Integrand size = 136, antiderivative size = 24 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\frac {1}{x-e^{\frac {1}{30} x \left (3+\frac {-2+x}{\log (x)}\right )} x} \]
Timed out. \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\text {\$Aborted} \]
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 |
\(\displaystyle -\frac {1}{30} \int \left (\frac {e^{\frac {x}{15 \log (x)}} \left (2 \log (x) x^2-x^2+3 \log ^2(x) x-2 \log (x) x+2 x+30 \log ^2(x)\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) x^2 \log ^2(x)}-\frac {e^{\frac {2 x}{15 \log (x)}} \left (3 \log ^2(x)+2 x \log (x)-2 \log (x)-x+2\right )}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log ^2(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{30} \left (-30 \int \frac {e^{\frac {x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) x^2}dx-\int \frac {e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log ^2(x)}dx+\int \frac {e^{\frac {x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) \log ^2(x)}dx+2 \int \frac {e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log ^2(x)}dx-2 \int \frac {e^{\frac {x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) x \log ^2(x)}dx+3 \int \frac {e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x}dx-3 \int \frac {e^{\frac {x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) x}dx+2 \int \frac {e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 \log (x)}dx-2 \int \frac {e^{\frac {x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) \log (x)}dx-2 \int \frac {e^{\frac {2 x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right )^2 x \log (x)}dx+2 \int \frac {e^{\frac {x}{15 \log (x)}}}{\left (e^{\frac {x}{15 \log (x)}}-e^{\frac {x (x+3 \log (x))}{30 \log (x)}}\right ) x \log (x)}dx\right )\) |
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]
3.26.11.3.1 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 = 0.52 (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 | \(-\frac {1}{x \left ({\mathrm e}^{\frac {x \left (3 \ln \left (x \right )+x -2\right )}{30 \ln \left (x \right )}}-1\right )}\) | \(25\) |
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)
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=-\frac {1}{x e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{30 \, \log \left (x\right )}\right )} - x} \]
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=\
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=- \frac {1}{x e^{\frac {\frac {x^{2}}{30} + \frac {x \log {\left (x \right )}}{10} - \frac {x}{15}}{\log {\left (x \right )}}} - x} \]
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)
\[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\int { \frac {{\left (3 \, {\left (x + 10\right )} \log \left (x\right )^{2} - x^{2} + 2 \, {\left (x^{2} - x\right )} \log \left (x\right ) + 2 \, x\right )} e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{30 \, \log \left (x\right )}\right )} - 30 \, \log \left (x\right )^{2}}{30 \, {\left (x^{2} e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{15 \, \log \left (x\right )}\right )} \log \left (x\right )^{2} - 2 \, x^{2} e^{\left (\frac {x^{2} + 3 \, x \log \left (x\right ) - 2 \, x}{30 \, \log \left (x\right )}\right )} \log \left (x\right )^{2} + x^{2} \log \left (x\right )^{2}\right )}} \,d x } \]
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=\
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. \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
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=\
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 = 14.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.21 \[ \int \frac {-30 \log ^2(x)+e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} \left (2 x-x^2+\left (-2 x+2 x^2\right ) \log (x)+(30+3 x) \log ^2(x)\right )}{30 x^2 \log ^2(x)-60 e^{\frac {-2 x+x^2+3 x \log (x)}{30 \log (x)}} x^2 \log ^2(x)+30 e^{\frac {-2 x+x^2+3 x \log (x)}{15 \log (x)}} x^2 \log ^2(x)} \, dx=-\frac {x\,\left (3\,{\ln \left (x\right )}^2-2\,\ln \left (x\right )+2\right )+x^2\,\left (2\,\ln \left (x\right )-1\right )}{x^2\,\left ({\mathrm {e}}^{\frac {x}{10}-\frac {x}{15\,\ln \left (x\right )}+\frac {x^2}{30\,\ln \left (x\right )}}-1\right )\,\left (3\,{\ln \left (x\right )}^2-2\,\ln \left (x\right )-x+2\,x\,\ln \left (x\right )+2\right )} \]
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)