Integrand size = 156, antiderivative size = 29 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=5 e^{(2+x+\log (x)) \left (2 x+\frac {x}{\log \left (\frac {1}{x (4+\log (x))}\right )}\right )} \] Output:
5*exp((2*x+x/ln(1/(ln(x)+4)/x))*(x+ln(x)+2))
Time = 0.12 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=5 e^{x \left (4+2 x+\frac {2+x+\log (x)}{\log \left (\frac {1}{4 x+x \log (x)}\right )}\right )} x^{2 x} \] Input:
Integrate[(E^((2*x + x^2 + x*Log[x] + (4*x + 2*x^2 + 2*x*Log[x])*Log[(4*x + x*Log[x])^(-1)])/Log[(4*x + x*Log[x])^(-1)])*(50 + 25*x + (35 + 5*x)*Log [x] + 5*Log[x]^2 + (60 + 40*x + (35 + 10*x)*Log[x] + 5*Log[x]^2)*Log[(4*x + x*Log[x])^(-1)] + (120 + 80*x + (70 + 20*x)*Log[x] + 10*Log[x]^2)*Log[(4 *x + x*Log[x])^(-1)]^2))/((4 + Log[x])*Log[(4*x + x*Log[x])^(-1)]^2),x]
Output:
5*E^(x*(4 + 2*x + (2 + x + Log[x])/Log[(4*x + x*Log[x])^(-1)]))*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.
| \(\displaystyle \int \frac {\left (25 x+5 \log ^2(x)+\left (80 x+10 \log ^2(x)+(20 x+70) \log (x)+120\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )+\left (40 x+5 \log ^2(x)+(10 x+35) \log (x)+60\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+(5 x+35) \log (x)+50\right ) \exp \left (\frac {x^2+\left (2 x^2+4 x+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+2 x+x \log (x)}{\log \left (\frac {1}{4 x+x \log (x)}\right )}\right )}{(\log (x)+4) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (25 x+5 \log ^2(x)+\left (80 x+10 \log ^2(x)+(20 x+70) \log (x)+120\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )+\left (40 x+5 \log ^2(x)+(10 x+35) \log (x)+60\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+(5 x+35) \log (x)+50\right ) \exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{4 x+x \log (x)}\right )+1\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}\right )}{(\log (x)+4) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {5 (\log (x)+5) (x+\log (x)+2) \exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{4 x+x \log (x)}\right )+1\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}\right )}{(\log (x)+4) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )}+10 (2 x+\log (x)+3) \exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{4 x+x \log (x)}\right )+1\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}\right )+\frac {5 (2 x+\log (x)+3) \exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{4 x+x \log (x)}\right )+1\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 50 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right )}{(\log (x)+4) \log ^2\left (\frac {1}{\log (x) x+4 x}\right )}dx+25 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) x}{(\log (x)+4) \log ^2\left (\frac {1}{\log (x) x+4 x}\right )}dx+35 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) \log (x)}{(\log (x)+4) \log ^2\left (\frac {1}{\log (x) x+4 x}\right )}dx+5 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) x \log (x)}{(\log (x)+4) \log ^2\left (\frac {1}{\log (x) x+4 x}\right )}dx+5 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) \log ^2(x)}{(\log (x)+4) \log ^2\left (\frac {1}{\log (x) x+4 x}\right )}dx+30 \int \exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right )dx+20 \int \exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) xdx+10 \int \exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) \log (x)dx+15 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}dx+10 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) x}{\log \left (\frac {1}{\log (x) x+4 x}\right )}dx+5 \int \frac {\exp \left (\frac {x (x+\log (x)+2) \left (2 \log \left (\frac {1}{\log (x) x+4 x}\right )+1\right )}{\log \left (\frac {1}{\log (x) x+4 x}\right )}\right ) \log (x)}{\log \left (\frac {1}{\log (x) x+4 x}\right )}dx\) |
Input:
Int[(E^((2*x + x^2 + x*Log[x] + (4*x + 2*x^2 + 2*x*Log[x])*Log[(4*x + x*Lo g[x])^(-1)])/Log[(4*x + x*Log[x])^(-1)])*(50 + 25*x + (35 + 5*x)*Log[x] + 5*Log[x]^2 + (60 + 40*x + (35 + 10*x)*Log[x] + 5*Log[x]^2)*Log[(4*x + x*Lo g[x])^(-1)] + (120 + 80*x + (70 + 20*x)*Log[x] + 10*Log[x]^2)*Log[(4*x + x *Log[x])^(-1)]^2))/((4 + Log[x])*Log[(4*x + x*Log[x])^(-1)]^2),x]
Output:
$Aborted
Time = 2.16 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90
| method | result | size |
| parallelrisch | \(5 \,{\mathrm e}^{\frac {\left (2 x \ln \left (x \right )+2 x^{2}+4 x \right ) \ln \left (\frac {1}{\left (\ln \left (x \right )+4\right ) x}\right )+x \ln \left (x \right )+x^{2}+2 x}{\ln \left (\frac {1}{\left (\ln \left (x \right )+4\right ) x}\right )}}\) | \(55\) |
| risch | \(5 \,{\mathrm e}^{\frac {2 x \left (x +\ln \left (x \right )+2\right ) \left (i \pi \operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+4}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+4}\right )+2 \ln \left (x \right )+2 \ln \left (\ln \left (x \right )+4\right )-1\right )}{i \pi \operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right )^{2} \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+4}\right )+i \pi \,\operatorname {csgn}\left (\frac {i}{x \left (\ln \left (x \right )+4\right )}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+4}\right )+2 \ln \left (x \right )+2 \ln \left (\ln \left (x \right )+4\right )}}\) | \(255\) |
Input:
int(((10*ln(x)^2+(20*x+70)*ln(x)+80*x+120)*ln(1/(x*ln(x)+4*x))^2+(5*ln(x)^ 2+(10*x+35)*ln(x)+40*x+60)*ln(1/(x*ln(x)+4*x))+5*ln(x)^2+(5*x+35)*ln(x)+25 *x+50)*exp(((2*x*ln(x)+2*x^2+4*x)*ln(1/(x*ln(x)+4*x))+x*ln(x)+x^2+2*x)/ln( 1/(x*ln(x)+4*x)))/(ln(x)+4)/ln(1/(x*ln(x)+4*x))^2,x,method=_RETURNVERBOSE)
Output:
5*exp(((2*x*ln(x)+2*x^2+4*x)*ln(1/(ln(x)+4)/x)+x*ln(x)+x^2+2*x)/ln(1/(ln(x )+4)/x))
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=5 \, e^{\left (\frac {x^{2} + x \log \left (x\right ) + 2 \, {\left (x^{2} + x \log \left (x\right ) + 2 \, x\right )} \log \left (\frac {1}{x \log \left (x\right ) + 4 \, x}\right ) + 2 \, x}{\log \left (\frac {1}{x \log \left (x\right ) + 4 \, x}\right )}\right )} \] Input:
integrate(((10*log(x)^2+(20*x+70)*log(x)+80*x+120)*log(1/(x*log(x)+4*x))^2 +(5*log(x)^2+(10*x+35)*log(x)+40*x+60)*log(1/(x*log(x)+4*x))+5*log(x)^2+(5 *x+35)*log(x)+25*x+50)*exp(((2*x*log(x)+2*x^2+4*x)*log(1/(x*log(x)+4*x))+x *log(x)+x^2+2*x)/log(1/(x*log(x)+4*x)))/(log(x)+4)/log(1/(x*log(x)+4*x))^2 ,x, algorithm="fricas")
Output:
5*e^((x^2 + x*log(x) + 2*(x^2 + x*log(x) + 2*x)*log(1/(x*log(x) + 4*x)) + 2*x)/log(1/(x*log(x) + 4*x)))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.71 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=5 e^{\frac {x^{2} + x \log {\left (x \right )} + 2 x + \left (2 x^{2} + 2 x \log {\left (x \right )} + 4 x\right ) \log {\left (\frac {1}{x \log {\left (x \right )} + 4 x} \right )}}{\log {\left (\frac {1}{x \log {\left (x \right )} + 4 x} \right )}}} \] Input:
integrate(((10*ln(x)**2+(20*x+70)*ln(x)+80*x+120)*ln(1/(x*ln(x)+4*x))**2+( 5*ln(x)**2+(10*x+35)*ln(x)+40*x+60)*ln(1/(x*ln(x)+4*x))+5*ln(x)**2+(5*x+35 )*ln(x)+25*x+50)*exp(((2*x*ln(x)+2*x**2+4*x)*ln(1/(x*ln(x)+4*x))+x*ln(x)+x **2+2*x)/ln(1/(x*ln(x)+4*x)))/(ln(x)+4)/ln(1/(x*ln(x)+4*x))**2,x)
Output:
5*exp((x**2 + x*log(x) + 2*x + (2*x**2 + 2*x*log(x) + 4*x)*log(1/(x*log(x) + 4*x)))/log(1/(x*log(x) + 4*x)))
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (28) = 56\).
Time = 0.20 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=5 \, e^{\left (2 \, x^{2} + 2 \, x \log \left (x\right ) + 4 \, x - \frac {x^{2}}{\log \left (x\right ) + \log \left (\log \left (x\right ) + 4\right )} - \frac {x \log \left (x\right )}{\log \left (x\right ) + \log \left (\log \left (x\right ) + 4\right )} - \frac {2 \, x}{\log \left (x\right ) + \log \left (\log \left (x\right ) + 4\right )}\right )} \] Input:
integrate(((10*log(x)^2+(20*x+70)*log(x)+80*x+120)*log(1/(x*log(x)+4*x))^2 +(5*log(x)^2+(10*x+35)*log(x)+40*x+60)*log(1/(x*log(x)+4*x))+5*log(x)^2+(5 *x+35)*log(x)+25*x+50)*exp(((2*x*log(x)+2*x^2+4*x)*log(1/(x*log(x)+4*x))+x *log(x)+x^2+2*x)/log(1/(x*log(x)+4*x)))/(log(x)+4)/log(1/(x*log(x)+4*x))^2 ,x, algorithm="maxima")
Output:
5*e^(2*x^2 + 2*x*log(x) + 4*x - x^2/(log(x) + log(log(x) + 4)) - x*log(x)/ (log(x) + log(log(x) + 4)) - 2*x/(log(x) + log(log(x) + 4)))
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (28) = 56\).
Time = 0.45 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.17 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=5 \, e^{\left (2 \, x^{2} + 2 \, x \log \left (x\right ) + 4 \, x - \frac {x^{2}}{\log \left (x \log \left (x\right ) + 4 \, x\right )} - \frac {x \log \left (x\right )}{\log \left (x \log \left (x\right ) + 4 \, x\right )} - \frac {2 \, x}{\log \left (x \log \left (x\right ) + 4 \, x\right )}\right )} \] Input:
integrate(((10*log(x)^2+(20*x+70)*log(x)+80*x+120)*log(1/(x*log(x)+4*x))^2 +(5*log(x)^2+(10*x+35)*log(x)+40*x+60)*log(1/(x*log(x)+4*x))+5*log(x)^2+(5 *x+35)*log(x)+25*x+50)*exp(((2*x*log(x)+2*x^2+4*x)*log(1/(x*log(x)+4*x))+x *log(x)+x^2+2*x)/log(1/(x*log(x)+4*x)))/(log(x)+4)/log(1/(x*log(x)+4*x))^2 ,x, algorithm="giac")
Output:
5*e^(2*x^2 + 2*x*log(x) + 4*x - x^2/log(x*log(x) + 4*x) - x*log(x)/log(x*l og(x) + 4*x) - 2*x/log(x*log(x) + 4*x))
Time = 2.67 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.38 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=5\,x^{2\,x}\,x^{\frac {x}{\ln \left (\frac {1}{4\,x+x\,\ln \left (x\right )}\right )}}\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{\frac {2\,x}{\ln \left (\frac {1}{4\,x+x\,\ln \left (x\right )}\right )}}\,{\mathrm {e}}^{2\,x^2}\,{\mathrm {e}}^{\frac {x^2}{\ln \left (\frac {1}{4\,x+x\,\ln \left (x\right )}\right )}} \] Input:
int((exp((2*x + log(1/(4*x + x*log(x)))*(4*x + 2*x*log(x) + 2*x^2) + x*log (x) + x^2)/log(1/(4*x + x*log(x))))*(25*x + 5*log(x)^2 + log(1/(4*x + x*lo g(x)))*(40*x + 5*log(x)^2 + log(x)*(10*x + 35) + 60) + log(x)*(5*x + 35) + log(1/(4*x + x*log(x)))^2*(80*x + 10*log(x)^2 + log(x)*(20*x + 70) + 120) + 50))/(log(1/(4*x + x*log(x)))^2*(log(x) + 4)),x)
Output:
5*x^(2*x)*x^(x/log(1/(4*x + x*log(x))))*exp(4*x)*exp((2*x)/log(1/(4*x + x* log(x))))*exp(2*x^2)*exp(x^2/log(1/(4*x + x*log(x))))
Time = 0.41 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {2 x+x^2+x \log (x)+\left (4 x+2 x^2+2 x \log (x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )}{\log \left (\frac {1}{4 x+x \log (x)}\right )}} \left (50+25 x+(35+5 x) \log (x)+5 \log ^2(x)+\left (60+40 x+(35+10 x) \log (x)+5 \log ^2(x)\right ) \log \left (\frac {1}{4 x+x \log (x)}\right )+\left (120+80 x+(70+20 x) \log (x)+10 \log ^2(x)\right ) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )\right )}{(4+\log (x)) \log ^2\left (\frac {1}{4 x+x \log (x)}\right )} \, dx=\frac {5 x^{2 x} e^{2 x^{2}+4 x}}{e^{\frac {\mathrm {log}\left (x \right ) x +x^{2}+2 x}{\mathrm {log}\left (\mathrm {log}\left (x \right ) x +4 x \right )}}} \] Input:
int(((10*log(x)^2+(20*x+70)*log(x)+80*x+120)*log(1/(x*log(x)+4*x))^2+(5*lo g(x)^2+(10*x+35)*log(x)+40*x+60)*log(1/(x*log(x)+4*x))+5*log(x)^2+(5*x+35) *log(x)+25*x+50)*exp(((2*x*log(x)+2*x^2+4*x)*log(1/(x*log(x)+4*x))+x*log(x )+x^2+2*x)/log(1/(x*log(x)+4*x)))/(log(x)+4)/log(1/(x*log(x)+4*x))^2,x)
Output:
(5*x**(2*x)*e**(2*x**2 + 4*x))/e**((log(x)*x + x**2 + 2*x)/log(log(x)*x + 4*x))