Integrand size = 107, antiderivative size = 25 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\log \left (\log (x)-\frac {3 \left (-e^x \log \left (5 e^x\right )+\log (x)\right )^2}{e}\right ) \]
\[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx \]
Integrate[(-E + 6*E^(2*x)*x*Log[5*E^x]^2 + (6 - 6*E^x*x)*Log[x] + Log[5*E^ x]*(-6*E^x + 6*E^(2*x)*x - 6*E^x*x*Log[x]))/(3*E^(2*x)*x*Log[5*E^x]^2 - E* x*Log[x] - 6*E^x*x*Log[5*E^x]*Log[x] + 3*x*Log[x]^2),x]
Integrate[(-E + 6*E^(2*x)*x*Log[5*E^x]^2 + (6 - 6*E^x*x)*Log[x] + Log[5*E^ x]*(-6*E^x + 6*E^(2*x)*x - 6*E^x*x*Log[x]))/(3*E^(2*x)*x*Log[5*E^x]^2 - E* x*Log[x] - 6*E^x*x*Log[5*E^x]*Log[x] + 3*x*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.
\(\displaystyle \int \frac {6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6 e^{2 x} x-6 e^x-6 e^x x \log (x)\right ) \log \left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)-e}{3 e^{2 x} x \log ^2\left (5 e^x\right )+3 x \log ^2(x)-6 e^x x \log (x) \log \left (5 e^x\right )-e x \log (x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {-6 e^x \log ^2\left (5 e^x\right )+6 e^x x \log (x) \log ^2\left (5 e^x\right )-6 x \log ^2(x) \log \left (5 e^x\right )-6 x \log ^2(x)+6 e^x x \log (x) \log \left (5 e^x\right )+2 e x \log (x) \log \left (5 e^x\right )+6 \log (x) \log \left (5 e^x\right )-e \log \left (5 e^x\right )+2 e x \log (x)}{x \log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )+3 \log ^2(x)-6 e^x \log (x) \log \left (5 e^x\right )-e \log (x)\right )}+\frac {2 \left (\log \left (5 e^x\right )+1\right )}{\log \left (5 e^x\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle e \int \frac {1}{x \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+6 e^x \log (x) \log \left (5 e^x\right )-3 \log ^2(x)+e \log (x)\right )}dx-2 e \int \frac {\log (x)}{-3 e^{2 x} \log ^2\left (5 e^x\right )+6 e^x \log (x) \log \left (5 e^x\right )-3 \log ^2(x)+e \log (x)}dx-2 e \int \frac {\log (x)}{\log \left (5 e^x\right ) \left (-3 e^{2 x} \log ^2\left (5 e^x\right )+6 e^x \log (x) \log \left (5 e^x\right )-3 \log ^2(x)+e \log (x)\right )}dx-6 \int \frac {e^x \log \left (5 e^x\right )}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-6 e^x \log (x) \log \left (5 e^x\right )+3 \log ^2(x)-e \log (x)\right )}dx+6 \int \frac {e^x \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-6 e^x \log (x) \log \left (5 e^x\right )+3 \log ^2(x)-e \log (x)}dx+6 \int \frac {\log (x)}{x \left (3 e^{2 x} \log ^2\left (5 e^x\right )-6 e^x \log (x) \log \left (5 e^x\right )+3 \log ^2(x)-e \log (x)\right )}dx+6 \int \frac {e^x \log \left (5 e^x\right ) \log (x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-6 e^x \log (x) \log \left (5 e^x\right )+3 \log ^2(x)-e \log (x)}dx-6 \int \frac {\log ^2(x)}{3 e^{2 x} \log ^2\left (5 e^x\right )-6 e^x \log (x) \log \left (5 e^x\right )+3 \log ^2(x)-e \log (x)}dx-6 \int \frac {\log ^2(x)}{\log \left (5 e^x\right ) \left (3 e^{2 x} \log ^2\left (5 e^x\right )-6 e^x \log (x) \log \left (5 e^x\right )+3 \log ^2(x)-e \log (x)\right )}dx+2 x+2 \log \left (\log \left (5 e^x\right )\right )\) |
Int[(-E + 6*E^(2*x)*x*Log[5*E^x]^2 + (6 - 6*E^x*x)*Log[x] + Log[5*E^x]*(-6 *E^x + 6*E^(2*x)*x - 6*E^x*x*Log[x]))/(3*E^(2*x)*x*Log[5*E^x]^2 - E*x*Log[ x] - 6*E^x*x*Log[5*E^x]*Log[x] + 3*x*Log[x]^2),x]
3.30.8.3.1 Defintions of rubi rules used
Time = 3.99 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44
method | result | size |
parallelrisch | \(\ln \left (\ln \left (5 \,{\mathrm e}^{x}\right )^{2} {\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} \ln \left (x \right ) \ln \left (5 \,{\mathrm e}^{x}\right )-\frac {{\mathrm e} \ln \left (x \right )}{3}+\ln \left (x \right )^{2}\right )\) | \(36\) |
risch | \(2 x +\ln \left (\ln \left ({\mathrm e}^{x}\right )^{2}+2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (5\right )-2 \,{\mathrm e}^{-x} \ln \left (x \right ) \ln \left ({\mathrm e}^{x}\right )-2 \ln \left (5\right ) \ln \left (x \right ) {\mathrm e}^{-x}+\ln \left (5\right )^{2}+{\mathrm e}^{-2 x} \ln \left (x \right )^{2}-\frac {\ln \left (x \right ) {\mathrm e}^{1-2 x}}{3}\right )\) | \(63\) |
int((6*x*exp(x)^2*ln(5*exp(x))^2+(-6*x*exp(x)*ln(x)+6*x*exp(x)^2-6*exp(x)) *ln(5*exp(x))+(-6*exp(x)*x+6)*ln(x)-exp(1))/(3*x*exp(x)^2*ln(5*exp(x))^2-6 *x*exp(x)*ln(x)*ln(5*exp(x))+3*x*ln(x)^2-x*exp(1)*ln(x)),x,method=_RETURNV ERBOSE)
Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\log \left (3 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )} e^{\left (2 \, x\right )} - {\left (6 \, {\left (x + \log \left (5\right )\right )} e^{x} + e\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2}\right ) \]
integrate((6*x*exp(x)^2*log(5*exp(x))^2+(-6*x*exp(x)*log(x)+6*x*exp(x)^2-6 *exp(x))*log(5*exp(x))+(-6*exp(x)*x+6)*log(x)-exp(1))/(3*x*exp(x)^2*log(5* exp(x))^2-6*x*exp(x)*log(x)*log(5*exp(x))+3*x*log(x)^2-x*exp(1)*log(x)),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 1.79 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=2 \log {\left (x + \log {\left (5 \right )} \right )} + \log {\left (\frac {3 \log {\left (x \right )}^{2} - e \log {\left (x \right )}}{3 x^{2} + 6 x \log {\left (5 \right )} + 3 \log {\left (5 \right )}^{2}} + e^{2 x} - \frac {2 e^{x} \log {\left (x \right )}}{x + \log {\left (5 \right )}} \right )} \]
integrate((6*x*exp(x)**2*ln(5*exp(x))**2+(-6*x*exp(x)*ln(x)+6*x*exp(x)**2- 6*exp(x))*ln(5*exp(x))+(-6*exp(x)*x+6)*ln(x)-exp(1))/(3*x*exp(x)**2*ln(5*e xp(x))**2-6*x*exp(x)*ln(x)*ln(5*exp(x))+3*x*ln(x)**2-x*exp(1)*ln(x)),x)
2*log(x + log(5)) + log((3*log(x)**2 - E*log(x))/(3*x**2 + 6*x*log(5) + 3* log(5)**2) + exp(2*x) - 2*exp(x)*log(x)/(x + log(5)))
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (23) = 46\).
Time = 0.37 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=2 \, \log \left (x + \log \left (5\right )\right ) + \log \left (-\frac {6 \, {\left (x + \log \left (5\right )\right )} e^{x} \log \left (x\right ) - 3 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )} e^{\left (2 \, x\right )} + e \log \left (x\right ) - 3 \, \log \left (x\right )^{2}}{3 \, {\left (x^{2} + 2 \, x \log \left (5\right ) + \log \left (5\right )^{2}\right )}}\right ) \]
integrate((6*x*exp(x)^2*log(5*exp(x))^2+(-6*x*exp(x)*log(x)+6*x*exp(x)^2-6 *exp(x))*log(5*exp(x))+(-6*exp(x)*x+6)*log(x)-exp(1))/(3*x*exp(x)^2*log(5* exp(x))^2-6*x*exp(x)*log(x)*log(5*exp(x))+3*x*log(x)^2-x*exp(1)*log(x)),x, algorithm=\
2*log(x + log(5)) + log(-1/3*(6*(x + log(5))*e^x*log(x) - 3*(x^2 + 2*x*log (5) + log(5)^2)*e^(2*x) + e*log(x) - 3*log(x)^2)/(x^2 + 2*x*log(5) + log(5 )^2))
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.37 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\log \left (-3 \, x^{2} e^{\left (2 \, x\right )} - 6 \, x e^{\left (2 \, x\right )} \log \left (5\right ) - 3 \, e^{\left (2 \, x\right )} \log \left (5\right )^{2} + 6 \, x e^{x} \log \left (x\right ) + 6 \, e^{x} \log \left (5\right ) \log \left (x\right ) + e \log \left (x\right ) - 3 \, \log \left (x\right )^{2}\right ) \]
integrate((6*x*exp(x)^2*log(5*exp(x))^2+(-6*x*exp(x)*log(x)+6*x*exp(x)^2-6 *exp(x))*log(5*exp(x))+(-6*exp(x)*x+6)*log(x)-exp(1))/(3*x*exp(x)^2*log(5* exp(x))^2-6*x*exp(x)*log(x)*log(5*exp(x))+3*x*log(x)^2-x*exp(1)*log(x)),x, algorithm=\
log(-3*x^2*e^(2*x) - 6*x*e^(2*x)*log(5) - 3*e^(2*x)*log(5)^2 + 6*x*e^x*log (x) + 6*e^x*log(5)*log(x) + e*log(x) - 3*log(x)^2)
Timed out. \[ \int \frac {-e+6 e^{2 x} x \log ^2\left (5 e^x\right )+\left (6-6 e^x x\right ) \log (x)+\log \left (5 e^x\right ) \left (-6 e^x+6 e^{2 x} x-6 e^x x \log (x)\right )}{3 e^{2 x} x \log ^2\left (5 e^x\right )-e x \log (x)-6 e^x x \log \left (5 e^x\right ) \log (x)+3 x \log ^2(x)} \, dx=\int -\frac {-6\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (5\,{\mathrm {e}}^x\right )}^2+\left (6\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^{2\,x}+6\,x\,{\mathrm {e}}^x\,\ln \left (x\right )\right )\,\ln \left (5\,{\mathrm {e}}^x\right )+\mathrm {e}+\ln \left (x\right )\,\left (6\,x\,{\mathrm {e}}^x-6\right )}{3\,x\,{\mathrm {e}}^{2\,x}\,{\ln \left (5\,{\mathrm {e}}^x\right )}^2-6\,x\,{\mathrm {e}}^x\,\ln \left (5\,{\mathrm {e}}^x\right )\,\ln \left (x\right )+3\,x\,{\ln \left (x\right )}^2-x\,\mathrm {e}\,\ln \left (x\right )} \,d x \]
int(-(exp(1) + log(5*exp(x))*(6*exp(x) - 6*x*exp(2*x) + 6*x*exp(x)*log(x)) + log(x)*(6*x*exp(x) - 6) - 6*x*exp(2*x)*log(5*exp(x))^2)/(3*x*log(x)^2 + 3*x*exp(2*x)*log(5*exp(x))^2 - x*exp(1)*log(x) - 6*x*exp(x)*log(5*exp(x)) *log(x)),x)