Integrand size = 136, antiderivative size = 24 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\frac {1}{x-e^{\frac {1}{30} x \left (3+\frac {-2+x}{\log (x)}\right )} x} \] Output:
1/(x-x*exp(1/30*x*(3+(-2+x)/ln(x))))
\[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx \] Input:
Integrate[(-30*Log[x]^2 + E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*(2*x - x^2 + (-2*x + 2*x^2)*Log[x] + (30 + 3*x)*Log[x]^2))/(30*x^2*Log[x]^2 - 60 *E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*x^2*Log[x]^2 + 30*E^((-2*x + x^ 2 + 3*x*Log[x])/(15*Log[x]))*x^2*Log[x]^2),x]
Output:
Integrate[(-30*Log[x]^2 + E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*(2*x - x^2 + (-2*x + 2*x^2)*Log[x] + (30 + 3*x)*Log[x]^2))/(30*x^2*Log[x]^2 - 60 *E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*x^2*Log[x]^2 + 30*E^((-2*x + x^ 2 + 3*x*Log[x])/(15*Log[x]))*x^2*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.
| \(\displaystyle \int \frac {e^{\frac {x^2-2 x+3 x \log (x)}{30 \log (x)}} \left (-x^2+\left (2 x^2-2 x\right ) \log (x)+2 x+(3 x+30) \log ^2(x)\right )-30 \log ^2(x)}{-60 x^2 e^{\frac {x^2-2 x+3 x \log (x)}{30 \log (x)}} \log ^2(x)+30 x^2 e^{\frac {x^2-2 x+3 x \log (x)}{15 \log (x)}} \log ^2(x)+30 x^2 \log ^2(x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {2 x}{15 \log (x)}} \left (e^{\frac {x^2-2 x+3 x \log (x)}{30 \log (x)}} \left (-x^2+\left (2 x^2-2 x\right ) \log (x)+2 x+(3 x+30) \log ^2(x)\right )-30 \log ^2(x)\right )}{30 x^2 \left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{30} \int -\frac {e^{\frac {2 x}{15 \log (x)}} \left (30 \log ^2(x)-e^{-\frac {2 x-x^2}{30 \log (x)}} x^{\frac {x}{10 \log (x)}} \left (-x^2+2 x+3 (x+10) \log ^2(x)-2 \left (x-x^2\right ) \log (x)\right )\right )}{x^2 \left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x^2}{30 \log (x)}} x^{\frac {x}{10 \log (x)}}\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{30} \int \frac {e^{\frac {2 x}{15 \log (x)}} \left (30 \log ^2(x)-e^{-\frac {2 x-x^2}{30 \log (x)}} x^{\frac {x}{10 \log (x)}} \left (-x^2+2 x+3 (x+10) \log ^2(x)-2 \left (x-x^2\right ) \log (x)\right )\right )}{x^2 \left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x^2}{30 \log (x)}} x^{\frac {x}{10 \log (x)}}\right )^2 \log ^2(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{30} \int \left (\frac {e^{\frac {x}{15 \log (x)}} \left (2 \log (x) x^2-x^2+3 \log ^2(x) x-2 \log (x) x+2 x+30 \log ^2(x)\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) x^2 \log ^2(x)}-\frac {e^{\frac {2 x}{15 \log (x)}} \left (3 \log ^2(x)+2 x \log (x)-2 \log (x)-x+2\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log ^2(x)}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{30} \int \left (\frac {e^{\frac {x}{15 \log (x)}} \left (2 \log (x) x^2-x^2+3 \log ^2(x) x-2 \log (x) x+2 x+30 \log ^2(x)\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) x^2 \log ^2(x)}-\frac {e^{\frac {2 x}{15 \log (x)}} \left (3 \log ^2(x)+2 x \log (x)-2 \log (x)-x+2\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log ^2(x)}\right )dx\) |
Input:
Int[(-30*Log[x]^2 + E^((-2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*(2*x - x^2 + (-2*x + 2*x^2)*Log[x] + (30 + 3*x)*Log[x]^2))/(30*x^2*Log[x]^2 - 60*E^((- 2*x + x^2 + 3*x*Log[x])/(30*Log[x]))*x^2*Log[x]^2 + 30*E^((-2*x + x^2 + 3* x*Log[x])/(15*Log[x]))*x^2*Log[x]^2),x]
Output:
$Aborted
Time = 0.66 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(-\frac {1}{x \left ({\mathrm e}^{\frac {x \left (3 \ln \left (x \right )+x -2\right )}{30 \ln \left (x \right )}}-1\right )}\) | \(25\) |
| parallelrisch | \(-\frac {1}{x \left ({\mathrm e}^{\frac {x \left (3 \ln \left (x \right )+x -2\right )}{30 \ln \left (x \right )}}-1\right )}\) | \(25\) |
Input:
int((((3*x+30)*ln(x)^2+(2*x^2-2*x)*ln(x)-x^2+2*x)*exp(1/30*(3*x*ln(x)+x^2- 2*x)/ln(x))-30*ln(x)^2)/(30*x^2*ln(x)^2*exp(1/30*(3*x*ln(x)+x^2-2*x)/ln(x) )^2-60*x^2*ln(x)^2*exp(1/30*(3*x*ln(x)+x^2-2*x)/ln(x))+30*x^2*ln(x)^2),x,m ethod=_RETURNVERBOSE)
Output:
-1/x/(exp(1/30*x*(3*ln(x)+x-2)/ln(x))-1)
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=-\frac {1}{x e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{30 \, \log \left (x\right )}\right )} - x} \] Input:
integrate((((3*x+30)*log(x)^2+(2*x^2-2*x)*log(x)-x^2+2*x)*exp(1/30*(3*x*lo g(x)+x^2-2*x)/log(x))-30*log(x)^2)/(30*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x ^2-2*x)/log(x))^2-60*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))+30 *x^2*log(x)^2),x, algorithm="fricas")
Output:
-1/(x*e^(1/30*(x^2 + 3*x*log(x) - 2*x)/log(x)) - x)
Time = 0.14 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=- \frac {1}{x e^{\frac {\frac {x^{2}}{30} + \frac {x \log {\left (x \right )}}{10} - \frac {x}{15}}{\log {\left (x \right )}}} - x} \] Input:
integrate((((3*x+30)*ln(x)**2+(2*x**2-2*x)*ln(x)-x**2+2*x)*exp(1/30*(3*x*l n(x)+x**2-2*x)/ln(x))-30*ln(x)**2)/(30*x**2*ln(x)**2*exp(1/30*(3*x*ln(x)+x **2-2*x)/ln(x))**2-60*x**2*ln(x)**2*exp(1/30*(3*x*ln(x)+x**2-2*x)/ln(x))+3 0*x**2*ln(x)**2),x)
Output:
-1/(x*exp((x**2/30 + x*log(x)/10 - x/15)/log(x)) - x)
\[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\int { \frac {{\left (3 \, {\left (x + 10\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 2 \, x\right )} e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{30 \, \log \left (x\right )}\right )} - 30 \, \log \left (x\right )^{2}}{30 \, {\left (x^{2} e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{15 \, \log \left (x\right )}\right )} \log \left (x\right )^{2} - 2 \, x^{2} e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{30 \, \log \left (x\right )}\right )} \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{2}\right )}} \,d x } \] Input:
integrate((((3*x+30)*log(x)^2+(2*x^2-2*x)*log(x)-x^2+2*x)*exp(1/30*(3*x*lo g(x)+x^2-2*x)/log(x))-30*log(x)^2)/(30*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x ^2-2*x)/log(x))^2-60*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))+30 *x^2*log(x)^2),x, algorithm="maxima")
Output:
1/30*integrate(((3*(x + 10)*log(x)^2 - x^2 + 2*(x^2 - x)*log(x) + 2*x)*e^( 1/30*(x^2 + 3*x*log(x) - 2*x)/log(x)) - 30*log(x)^2)/(x^2*e^(1/15*(x^2 + 3 *x*log(x) - 2*x)/log(x))*log(x)^2 - 2*x^2*e^(1/30*(x^2 + 3*x*log(x) - 2*x) /log(x))*log(x)^2 + x^2*log(x)^2), x)
Exception generated. \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((((3*x+30)*log(x)^2+(2*x^2-2*x)*log(x)-x^2+2*x)*exp(1/30*(3*x*lo g(x)+x^2-2*x)/log(x))-30*log(x)^2)/(30*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x ^2-2*x)/log(x))^2-60*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))+30 *x^2*log(x)^2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{50625,[0,19]%%%}+%%%{-607500,[0,18]%%%}+%%%{3037500,[0,17] %%%}+%%%{
Time = 3.85 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.21 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=-\frac {x\,\left (3\,{\ln \left (x\right )}^2-2\,\ln \left (x\right )+2\right )+x^2\,\left (2\,\ln \left (x\right )-1\right )}{x^2\,\left ({\mathrm {e}}^{\frac {x}{10}-\frac {x}{15\,\ln \left (x\right )}+\frac {x^2}{30\,\ln \left (x\right )}}-1\right )\,\left (3\,{\ln \left (x\right )}^2-2\,\ln \left (x\right )-x+2\,x\,\ln \left (x\right )+2\right )} \] Input:
int(-(30*log(x)^2 - exp(((x*log(x))/10 - x/15 + x^2/30)/log(x))*(2*x - log (x)*(2*x - 2*x^2) - x^2 + log(x)^2*(3*x + 30)))/(30*x^2*log(x)^2 - 60*x^2* exp(((x*log(x))/10 - x/15 + x^2/30)/log(x))*log(x)^2 + 30*x^2*exp((2*((x*l og(x))/10 - x/15 + x^2/30))/log(x))*log(x)^2),x)
Output:
-(x*(3*log(x)^2 - 2*log(x) + 2) + x^2*(2*log(x) - 1))/(x^2*(exp(x/10 - x/( 15*log(x)) + x^2/(30*log(x))) - 1)*(3*log(x)^2 - 2*log(x) - x + 2*x*log(x) + 2))
Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.88 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=-\frac {e^{\frac {x}{15 \,\mathrm {log}\left (x \right )}}}{x \left (e^{\frac {3 \,\mathrm {log}\left (x \right ) x +x^{2}}{30 \,\mathrm {log}\left (x \right )}}-e^{\frac {x}{15 \,\mathrm {log}\left (x \right )}}\right )} \] Input:
int((((3*x+30)*log(x)^2+(2*x^2-2*x)*log(x)-x^2+2*x)*exp(1/30*(3*x*log(x)+x ^2-2*x)/log(x))-30*log(x)^2)/(30*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x )/log(x))^2-60*x^2*log(x)^2*exp(1/30*(3*x*log(x)+x^2-2*x)/log(x))+30*x^2*l og(x)^2),x)
Output:
( - e**(x/(15*log(x))))/(x*(e**((3*log(x)*x + x**2)/(30*log(x))) - e**(x/( 15*log(x)))))