Integrand size = 105, antiderivative size = 31 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=4+\frac {4 e^{6+2 e^3} x^2}{\left (-\frac {3}{x}+\log \left (\frac {3}{\log (x)}\right )\right )^2} \] Output:
4+4*exp(exp(3)+3)^2/(ln(3/ln(x))-3/x)^2*x^2
Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=\frac {4 e^{6+2 e^3} x^4}{\left (-3+x \log \left (\frac {3}{\log (x)}\right )\right )^2} \] Input:
Integrate[(8*E^(6 + 2*E^3)*x^4 - 48*E^(6 + 2*E^3)*x^3*Log[x] + 8*E^(6 + 2* E^3)*x^4*Log[x]*Log[3/Log[x]])/(-27*Log[x] + 27*x*Log[x]*Log[3/Log[x]] - 9 *x^2*Log[x]*Log[3/Log[x]]^2 + x^3*Log[x]*Log[3/Log[x]]^3),x]
Output:
(4*E^(6 + 2*E^3)*x^4)/(-3 + x*Log[3/Log[x]])^2
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.
\(\displaystyle \int \frac {8 e^{6+2 e^3} x^4+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )-48 e^{6+2 e^3} x^3 \log (x)}{x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-27 \log (x)} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {8 e^{6+2 e^3} x^3 \left (-x-\log (x) \left (x \log \left (\frac {3}{\log (x)}\right )-6\right )\right )}{\log (x) \left (3-x \log \left (\frac {3}{\log (x)}\right )\right )^3}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 8 e^{6+2 e^3} \int -\frac {x^3 \left (x-\log (x) \left (6-x \log \left (\frac {3}{\log (x)}\right )\right )\right )}{\log (x) \left (3-x \log \left (\frac {3}{\log (x)}\right )\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -8 e^{6+2 e^3} \int \frac {x^3 \left (x-\log (x) \left (6-x \log \left (\frac {3}{\log (x)}\right )\right )\right )}{\log (x) \left (3-x \log \left (\frac {3}{\log (x)}\right )\right )^3}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -8 e^{6+2 e^3} \int \left (-\frac {x^3}{\left (x \log \left (\frac {3}{\log (x)}\right )-3\right )^2}-\frac {(x-3 \log (x)) x^3}{\log (x) \left (x \log \left (\frac {3}{\log (x)}\right )-3\right )^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -8 e^{6+2 e^3} \left (-\int \frac {x^4}{\log (x) \left (x \log \left (\frac {3}{\log (x)}\right )-3\right )^3}dx+3 \int \frac {x^3}{\left (x \log \left (\frac {3}{\log (x)}\right )-3\right )^3}dx-\int \frac {x^3}{\left (x \log \left (\frac {3}{\log (x)}\right )-3\right )^2}dx\right )\) |
Input:
Int[(8*E^(6 + 2*E^3)*x^4 - 48*E^(6 + 2*E^3)*x^3*Log[x] + 8*E^(6 + 2*E^3)*x ^4*Log[x]*Log[3/Log[x]])/(-27*Log[x] + 27*x*Log[x]*Log[3/Log[x]] - 9*x^2*L og[x]*Log[3/Log[x]]^2 + x^3*Log[x]*Log[3/Log[x]]^3),x]
Output:
$Aborted
Result contains complex when optimal does not.
Time = 3.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {16 x^{4} {\mathrm e}^{2 \,{\mathrm e}^{3}+6}}{\left (-2 i x \ln \left (3\right )+2 i x \ln \left (\ln \left (x \right )\right )+6 i\right )^{2}}\) | \(31\) |
parallelrisch | \(\frac {4 x^{4} {\mathrm e}^{2 \,{\mathrm e}^{3}+6}}{x^{2} \ln \left (\frac {3}{\ln \left (x \right )}\right )^{2}-6 x \ln \left (\frac {3}{\ln \left (x \right )}\right )+9}\) | \(40\) |
Input:
int((8*x^4*exp(exp(3)+3)^2*ln(x)*ln(3/ln(x))-48*x^3*exp(exp(3)+3)^2*ln(x)+ 8*x^4*exp(exp(3)+3)^2)/(x^3*ln(x)*ln(3/ln(x))^3-9*x^2*ln(x)*ln(3/ln(x))^2+ 27*x*ln(x)*ln(3/ln(x))-27*ln(x)),x,method=_RETURNVERBOSE)
Output:
-16/(-2*I*x*ln(3)+2*I*x*ln(ln(x))+6*I)^2*x^4*exp(2*exp(3)+6)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=\frac {4 \, x^{4} e^{\left (2 \, e^{3} + 6\right )}}{x^{2} \log \left (\frac {3}{\log \left (x\right )}\right )^{2} - 6 \, x \log \left (\frac {3}{\log \left (x\right )}\right ) + 9} \] Input:
integrate((8*x^4*exp(exp(3)+3)^2*log(x)*log(3/log(x))-48*x^3*exp(exp(3)+3) ^2*log(x)+8*x^4*exp(exp(3)+3)^2)/(x^3*log(x)*log(3/log(x))^3-9*x^2*log(x)* log(3/log(x))^2+27*x*log(x)*log(3/log(x))-27*log(x)),x, algorithm="fricas" )
Output:
4*x^4*e^(2*e^3 + 6)/(x^2*log(3/log(x))^2 - 6*x*log(3/log(x)) + 9)
Time = 0.10 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.19 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=\frac {4 x^{4} e^{6} e^{2 e^{3}}}{x^{2} \log {\left (\frac {3}{\log {\left (x \right )}} \right )}^{2} - 6 x \log {\left (\frac {3}{\log {\left (x \right )}} \right )} + 9} \] Input:
integrate((8*x**4*exp(exp(3)+3)**2*ln(x)*ln(3/ln(x))-48*x**3*exp(exp(3)+3) **2*ln(x)+8*x**4*exp(exp(3)+3)**2)/(x**3*ln(x)*ln(3/ln(x))**3-9*x**2*ln(x) *ln(3/ln(x))**2+27*x*ln(x)*ln(3/ln(x))-27*ln(x)),x)
Output:
4*x**4*exp(6)*exp(2*exp(3))/(x**2*log(3/log(x))**2 - 6*x*log(3/log(x)) + 9 )
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.71 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=\frac {4 \, x^{4} e^{\left (2 \, e^{3} + 6\right )}}{x^{2} \log \left (3\right )^{2} + x^{2} \log \left (\log \left (x\right )\right )^{2} - 6 \, x \log \left (3\right ) - 2 \, {\left (x^{2} \log \left (3\right ) - 3 \, x\right )} \log \left (\log \left (x\right )\right ) + 9} \] Input:
integrate((8*x^4*exp(exp(3)+3)^2*log(x)*log(3/log(x))-48*x^3*exp(exp(3)+3) ^2*log(x)+8*x^4*exp(exp(3)+3)^2)/(x^3*log(x)*log(3/log(x))^3-9*x^2*log(x)* log(3/log(x))^2+27*x*log(x)*log(3/log(x))-27*log(x)),x, algorithm="maxima" )
Output:
4*x^4*e^(2*e^3 + 6)/(x^2*log(3)^2 + x^2*log(log(x))^2 - 6*x*log(3) - 2*(x^ 2*log(3) - 3*x)*log(log(x)) + 9)
Time = 0.16 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=\frac {4 \, x^{4} e^{\left (2 \, e^{3} + 6\right )}}{x^{2} \log \left (3\right )^{2} - 2 \, x^{2} \log \left (3\right ) \log \left (\log \left (x\right )\right ) + x^{2} \log \left (\log \left (x\right )\right )^{2} - 6 \, x \log \left (3\right ) + 6 \, x \log \left (\log \left (x\right )\right ) + 9} \] Input:
integrate((8*x^4*exp(exp(3)+3)^2*log(x)*log(3/log(x))-48*x^3*exp(exp(3)+3) ^2*log(x)+8*x^4*exp(exp(3)+3)^2)/(x^3*log(x)*log(3/log(x))^3-9*x^2*log(x)* log(3/log(x))^2+27*x*log(x)*log(3/log(x))-27*log(x)),x, algorithm="giac")
Output:
4*x^4*e^(2*e^3 + 6)/(x^2*log(3)^2 - 2*x^2*log(3)*log(log(x)) + x^2*log(log (x))^2 - 6*x*log(3) + 6*x*log(log(x)) + 9)
Time = 1.70 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.39 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=\frac {4\,\left (3\,x^4\,{\mathrm {e}}^{2\,{\mathrm {e}}^3+6}\,{\ln \left (x\right )}^2-x^5\,{\mathrm {e}}^{2\,{\mathrm {e}}^3+6}\,\ln \left (x\right )\right )}{\left (3\,{\ln \left (x\right )}^2-x\,\ln \left (x\right )\right )\,\left (x^2\,{\ln \left (\frac {3}{\ln \left (x\right )}\right )}^2-6\,x\,\ln \left (\frac {3}{\ln \left (x\right )}\right )+9\right )} \] Input:
int(-(8*x^4*exp(2*exp(3) + 6) - 48*x^3*exp(2*exp(3) + 6)*log(x) + 8*x^4*lo g(3/log(x))*exp(2*exp(3) + 6)*log(x))/(27*log(x) + 9*x^2*log(3/log(x))^2*l og(x) - x^3*log(3/log(x))^3*log(x) - 27*x*log(3/log(x))*log(x)),x)
Output:
(4*(3*x^4*exp(2*exp(3) + 6)*log(x)^2 - x^5*exp(2*exp(3) + 6)*log(x)))/((3* log(x)^2 - x*log(x))*(x^2*log(3/log(x))^2 - 6*x*log(3/log(x)) + 9))
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35 \[ \int \frac {8 e^{6+2 e^3} x^4-48 e^{6+2 e^3} x^3 \log (x)+8 e^{6+2 e^3} x^4 \log (x) \log \left (\frac {3}{\log (x)}\right )}{-27 \log (x)+27 x \log (x) \log \left (\frac {3}{\log (x)}\right )-9 x^2 \log (x) \log ^2\left (\frac {3}{\log (x)}\right )+x^3 \log (x) \log ^3\left (\frac {3}{\log (x)}\right )} \, dx=\frac {4 e^{2 e^{3}} e^{6} x^{4}}{\mathrm {log}\left (\frac {3}{\mathrm {log}\left (x \right )}\right )^{2} x^{2}-6 \,\mathrm {log}\left (\frac {3}{\mathrm {log}\left (x \right )}\right ) x +9} \] Input:
int((8*x^4*exp(exp(3)+3)^2*log(x)*log(3/log(x))-48*x^3*exp(exp(3)+3)^2*log (x)+8*x^4*exp(exp(3)+3)^2)/(x^3*log(x)*log(3/log(x))^3-9*x^2*log(x)*log(3/ log(x))^2+27*x*log(x)*log(3/log(x))-27*log(x)),x)
Output:
(4*e**(2*e**3)*e**6*x**4)/(log(3/log(x))**2*x**2 - 6*log(3/log(x))*x + 9)