Integrand size = 145, antiderivative size = 22 \[ \int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx=\frac {\log \left (x+\frac {x \log (-3-x-\log (x))}{e^5}\right )}{x^2} \]
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx=\frac {\log \left (x+\frac {x \log (-3-x-\log (x))}{e^5}\right )}{x^2} \]
Integrate[(1 + x + E^5*(3 + x) + E^5*Log[x] + (3 + x + Log[x])*Log[-3 - x - Log[x]] + (E^5*(-6 - 2*x) - 2*E^5*Log[x] + (-6 - 2*x - 2*Log[x])*Log[-3 - x - Log[x]])*Log[(E^5*x + x*Log[-3 - x - Log[x]])/E^5])/(E^5*(3*x^3 + x^ 4) + E^5*x^3*Log[x] + (3*x^3 + x^4 + x^3*Log[x])*Log[-3 - x - Log[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.
\(\displaystyle \int \frac {x+e^5 (x+3)+e^5 \log (x)+(x+\log (x)+3) \log (-x-\log (x)-3)+\left (e^5 (-2 x-6)-2 e^5 \log (x)+(-2 x-2 \log (x)-6) \log (-x-\log (x)-3)\right ) \log \left (\frac {e^5 x+x \log (-x-\log (x)-3)}{e^5}\right )+1}{e^5 x^3 \log (x)+e^5 \left (x^4+3 x^3\right )+\left (x^4+3 x^3+x^3 \log (x)\right ) \log (-x-\log (x)-3)} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {x+e^5 (x+3)+e^5 \log (x)+(x+\log (x)+3) \log (-x-\log (x)-3)+\left (e^5 (-2 x-6)-2 e^5 \log (x)+(-2 x-2 \log (x)-6) \log (-x-\log (x)-3)\right ) \log \left (\frac {e^5 x+x \log (-x-\log (x)-3)}{e^5}\right )+1}{x^3 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^5 (x+3)}{x^3 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}-\frac {2 \log \left (\frac {x \left (\log (-x-\log (x)-3)+e^5\right )}{e^5}\right )}{x^3}+\frac {\log (-x-\log (x)-3)}{x^3 \left (\log (-x-\log (x)-3)+e^5\right )}+\frac {e^5 \log (x)}{x^3 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}+\frac {1}{x^3 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}+\frac {1}{x^2 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -e^5 \int \frac {1}{x^3 \left (\log (-x-\log (x)-3)+e^5\right )}dx+3 e^5 \int \frac {1}{x^3 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}dx+e^5 \int \frac {\log (x)}{x^3 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}dx+e^5 \int \frac {1}{x^2 (x+\log (x)+3) \left (\log (-x-\log (x)-3)+e^5\right )}dx+\frac {\log \left (\frac {x \left (\log (-x-\log (x)-3)+e^5\right )}{e^5}\right )}{x^2}\) |
Int[(1 + x + E^5*(3 + x) + E^5*Log[x] + (3 + x + Log[x])*Log[-3 - x - Log[ x]] + (E^5*(-6 - 2*x) - 2*E^5*Log[x] + (-6 - 2*x - 2*Log[x])*Log[-3 - x - Log[x]])*Log[(E^5*x + x*Log[-3 - x - Log[x]])/E^5])/(E^5*(3*x^3 + x^4) + E ^5*x^3*Log[x] + (3*x^3 + x^4 + x^3*Log[x])*Log[-3 - x - Log[x]]),x]
3.7.46.3.1 Defintions of rubi rules used
Time = 33.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
parallelrisch | \(\frac {\ln \left (x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right ) {\mathrm e}^{-5}\right )}{x^{2}}\) | \(25\) |
risch | \(\frac {\ln \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )}{x^{2}}+\frac {-i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right ) \operatorname {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )+i \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )}^{2}+i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right ) {\operatorname {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )}^{2}-10-i \pi {\operatorname {csgn}\left (i x \left ({\mathrm e}^{5}+\ln \left (-\ln \left (x \right )-3-x \right )\right )\right )}^{3}+2 \ln \left (x \right )}{2 x^{2}}\) | \(169\) |
int((((-2*ln(x)-2*x-6)*ln(-ln(x)-3-x)-2*exp(5)*ln(x)+(-2*x-6)*exp(5))*ln(( x*ln(-ln(x)-3-x)+x*exp(5))/exp(5))+(3+x+ln(x))*ln(-ln(x)-3-x)+exp(5)*ln(x) +(3+x)*exp(5)+x+1)/((x^3*ln(x)+x^4+3*x^3)*ln(-ln(x)-3-x)+x^3*exp(5)*ln(x)+ (x^4+3*x^3)*exp(5)),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx=\frac {\log \left ({\left (x e^{5} + x \log \left (-x - \log \left (x\right ) - 3\right )\right )} e^{\left (-5\right )}\right )}{x^{2}} \]
integrate((((-2*log(x)-2*x-6)*log(-log(x)-3-x)-2*exp(5)*log(x)+(-2*x-6)*ex p(5))*log((x*log(-log(x)-3-x)+x*exp(5))/exp(5))+(3+x+log(x))*log(-log(x)-3 -x)+exp(5)*log(x)+(3+x)*exp(5)+x+1)/((x^3*log(x)+x^4+3*x^3)*log(-log(x)-3- x)+x^3*exp(5)*log(x)+(x^4+3*x^3)*exp(5)),x, algorithm=\
Time = 3.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx=\frac {\log {\left (\frac {x \log {\left (- x - \log {\left (x \right )} - 3 \right )} + x e^{5}}{e^{5}} \right )}}{x^{2}} \]
integrate((((-2*ln(x)-2*x-6)*ln(-ln(x)-3-x)-2*exp(5)*ln(x)+(-2*x-6)*exp(5) )*ln((x*ln(-ln(x)-3-x)+x*exp(5))/exp(5))+(3+x+ln(x))*ln(-ln(x)-3-x)+exp(5) *ln(x)+(3+x)*exp(5)+x+1)/((x**3*ln(x)+x**4+3*x**3)*ln(-ln(x)-3-x)+x**3*exp (5)*ln(x)+(x**4+3*x**3)*exp(5)),x)
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx=\frac {\log \left (x\right ) + \log \left (e^{5} + \log \left (-x - \log \left (x\right ) - 3\right )\right ) - 5}{x^{2}} \]
integrate((((-2*log(x)-2*x-6)*log(-log(x)-3-x)-2*exp(5)*log(x)+(-2*x-6)*ex p(5))*log((x*log(-log(x)-3-x)+x*exp(5))/exp(5))+(3+x+log(x))*log(-log(x)-3 -x)+exp(5)*log(x)+(3+x)*exp(5)+x+1)/((x^3*log(x)+x^4+3*x^3)*log(-log(x)-3- x)+x^3*exp(5)*log(x)+(x^4+3*x^3)*exp(5)),x, algorithm=\
Time = 0.36 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx=\frac {\log \left (x\right ) + \log \left (e^{5} + \log \left (-x - \log \left (x\right ) - 3\right )\right ) - 5}{x^{2}} \]
integrate((((-2*log(x)-2*x-6)*log(-log(x)-3-x)-2*exp(5)*log(x)+(-2*x-6)*ex p(5))*log((x*log(-log(x)-3-x)+x*exp(5))/exp(5))+(3+x+log(x))*log(-log(x)-3 -x)+exp(5)*log(x)+(3+x)*exp(5)+x+1)/((x^3*log(x)+x^4+3*x^3)*log(-log(x)-3- x)+x^3*exp(5)*log(x)+(x^4+3*x^3)*exp(5)),x, algorithm=\
Time = 9.64 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1+x+e^5 (3+x)+e^5 \log (x)+(3+x+\log (x)) \log (-3-x-\log (x))+\left (e^5 (-6-2 x)-2 e^5 \log (x)+(-6-2 x-2 \log (x)) \log (-3-x-\log (x))\right ) \log \left (\frac {e^5 x+x \log (-3-x-\log (x))}{e^5}\right )}{e^5 \left (3 x^3+x^4\right )+e^5 x^3 \log (x)+\left (3 x^3+x^4+x^3 \log (x)\right ) \log (-3-x-\log (x))} \, dx=\frac {\ln \left (x\,\left ({\mathrm {e}}^5+\ln \left (-x-\ln \left (x\right )-3\right )\right )\right )-5}{x^2} \]
int((x + log(- x - log(x) - 3)*(x + log(x) + 3) + exp(5)*(x + 3) + exp(5)* log(x) - log(exp(-5)*(x*exp(5) + x*log(- x - log(x) - 3)))*(2*exp(5)*log(x ) + log(- x - log(x) - 3)*(2*x + 2*log(x) + 6) + exp(5)*(2*x + 6)) + 1)/(e xp(5)*(3*x^3 + x^4) + log(- x - log(x) - 3)*(x^3*log(x) + 3*x^3 + x^4) + x ^3*exp(5)*log(x)),x)