Integrand size = 60, antiderivative size = 24 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\left (\frac {4}{3} \left (-5+e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \]
\[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx \]
Integrate[(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2)*(1 - Log[x] - 2*Log[ x]*Log[((-20 + 4*E^2)*Log[x])/(3*x)]))/(x^3*Log[x]),x]
Integrate[(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2)*(1 - Log[x] - 2*Log[ x]*Log[((-20 + 4*E^2)*Log[x])/(3*x)]))/(x^3*Log[x]), 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.
\(\displaystyle \int \frac {\left (\frac {1}{3} \left (4 e^2-20\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (-2 \log \left (\frac {\left (4 e^2-20\right ) \log (x)}{3 x}\right ) \log (x)-\log (x)+1\right )}{x^3 \log (x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\left (\frac {1}{3} \left (4 e^2-20\right )\right )^{\frac {1}{x^2}} (1-\log (x)) \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)}-\frac {2 \left (\frac {1}{3} \left (4 e^2-20\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (e^2-5\right ) \log (x)}{3 x}\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3}dx+\int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}}}{x^3 \log (x)}dx-2 \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \log \left (\frac {4 \left (-5+e^2\right ) \log (x)}{3 x}\right )}{x^3}dx\) |
Int[(((-20 + 4*E^2)/3)^x^(-2)*(Log[x]/x)^x^(-2)*(1 - Log[x] - 2*Log[x]*Log [((-20 + 4*E^2)*Log[x])/(3*x)]))/(x^3*Log[x]),x]
3.13.44.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.42
method | result | size |
risch | \(\left (\frac {4}{3}\right )^{\frac {1}{x^{2}}} x^{-\frac {1}{x^{2}}} \ln \left (x \right )^{\frac {1}{x^{2}}} \left ({\mathrm e}^{2}-5\right )^{\frac {1}{x^{2}}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right ) \left (-\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )+\operatorname {csgn}\left (i \ln \left (x \right )\right )\right ) \left (\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )-\operatorname {csgn}\left (\frac {i}{x}\right )\right )}{2 x^{2}}}\) | \(82\) |
int((-2*ln(x)*ln(1/3*(4*exp(2)-20)*ln(x)/x)+1-ln(x))*exp(ln(1/3*(4*exp(2)- 20)*ln(x)/x)/x^2)/x^3/ln(x),x,method=_RETURNVERBOSE)
(4/3)^(1/x^2)*x^(-1/x^2)*ln(x)^(1/x^2)*(exp(2)-5)^(1/x^2)*exp(1/2*I*Pi*csg n(I*ln(x)/x)*(-csgn(I*ln(x)/x)+csgn(I*ln(x)))*(csgn(I*ln(x)/x)-csgn(I/x))/ x^2)
Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.62 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\left (\frac {4 \, {\left (e^{2} - 5\right )} \log \left (x\right )}{3 \, x}\right )^{\left (\frac {1}{x^{2}}\right )} \]
integrate((-2*log(x)*log(1/3*(4*exp(2)-20)*log(x)/x)+1-log(x))*exp(log(1/3 *(4*exp(2)-20)*log(x)/x)/x^2)/x^3/log(x),x, algorithm=\
Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=e^{\frac {\log {\left (\frac {\left (- \frac {20}{3} + \frac {4 e^{2}}{3}\right ) \log {\left (x \right )}}{x} \right )}}{x^{2}}} \]
integrate((-2*ln(x)*ln(1/3*(4*exp(2)-20)*ln(x)/x)+1-ln(x))*exp(ln(1/3*(4*e xp(2)-20)*ln(x)/x)/x**2)/x**3/ln(x),x)
Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (15) = 30\).
Time = 0.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.62 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=e^{\left (-\frac {\log \left (3\right )}{x^{2}} + \frac {2 \, \log \left (2\right )}{x^{2}} - \frac {\log \left (x\right )}{x^{2}} + \frac {\log \left (e^{2} - 5\right )}{x^{2}} + \frac {\log \left (\log \left (x\right )\right )}{x^{2}}\right )} \]
integrate((-2*log(x)*log(1/3*(4*exp(2)-20)*log(x)/x)+1-log(x))*exp(log(1/3 *(4*exp(2)-20)*log(x)/x)/x^2)/x^3/log(x),x, algorithm=\
\[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx=\int { -\frac {{\left (2 \, \log \left (x\right ) \log \left (\frac {4 \, {\left (e^{2} - 5\right )} \log \left (x\right )}{3 \, x}\right ) + \log \left (x\right ) - 1\right )} \left (\frac {4 \, {\left (e^{2} - 5\right )} \log \left (x\right )}{3 \, x}\right )^{\left (\frac {1}{x^{2}}\right )}}{x^{3} \log \left (x\right )} \,d x } \]
integrate((-2*log(x)*log(1/3*(4*exp(2)-20)*log(x)/x)+1-log(x))*exp(log(1/3 *(4*exp(2)-20)*log(x)/x)/x^2)/x^3/log(x),x, algorithm=\
integrate(-(2*log(x)*log(4/3*(e^2 - 5)*log(x)/x) + log(x) - 1)*(4/3*(e^2 - 5)*log(x)/x)^(x^(-2))/(x^3*log(x)), x)
Time = 12.70 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\left (\frac {1}{3} \left (-20+4 e^2\right )\right )^{\frac {1}{x^2}} \left (\frac {\log (x)}{x}\right )^{\frac {1}{x^2}} \left (1-\log (x)-2 \log (x) \log \left (\frac {\left (-20+4 e^2\right ) \log (x)}{3 x}\right )\right )}{x^3 \log (x)} \, dx={\left (-\frac {20\,\ln \left (x\right )-4\,{\mathrm {e}}^2\,\ln \left (x\right )}{3\,x}\right )}^{\frac {1}{x^2}} \]