Integrand size = 84, antiderivative size = 23 \begin {dmath*} \int \frac {-x^2+x^5+\left (x^2+x^3\right ) \log (x)+\left (3 x^3+3 x^4+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )}{\left (x+x^2+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \, dx=\frac {x^3}{\log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \end {dmath*}
Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {-x^2+x^5+\left (x^2+x^3\right ) \log (x)+\left (3 x^3+3 x^4+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )}{\left (x+x^2+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \, dx=\frac {x^3}{\log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \end {dmath*}
Integrate[(-x^2 + x^5 + (x^2 + x^3)*Log[x] + (3*x^3 + 3*x^4 + 3*x^2*Log[x] )*Log[(x + x^2 + Log[x])/(E^x*x)])/((x + x^2 + Log[x])*Log[(x + x^2 + Log[ x])/(E^x*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.
\(\displaystyle \int \frac {x^5-x^2+\left (x^3+x^2\right ) \log (x)+\left (3 x^4+3 x^3+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}{\left (x^2+x+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x^2}{\log \left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}+\frac {x^2 \left (x^3+x \log (x)+\log (x)-1\right )}{\left (x^2+x+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {x^2}{\left (x^2+x+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}dx+\int \frac {x^2 \log (x)}{\left (x^2+x+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}dx+3 \int \frac {x^2}{\log \left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}dx+\int \frac {x^5}{\left (x^2+x+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}dx+\int \frac {x^3 \log (x)}{\left (x^2+x+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x^2+x+\log (x)\right )}{x}\right )}dx\) |
Int[(-x^2 + x^5 + (x^2 + x^3)*Log[x] + (3*x^3 + 3*x^4 + 3*x^2*Log[x])*Log[ (x + x^2 + Log[x])/(E^x*x)])/((x + x^2 + Log[x])*Log[(x + x^2 + Log[x])/(E ^x*x)]^2),x]
3.12.87.3.1 Defintions of rubi rules used
Time = 1.82 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00
method | result | size |
parallelrisch | \(\frac {x^{3}}{\ln \left (\frac {\left (\ln \left (x \right )+x^{2}+x \right ) {\mathrm e}^{-x}}{x}\right )}\) | \(23\) |
risch | \(-\frac {2 i x^{3}}{\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+x^{2}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\ln \left (x \right )+x^{2}+x \right )\right )^{2}-\pi \,\operatorname {csgn}\left (i \left (\ln \left (x \right )+x^{2}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\ln \left (x \right )+x^{2}+x \right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )-\pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\ln \left (x \right )+x^{2}+x \right )\right )^{3}+\pi \operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\ln \left (x \right )+x^{2}+x \right )\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\ln \left (x \right )+x^{2}+x \right )\right ) {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2}+x \right ) {\mathrm e}^{-x}}{x}\right )}^{2}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{-x} \left (\ln \left (x \right )+x^{2}+x \right )\right ) \operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2}+x \right ) {\mathrm e}^{-x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-\pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2}+x \right ) {\mathrm e}^{-x}}{x}\right )}^{3}+\pi {\operatorname {csgn}\left (\frac {i \left (\ln \left (x \right )+x^{2}+x \right ) {\mathrm e}^{-x}}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )+2 i \ln \left (x \right )-2 i \ln \left (\ln \left (x \right )+x^{2}+x \right )+2 i \ln \left ({\mathrm e}^{x}\right )}\) | \(278\) |
int(((3*x^2*ln(x)+3*x^4+3*x^3)*ln((ln(x)+x^2+x)/exp(x)/x)+(x^3+x^2)*ln(x)+ x^5-x^2)/(ln(x)+x^2+x)/ln((ln(x)+x^2+x)/exp(x)/x)^2,x,method=_RETURNVERBOS E)
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \begin {dmath*} \int \frac {-x^2+x^5+\left (x^2+x^3\right ) \log (x)+\left (3 x^3+3 x^4+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )}{\left (x+x^2+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \, dx=\frac {x^{3}}{\log \left (\frac {{\left (x^{2} + x\right )} e^{\left (-x\right )} + e^{\left (-x\right )} \log \left (x\right )}{x}\right )} \end {dmath*}
integrate(((3*x^2*log(x)+3*x^4+3*x^3)*log((log(x)+x^2+x)/exp(x)/x)+(x^3+x^ 2)*log(x)+x^5-x^2)/(log(x)+x^2+x)/log((log(x)+x^2+x)/exp(x)/x)^2,x, algori thm=\
Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \begin {dmath*} \int \frac {-x^2+x^5+\left (x^2+x^3\right ) \log (x)+\left (3 x^3+3 x^4+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )}{\left (x+x^2+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \, dx=\frac {x^{3}}{\log {\left (\frac {\left (x^{2} + x + \log {\left (x \right )}\right ) e^{- x}}{x} \right )}} \end {dmath*}
integrate(((3*x**2*ln(x)+3*x**4+3*x**3)*ln((ln(x)+x**2+x)/exp(x)/x)+(x**3+ x**2)*ln(x)+x**5-x**2)/(ln(x)+x**2+x)/ln((ln(x)+x**2+x)/exp(x)/x)**2,x)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \begin {dmath*} \int \frac {-x^2+x^5+\left (x^2+x^3\right ) \log (x)+\left (3 x^3+3 x^4+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )}{\left (x+x^2+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \, dx=-\frac {x^{3}}{x - \log \left (x^{2} + x + \log \left (x\right )\right ) + \log \left (x\right )} \end {dmath*}
integrate(((3*x^2*log(x)+3*x^4+3*x^3)*log((log(x)+x^2+x)/exp(x)/x)+(x^3+x^ 2)*log(x)+x^5-x^2)/(log(x)+x^2+x)/log((log(x)+x^2+x)/exp(x)/x)^2,x, algori thm=\
Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \begin {dmath*} \int \frac {-x^2+x^5+\left (x^2+x^3\right ) \log (x)+\left (3 x^3+3 x^4+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )}{\left (x+x^2+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \, dx=-\frac {x^{3}}{x - \log \left (x^{2} + x + \log \left (x\right )\right ) + \log \left (x\right )} \end {dmath*}
integrate(((3*x^2*log(x)+3*x^4+3*x^3)*log((log(x)+x^2+x)/exp(x)/x)+(x^3+x^ 2)*log(x)+x^5-x^2)/(log(x)+x^2+x)/log((log(x)+x^2+x)/exp(x)/x)^2,x, algori thm=\
Time = 15.48 (sec) , antiderivative size = 156, normalized size of antiderivative = 6.78 \begin {dmath*} \int \frac {-x^2+x^5+\left (x^2+x^3\right ) \log (x)+\left (3 x^3+3 x^4+3 x^2 \log (x)\right ) \log \left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )}{\left (x+x^2+\log (x)\right ) \log ^2\left (\frac {e^{-x} \left (x+x^2+\log (x)\right )}{x}\right )} \, dx=3\,x+\frac {3}{x+1}-3\,x^2+\frac {x^3+\frac {3\,x^3\,\ln \left (\frac {{\mathrm {e}}^{-x}\,\left (x+\ln \left (x\right )+x^2\right )}{x}\right )\,\left (x+\ln \left (x\right )+x^2\right )}{\ln \left (x\right )+x\,\ln \left (x\right )+x^3-1}}{\ln \left (\frac {{\mathrm {e}}^{-x}\,\left (x+\ln \left (x\right )+x^2\right )}{x}\right )}-\frac {3\,\left (4\,x^{10}+8\,x^9+7\,x^8+10\,x^7+6\,x^6+4\,x^5+x^4\right )}{\left (x+1\right )\,\left (\ln \left (x\right )\,\left (x+1\right )+x^3-1\right )\,\left (2\,x^5+3\,x^4+x^3+3\,x^2+x\right )} \end {dmath*}
int((log(x)*(x^2 + x^3) + log((exp(-x)*(x + log(x) + x^2))/x)*(3*x^2*log(x ) + 3*x^3 + 3*x^4) - x^2 + x^5)/(log((exp(-x)*(x + log(x) + x^2))/x)^2*(x + log(x) + x^2)),x)
3*x + 3/(x + 1) - 3*x^2 + (x^3 + (3*x^3*log((exp(-x)*(x + log(x) + x^2))/x )*(x + log(x) + x^2))/(log(x) + x*log(x) + x^3 - 1))/log((exp(-x)*(x + log (x) + x^2))/x) - (3*(x^4 + 4*x^5 + 6*x^6 + 10*x^7 + 7*x^8 + 8*x^9 + 4*x^10 ))/((x + 1)*(log(x)*(x + 1) + x^3 - 1)*(x + 3*x^2 + x^3 + 3*x^4 + 2*x^5))