Integrand size = 59, antiderivative size = 23 \begin {dmath*} \int \frac {\left (-20+12 x+(4-2 x) \log \left (e^{8+x}\right )\right ) \log \left (\frac {1}{5} e^{-x} \left (-10 x^2+2 x^2 \log \left (e^{8+x}\right )\right )\right )}{-5 x+x \log \left (e^{8+x}\right )} \, dx=\log ^2\left (\frac {2}{5} e^{-x} x^2 \left (-5+\log \left (e^{8+x}\right )\right )\right ) \end {dmath*}
Time = 0.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {\left (-20+12 x+(4-2 x) \log \left (e^{8+x}\right )\right ) \log \left (\frac {1}{5} e^{-x} \left (-10 x^2+2 x^2 \log \left (e^{8+x}\right )\right )\right )}{-5 x+x \log \left (e^{8+x}\right )} \, dx=\log ^2\left (\frac {2}{5} e^{-x} x^2 \left (-5+\log \left (e^{8+x}\right )\right )\right ) \end {dmath*}
Integrate[((-20 + 12*x + (4 - 2*x)*Log[E^(8 + x)])*Log[(-10*x^2 + 2*x^2*Lo g[E^(8 + x)])/(5*E^x)])/(-5*x + x*Log[E^(8 + 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.
\(\displaystyle \int \frac {\left (12 x+(4-2 x) \log \left (e^{x+8}\right )-20\right ) \log \left (\frac {1}{5} e^{-x} \left (2 x^2 \log \left (e^{x+8}\right )-10 x^2\right )\right )}{x \log \left (e^{x+8}\right )-5 x} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {\left (-12 x-(4-2 x) \log \left (e^{x+8}\right )+20\right ) \log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{5 x-x \log \left (e^{x+8}\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 \log \left (e^{x+8}\right ) \log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{x \left (\log \left (e^{x+8}\right )-5\right )}-\frac {2 \log \left (e^{x+8}\right ) \log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{\log \left (e^{x+8}\right )-5}-\frac {20 \log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{x \left (\log \left (e^{x+8}\right )-5\right )}+\frac {12 \log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{\log \left (e^{x+8}\right )-5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 12 \int \frac {\log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{\log \left (e^{x+8}\right )-5}dx-20 \int \frac {\log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{x \left (\log \left (e^{x+8}\right )-5\right )}dx-2 \int \frac {\log \left (e^{x+8}\right ) \log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{\log \left (e^{x+8}\right )-5}dx+4 \int \frac {\log \left (e^{x+8}\right ) \log \left (\frac {2}{5} e^{-x} x^2 \left (\log \left (e^{x+8}\right )-5\right )\right )}{x \left (\log \left (e^{x+8}\right )-5\right )}dx\) |
Int[((-20 + 12*x + (4 - 2*x)*Log[E^(8 + x)])*Log[(-10*x^2 + 2*x^2*Log[E^(8 + x)])/(5*E^x)])/(-5*x + x*Log[E^(8 + x)]),x]
3.11.39.3.1 Defintions of rubi rules used
Time = 0.56 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \({\ln \left (\frac {2 \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5\right ) x^{2} {\mathrm e}^{-x}}{5}\right )}^{2}\) | \(23\) |
default | \(4 \ln \left (\frac {\left (2 x^{2} \ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-10 x^{2}\right ) {\mathrm e}^{-x}}{5}\right ) \ln \left (x \right )+2 \ln \left (\frac {\left (2 x^{2} \ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-10 x^{2}\right ) {\mathrm e}^{-x}}{5}\right ) \ln \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5\right )-2 \ln \left (\frac {\left (2 x^{2} \ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-10 x^{2}\right ) {\mathrm e}^{-x}}{5}\right ) x -x^{2}-2 \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5-x \right ) \ln \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5\right )+4 x \ln \left (x \right )-4 \ln \left (x \right )^{2}-4 \operatorname {dilog}\left (\frac {\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5}{\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5-x}\right )-4 \ln \left (x \right ) \ln \left (\frac {\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5}{\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5-x}\right )+2 \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5\right ) \ln \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5\right )-2 \ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )+10+2 x -{\ln \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5\right )}^{2}-4 \operatorname {dilog}\left (-\frac {x}{\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5-x}\right )-4 \ln \left (\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5\right ) \ln \left (-\frac {x}{\ln \left ({\mathrm e}^{3} {\mathrm e}^{5+x}\right )-5-x}\right )\) | \(307\) |
int(((4-2*x)*ln(exp(3)*exp(5+x))+12*x-20)*ln(1/5*(2*x^2*ln(exp(3)*exp(5+x) )-10*x^2)/exp(x))/(x*ln(exp(3)*exp(5+x))-5*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \begin {dmath*} \int \frac {\left (-20+12 x+(4-2 x) \log \left (e^{8+x}\right )\right ) \log \left (\frac {1}{5} e^{-x} \left (-10 x^2+2 x^2 \log \left (e^{8+x}\right )\right )\right )}{-5 x+x \log \left (e^{8+x}\right )} \, dx=\log \left (\frac {2}{5} \, {\left (x^{3} + 3 \, x^{2}\right )} e^{\left (-x\right )}\right )^{2} \end {dmath*}
integrate(((4-2*x)*log(exp(3)*exp(5+x))+12*x-20)*log(1/5*(2*x^2*log(exp(3) *exp(5+x))-10*x^2)/exp(x))/(x*log(exp(3)*exp(5+x))-5*x),x, algorithm=\
Timed out. \begin {dmath*} \int \frac {\left (-20+12 x+(4-2 x) \log \left (e^{8+x}\right )\right ) \log \left (\frac {1}{5} e^{-x} \left (-10 x^2+2 x^2 \log \left (e^{8+x}\right )\right )\right )}{-5 x+x \log \left (e^{8+x}\right )} \, dx=\text {Timed out} \end {dmath*}
integrate(((4-2*x)*ln(exp(3)*exp(5+x))+12*x-20)*ln(1/5*(2*x**2*ln(exp(3)*e xp(5+x))-10*x**2)/exp(x))/(x*ln(exp(3)*exp(5+x))-5*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (19) = 38\).
Time = 0.23 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.39 \begin {dmath*} \int \frac {\left (-20+12 x+(4-2 x) \log \left (e^{8+x}\right )\right ) \log \left (\frac {1}{5} e^{-x} \left (-10 x^2+2 x^2 \log \left (e^{8+x}\right )\right )\right )}{-5 x+x \log \left (e^{8+x}\right )} \, dx=-x^{2} - 2 \, {\left (x - \log \left (x + 3\right ) - 2 \, \log \left (x\right )\right )} \log \left (\frac {2}{5} \, {\left ({\left (x + 8\right )} x^{2} - 5 \, x^{2}\right )} e^{\left (-x\right )}\right ) + 2 \, {\left (x - 2 \, \log \left (x\right ) + 3\right )} \log \left (x + 3\right ) - \log \left (x + 3\right )^{2} + 4 \, x \log \left (x\right ) - 4 \, \log \left (x\right )^{2} - 6 \, \log \left (x + 3\right ) \end {dmath*}
integrate(((4-2*x)*log(exp(3)*exp(5+x))+12*x-20)*log(1/5*(2*x^2*log(exp(3) *exp(5+x))-10*x^2)/exp(x))/(x*log(exp(3)*exp(5+x))-5*x),x, algorithm=\
-x^2 - 2*(x - log(x + 3) - 2*log(x))*log(2/5*((x + 8)*x^2 - 5*x^2)*e^(-x)) + 2*(x - 2*log(x) + 3)*log(x + 3) - log(x + 3)^2 + 4*x*log(x) - 4*log(x)^ 2 - 6*log(x + 3)
Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \begin {dmath*} \int \frac {\left (-20+12 x+(4-2 x) \log \left (e^{8+x}\right )\right ) \log \left (\frac {1}{5} e^{-x} \left (-10 x^2+2 x^2 \log \left (e^{8+x}\right )\right )\right )}{-5 x+x \log \left (e^{8+x}\right )} \, dx=\log \left (\frac {2}{5} \, x^{3} e^{\left (-x\right )} + \frac {6}{5} \, x^{2} e^{\left (-x\right )}\right )^{2} \end {dmath*}
integrate(((4-2*x)*log(exp(3)*exp(5+x))+12*x-20)*log(1/5*(2*x^2*log(exp(3) *exp(5+x))-10*x^2)/exp(x))/(x*log(exp(3)*exp(5+x))-5*x),x, algorithm=\
Time = 18.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \begin {dmath*} \int \frac {\left (-20+12 x+(4-2 x) \log \left (e^{8+x}\right )\right ) \log \left (\frac {1}{5} e^{-x} \left (-10 x^2+2 x^2 \log \left (e^{8+x}\right )\right )\right )}{-5 x+x \log \left (e^{8+x}\right )} \, dx={\left (x-\ln \left (\frac {2\,x^2\,\left (x+8\right )}{5}-2\,x^2\right )\right )}^2 \end {dmath*}