Integrand size = 108, antiderivative size = 31 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}}+\log (x)}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]
Time = 0.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{8/3} \left (x^2\right )^{-1/x}+\log (x)}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]
Integrate[(E^(5/3 + (x - Log[x^2])/x)*x + x*Log[x] + (x*Log[5/x] + E^((x - Log[x^2])/x)*(-2*E^(5/3)*Log[5/x] + E^(5/3)*Log[5/x]*Log[x^2]))*Log[Log[5 /x]])/(x^2*Log[5/x]*Log[Log[5/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.
\(\displaystyle \int \frac {x e^{\frac {x-\log \left (x^2\right )}{x}+\frac {5}{3}}+\left (e^{\frac {x-\log \left (x^2\right )}{x}} \left (e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )-2 e^{5/3} \log \left (\frac {5}{x}\right )\right )+x \log \left (\frac {5}{x}\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )+x \log (x)}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{8/3} \left (x^2\right )^{-\frac {1}{x}-1} \left (\log \left (\frac {5}{x}\right ) \log \left (x^2\right ) \log \left (\log \left (\frac {5}{x}\right )\right )+x-2 \log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )\right )}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}+\frac {\log (x)+\log \left (\frac {5}{x}\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e^{8/3} \int \frac {x \left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}dx-2 e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}}}{\log \left (\log \left (\frac {5}{x}\right )\right )}dx+e^{8/3} \int \frac {\left (x^2\right )^{-1-\frac {1}{x}} \log \left (x^2\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )}dx+\int \frac {\log (x)}{x \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )}dx-\operatorname {LogIntegral}\left (\log \left (\frac {5}{x}\right )\right )\) |
Int[(E^(5/3 + (x - Log[x^2])/x)*x + x*Log[x] + (x*Log[5/x] + E^((x - Log[x ^2])/x)*(-2*E^(5/3)*Log[5/x] + E^(5/3)*Log[5/x]*Log[x^2]))*Log[Log[5/x]])/ (x^2*Log[5/x]*Log[Log[5/x]]^2),x]
3.7.4.3.1 Defintions of rubi rules used
Time = 274.35 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{\frac {5}{3}} {\mathrm e}^{-\frac {\ln \left (x^{2}\right )-x}{x}}+\ln \left (x \right )}{\ln \left (\ln \left (\frac {5}{x}\right )\right )}\) | \(31\) |
risch | \(\frac {x^{-\frac {2}{x}} {\mathrm e}^{\frac {3 i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-6 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+3 i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+16 x}{6 x}}+\ln \left (x \right )}{\ln \left (\ln \left (5\right )-\ln \left (x \right )\right )}\) | \(82\) |
int((((exp(5/3)*ln(5/x)*ln(x^2)-2*exp(5/3)*ln(5/x))*exp((-ln(x^2)+x)/x)+x* ln(5/x))*ln(ln(5/x))+x*exp(5/3)*exp((-ln(x^2)+x)/x)+x*ln(x))/x^2/ln(5/x)/l n(ln(5/x))^2,x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{\left (\frac {2 \, {\left (4 \, x - 3 \, \log \left (5\right ) + 3 \, \log \left (\frac {5}{x}\right )\right )}}{3 \, x}\right )} + \log \left (5\right ) - \log \left (\frac {5}{x}\right )}{\log \left (\log \left (\frac {5}{x}\right )\right )} \]
integrate((((exp(5/3)*log(5/x)*log(x^2)-2*exp(5/3)*log(5/x))*exp((-log(x^2 )+x)/x)+x*log(5/x))*log(log(5/x))+x*exp(5/3)*exp((-log(x^2)+x)/x)+x*log(x) )/x^2/log(5/x)/log(log(5/x))^2,x, algorithm=\
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {e^{\frac {5}{3}} e^{\frac {x - 2 \log {\left (x \right )}}{x}}}{\log {\left (- \log {\left (x \right )} + \log {\left (5 \right )} \right )}} + \frac {\log {\left (x \right )}}{\log {\left (- \log {\left (x \right )} + \log {\left (5 \right )} \right )}} \]
integrate((((exp(5/3)*ln(5/x)*ln(x**2)-2*exp(5/3)*ln(5/x))*exp((-ln(x**2)+ x)/x)+x*ln(5/x))*ln(ln(5/x))+x*exp(5/3)*exp((-ln(x**2)+x)/x)+x*ln(x))/x**2 /ln(5/x)/ln(ln(5/x))**2,x)
Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {x^{\frac {2}{x}} \log \left (x\right ) + e^{\frac {8}{3}}}{x^{\frac {2}{x}} \log \left (\log \left (5\right ) - \log \left (x\right )\right )} \]
integrate((((exp(5/3)*log(5/x)*log(x^2)-2*exp(5/3)*log(5/x))*exp((-log(x^2 )+x)/x)+x*log(5/x))*log(log(5/x))+x*exp(5/3)*exp((-log(x^2)+x)/x)+x*log(x) )/x^2/log(5/x)/log(log(5/x))^2,x, algorithm=\
Time = 0.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\frac {\log \left (x\right )}{\log \left (\log \left (5\right ) - \log \left (x\right )\right )} + \frac {e^{\frac {8}{3}}}{x^{\frac {2}{x}} \log \left (\log \left (5\right ) - \log \left (x\right )\right )} \]
integrate((((exp(5/3)*log(5/x)*log(x^2)-2*exp(5/3)*log(5/x))*exp((-log(x^2 )+x)/x)+x*log(5/x))*log(log(5/x))+x*exp(5/3)*exp((-log(x^2)+x)/x)+x*log(x) )/x^2/log(5/x)/log(log(5/x))^2,x, algorithm=\
Time = 8.59 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {e^{\frac {5}{3}+\frac {x-\log \left (x^2\right )}{x}} x+x \log (x)+\left (x \log \left (\frac {5}{x}\right )+e^{\frac {x-\log \left (x^2\right )}{x}} \left (-2 e^{5/3} \log \left (\frac {5}{x}\right )+e^{5/3} \log \left (\frac {5}{x}\right ) \log \left (x^2\right )\right )\right ) \log \left (\log \left (\frac {5}{x}\right )\right )}{x^2 \log \left (\frac {5}{x}\right ) \log ^2\left (\log \left (\frac {5}{x}\right )\right )} \, dx=\ln \left (\frac {1}{x}\right )+\ln \left (x\right )+\frac {\ln \left (x\right )}{\ln \left (\ln \left (\frac {1}{x}\right )+\ln \left (5\right )\right )}+\frac {{\mathrm {e}}^{8/3}}{\ln \left (\ln \left (\frac {1}{x}\right )+\ln \left (5\right )\right )\,{\left (x^2\right )}^{1/x}} \]
int((log(log(5/x))*(x*log(5/x) - exp((x - log(x^2))/x)*(2*exp(5/3)*log(5/x ) - log(x^2)*exp(5/3)*log(5/x))) + x*log(x) + x*exp((x - log(x^2))/x)*exp( 5/3))/(x^2*log(log(5/x))^2*log(5/x)),x)