Integrand size = 125, antiderivative size = 33 \begin {dmath*} \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \left (1-\frac {x+\log (x)}{x}\right )^2}{x^2}\right )}{x} \end {dmath*}
Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.79 \begin {dmath*} \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=2+\frac {\log \left (6-\log (x)+\frac {e^{-2 x} \log ^2(x)}{x^4}\right )}{x} \end {dmath*}
Integrate[(-(E^(2*x)*x^4) + 2*Log[x] + (-4 - 2*x)*Log[x]^2 + (-6*E^(2*x)*x ^4 + E^(2*x)*x^4*Log[x] - Log[x]^2)*Log[(6*E^(2*x)*x^4 - E^(2*x)*x^4*Log[x ] + Log[x]^2)/(E^(2*x)*x^4)])/(6*E^(2*x)*x^6 - E^(2*x)*x^6*Log[x] + x^2*Lo g[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.
\(\displaystyle \int \frac {-e^{2 x} x^4+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )+(-2 x-4) \log ^2(x)+2 \log (x)}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {\log (x) \left (2 x \log ^2(x)+4 \log ^2(x)-12 x \log (x)-25 \log (x)+12\right )}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right )}+\frac {-\log (x) \log \left (\frac {e^{-2 x} \log ^2(x)}{x^4}-\log (x)+6\right )+6 \log \left (\frac {e^{-2 x} \log ^2(x)}{x^4}-\log (x)+6\right )+1}{x^2 (\log (x)-6)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -12 \int \frac {\log ^2(x)}{x (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx+2 \int \frac {\log ^3(x)}{x (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx+12 \int \frac {\log (x)}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx-25 \int \frac {\log ^2(x)}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx-\int \frac {\log \left (\frac {e^{-2 x} \log ^2(x)}{x^4}-\log (x)+6\right )}{x^2}dx+4 \int \frac {\log ^3(x)}{x^2 (\log (x)-6) \left (-6 e^{2 x} x^4+e^{2 x} \log (x) x^4-\log ^2(x)\right )}dx+\frac {\operatorname {ExpIntegralEi}(6-\log (x))}{e^6}\) |
Int[(-(E^(2*x)*x^4) + 2*Log[x] + (-4 - 2*x)*Log[x]^2 + (-6*E^(2*x)*x^4 + E ^(2*x)*x^4*Log[x] - Log[x]^2)*Log[(6*E^(2*x)*x^4 - E^(2*x)*x^4*Log[x] + Lo g[x]^2)/(E^(2*x)*x^4)])/(6*E^(2*x)*x^6 - E^(2*x)*x^6*Log[x] + x^2*Log[x]^2 ),x]
3.1.66.3.1 Defintions of rubi rules used
Time = 96.03 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18
method | result | size |
parallelrisch | \(\frac {\ln \left (\frac {\left (\ln \left (x \right )^{2}-x^{4} {\mathrm e}^{2 x} \ln \left (x \right )+6 \,{\mathrm e}^{2 x} x^{4}\right ) {\mathrm e}^{-2 x}}{x^{4}}\right )}{x}\) | \(39\) |
risch | \(\text {Expression too large to display}\) | \(722\) |
int(((-ln(x)^2+x^4*exp(x)^2*ln(x)-6*exp(x)^2*x^4)*ln((ln(x)^2-x^4*exp(x)^2 *ln(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)+(-2*x-4)*ln(x)^2+2*ln(x)-exp(x)^2*x^4 )/(x^2*ln(x)^2-x^6*exp(x)^2*ln(x)+6*x^6*exp(x)^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.21 \begin {dmath*} \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-\frac {{\left (x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )}}{x^{4}}\right )}{x} \end {dmath*}
integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^ 4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x) -exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algori thm=\
Time = 0.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \begin {dmath*} \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log {\left (\frac {\left (- x^{4} e^{2 x} \log {\left (x \right )} + 6 x^{4} e^{2 x} + \log {\left (x \right )}^{2}\right ) e^{- 2 x}}{x^{4}} \right )}}{x} \end {dmath*}
integrate(((-ln(x)**2+x**4*exp(x)**2*ln(x)-6*exp(x)**2*x**4)*ln((ln(x)**2- x**4*exp(x)**2*ln(x)+6*exp(x)**2*x**4)/exp(x)**2/x**4)+(-2*x-4)*ln(x)**2+2 *ln(x)-exp(x)**2*x**4)/(x**2*ln(x)**2-x**6*exp(x)**2*ln(x)+6*x**6*exp(x)** 2),x)
Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \begin {dmath*} \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) + 6 \, x^{4} e^{\left (2 \, x\right )} + \log \left (x\right )^{2}\right ) - 4 \, \log \left (x\right )}{x} \end {dmath*}
integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^ 4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x) -exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algori thm=\
Time = 0.38 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.27 \begin {dmath*} \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\log \left (-{\left (x^{4} e^{\left (2 \, x\right )} \log \left (x\right ) - 6 \, x^{4} e^{\left (2 \, x\right )} - \log \left (x\right )^{2}\right )} e^{\left (-2 \, x\right )}\right ) - 4 \, \log \left (x\right )}{x} \end {dmath*}
integrate(((-log(x)^2+x^4*exp(x)^2*log(x)-6*exp(x)^2*x^4)*log((log(x)^2-x^ 4*exp(x)^2*log(x)+6*exp(x)^2*x^4)/exp(x)^2/x^4)+(-2*x-4)*log(x)^2+2*log(x) -exp(x)^2*x^4)/(x^2*log(x)^2-x^6*exp(x)^2*log(x)+6*x^6*exp(x)^2),x, algori thm=\
Time = 13.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {-e^{2 x} x^4+2 \log (x)+(-4-2 x) \log ^2(x)+\left (-6 e^{2 x} x^4+e^{2 x} x^4 \log (x)-\log ^2(x)\right ) \log \left (\frac {e^{-2 x} \left (6 e^{2 x} x^4-e^{2 x} x^4 \log (x)+\log ^2(x)\right )}{x^4}\right )}{6 e^{2 x} x^6-e^{2 x} x^6 \log (x)+x^2 \log ^2(x)} \, dx=\frac {\ln \left (\frac {1}{x^4}\right )+\ln \left (6\,x^4-x^4\,\ln \left (x\right )+{\mathrm {e}}^{-2\,x}\,{\ln \left (x\right )}^2\right )}{x} \end {dmath*}