log[x]^2 + 2*x^6*Log[x]^3))/((-1 00*x^5*Log[x]^3 + 20*E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^ 4*Log[x]^2))*x^5*Log[x]^3 + Log[5 - E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6 )*Log[x]^2)/(x^4*Log[x]^2))]*(-5*x^5*Log[x]^3 + E^((25 + 10*x^3*Log[x] + ( -5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))*x^5*Log[x]^3))*Log[20 + Log[5 - E^ ((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[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.
| \(\displaystyle \int \frac {\left (2 x^6 \log ^3(x)-10 x^3 \log ^2(x)+\left (-10 x^3-100\right ) \log (x)-50\right ) \exp \left (\frac {10 x^3 \log (x)+\left (x^6-5 x^4\right ) \log ^2(x)+25}{x^4 \log ^2(x)}\right )}{\left (20 x^5 \log ^3(x) \exp \left (\frac {10 x^3 \log (x)+\left (x^6-5 x^4\right ) \log ^2(x)+25}{x^4 \log ^2(x)}\right )+\log \left (5-\exp \left (\frac {10 x^3 \log (x)+\left (x^6-5 x^4\right ) \log ^2(x)+25}{x^4 \log ^2(x)}\right )\right ) \left (x^5 \log ^3(x) \exp \left (\frac {10 x^3 \log (x)+\left (x^6-5 x^4\right ) \log ^2(x)+25}{x^4 \log ^2(x)}\right )-5 x^5 \log ^3(x)\right )-100 x^5 \log ^3(x)\right ) \log \left (\log \left (5-\exp \left (\frac {10 x^3 \log (x)+\left (x^6-5 x^4\right ) \log ^2(x)+25}{x^4 \log ^2(x)}\right )\right )+20\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 \left (x^6 \left (-\log ^3(x)\right )+5 x^3 \log ^2(x)+5 x^3 \log (x)+50 \log (x)+25\right ) \exp \left (\frac {10 x^3 \log (x)+\left (x^6-5 x^4\right ) \log ^2(x)+25}{x^4 \log ^2(x)}+5\right )}{x^5 \left (5 e^5-e^{\frac {\left (x^3 \log (x)+5\right )^2}{x^4 \log ^2(x)}}\right ) \left (\log \left (5-e^{\frac {\left (x^3 \log (x)+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^3(x) \log \left (\log \left (5-\exp \left (\frac {10 x^3 \log (x)+\left (x^6-5 x^4\right ) \log ^2(x)+25}{x^4 \log ^2(x)}\right )\right )+20\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\exp \left (\frac {10 \log (x) x^3-\left (5 x^4-x^6\right ) \log ^2(x)+25}{x^4 \log ^2(x)}+5\right ) \left (-\log ^3(x) x^6+5 \log ^2(x) x^3+5 \log (x) x^3+50 \log (x)+25\right )}{\left (5 e^5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^3(x) \log \left (\log \left (5-e^{\frac {25-\left (5 x^4-x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^{\frac {10}{x \log ^2(x)}}\right )+20\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 2 \int \frac {e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}} \left (-\log ^3(x) x^6+5 \log ^2(x) x^3+5 \log (x) x^3+50 \log (x)+25\right )}{\left (5 e^5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^3(x) \log \left (\log \left (5-e^{\frac {25-\left (5 x^4-x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^{\frac {10}{x \log ^2(x)}}\right )+20\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}} x}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right )}-\frac {5 e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log (x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) x^2}-\frac {5 e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^2(x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) x^2}-\frac {50 e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^2(x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) x^5}-\frac {25 e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^3(x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (\int \frac {e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}} x}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right )}dx-50 \int \frac {e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^2(x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right )}dx-25 \int \frac {e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) x^5 \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^3(x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right )}dx-5 \int \frac {e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) x^2 \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log ^2(x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right )}dx-5 \int \frac {e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}}{\left (-5 e^5+e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}}\right ) x^2 \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right ) \log (x) \log \left (\log \left (5-e^{\frac {\left (\log (x) x^3+5\right )^2}{x^4 \log ^2(x)}-5}\right )+20\right )}dx\right )\) |
Int[(E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))*(-5 0 + (-100 - 10*x^3)*Log[x] - 10*x^3*Log[x]^2 + 2*x^6*Log[x]^3))/((-100*x^5 *Log[x]^3 + 20*E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[ x]^2))*x^5*Log[x]^3 + Log[5 - E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[ x]^2)/(x^4*Log[x]^2))]*(-5*x^5*Log[x]^3 + E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))*x^5*Log[x]^3))*Log[20 + Log[5 - E^((25 + 10*x^3*Log[x] + (-5*x^4 + x^6)*Log[x]^2)/(x^4*Log[x]^2))]]),x]
3.21.76.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73
\[\ln \left (\ln \left (\ln \left (-{\mathrm e}^{\frac {x^{6} \ln \left (x \right )^{2}-5 x^{4} \ln \left (x \right )^{2}+10 x^{3} \ln \left (x \right )+25}{x^{4} \ln \left (x \right )^{2}}}+5\right )+20\right )\right )\]
int((2*x^6*ln(x)^3-10*x^3*ln(x)^2+(-10*x^3-100)*ln(x)-50)*exp(((x^6-5*x^4) *ln(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)/((x^5*ln(x)^3*exp(((x^6-5*x^4)*ln(x )^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)-5*x^5*ln(x)^3)*ln(-exp(((x^6-5*x^4)*ln(x )^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)+5)+20*x^5*ln(x)^3*exp(((x^6-5*x^4)*ln(x) ^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)-100*x^5*ln(x)^3)/ln(ln(-exp(((x^6-5*x^4)* ln(x)^2+10*x^3*ln(x)+25)/x^4/ln(x)^2)+5)+20),x)
Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right ) \left (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right )\right )} \, dx=\log \left (\log \left (\log \left (-e^{\left (\frac {10 \, x^{3} \log \left (x\right ) + {\left (x^{6} - 5 \, x^{4}\right )} \log \left (x\right )^{2} + 25}{x^{4} \log \left (x\right )^{2}}\right )} + 5\right ) + 20\right )\right ) \]
integrate((2*x^6*log(x)^3-10*x^3*log(x)^2+(-10*x^3-100)*log(x)-50)*exp(((x ^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)/((x^5*log(x)^3*exp(((x^ 6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-5*x^5*log(x)^3)*log(-exp (((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20*x^5*log(x)^3* exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-100*x^5*log(x)^3 )/log(log(-exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20 ),x, algorithm=\
Timed out. \[ \int \frac {e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right ) \left (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right )\right )} \, dx=\text {Timed out} \]
integrate((2*x**6*ln(x)**3-10*x**3*ln(x)**2+(-10*x**3-100)*ln(x)-50)*exp(( (x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25)/x**4/ln(x)**2)/((x**5*ln(x)**3*ex p(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25)/x**4/ln(x)**2)-5*x**5*ln(x)**3 )*ln(-exp(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25)/x**4/ln(x)**2)+5)+20*x **5*ln(x)**3*exp(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25)/x**4/ln(x)**2)- 100*x**5*ln(x)**3)/ln(ln(-exp(((x**6-5*x**4)*ln(x)**2+10*x**3*ln(x)+25)/x* *4/ln(x)**2)+5)+20),x)
Time = 0.41 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right ) \left (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right )\right )} \, dx=\log \left (\log \left (\log \left (5 \, e^{5} - e^{\left (x^{2} + \frac {10}{x \log \left (x\right )} + \frac {25}{x^{4} \log \left (x\right )^{2}}\right )}\right ) + 15\right )\right ) \]
integrate((2*x^6*log(x)^3-10*x^3*log(x)^2+(-10*x^3-100)*log(x)-50)*exp(((x ^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)/((x^5*log(x)^3*exp(((x^ 6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-5*x^5*log(x)^3)*log(-exp (((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20*x^5*log(x)^3* exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-100*x^5*log(x)^3 )/log(log(-exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20 ),x, algorithm=\
Time = 0.61 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right ) \left (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right )\right )} \, dx=\log \left (\log \left (\log \left (-e^{\left (\frac {x^{6} \log \left (x\right )^{2} - 5 \, x^{4} \log \left (x\right )^{2} + 10 \, x^{3} \log \left (x\right ) + 25}{x^{4} \log \left (x\right )^{2}}\right )} + 5\right ) + 20\right )\right ) \]
integrate((2*x^6*log(x)^3-10*x^3*log(x)^2+(-10*x^3-100)*log(x)-50)*exp(((x ^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)/((x^5*log(x)^3*exp(((x^ 6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-5*x^5*log(x)^3)*log(-exp (((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20*x^5*log(x)^3* exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)-100*x^5*log(x)^3 )/log(log(-exp(((x^6-5*x^4)*log(x)^2+10*x^3*log(x)+25)/x^4/log(x)^2)+5)+20 ),x, algorithm=\
log(log(log(-e^((x^6*log(x)^2 - 5*x^4*log(x)^2 + 10*x^3*log(x) + 25)/(x^4* log(x)^2)) + 5) + 20))
Time = 12.87 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} \left (-50+\left (-100-10 x^3\right ) \log (x)-10 x^3 \log ^2(x)+2 x^6 \log ^3(x)\right )}{\left (-100 x^5 \log ^3(x)+20 e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right ) \left (-5 x^5 \log ^3(x)+e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}} x^5 \log ^3(x)\right )\right ) \log \left (20+\log \left (5-e^{\frac {25+10 x^3 \log (x)+\left (-5 x^4+x^6\right ) \log ^2(x)}{x^4 \log ^2(x)}}\right )\right )} \, dx=\ln \left (\ln \left (\ln \left (5-{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{\frac {10}{x\,\ln \left (x\right )}}\,{\mathrm {e}}^{\frac {25}{x^4\,{\ln \left (x\right )}^2}}\right )+20\right )\right ) \]
int((exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2))*(10 *x^3*log(x)^2 - 2*x^6*log(x)^3 + log(x)*(10*x^3 + 100) + 50))/(log(log(5 - exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2))) + 20)* (log(5 - exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2)) )*(5*x^5*log(x)^3 - x^5*exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/ (x^4*log(x)^2))*log(x)^3) + 100*x^5*log(x)^3 - 20*x^5*exp((10*x^3*log(x) - log(x)^2*(5*x^4 - x^6) + 25)/(x^4*log(x)^2))*log(x)^3)),x)