Integrand size = 82, antiderivative size = 20 \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx=e^{3 \left (-5+x-\frac {e^5 x}{\log (3+\log (x))}\right )} \] Output:
exp(3*x-15+3*x/exp(ln(-ln(3+ln(x)))-5))
Time = 0.40 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx=e^{-15+3 x-\frac {3 e^5 x}{\log (3+\log (x))}} \] Input:
Integrate[-((E^(5 - (E^5*(3*x - ((-15 + 3*x)*Log[3 + Log[x]])/E^5))/Log[3 + Log[x]])*(-3 + (9 + 3*Log[x])*Log[3 + Log[x]] - ((9 + 3*Log[x])*Log[3 + Log[x]]^2)/E^5))/((3 + Log[x])*Log[3 + Log[x]]^2)),x]
Output:
E^(-15 + 3*x - (3*E^5*x)/Log[3 + Log[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.
\(\displaystyle \int -\frac {\left (-\frac {(3 \log (x)+9) \log ^2(\log (x)+3)}{e^5}+(3 \log (x)+9) \log (\log (x)+3)-3\right ) \exp \left (5-\frac {e^5 \left (3 x-\frac {(3 x-15) \log (\log (x)+3)}{e^5}\right )}{\log (\log (x)+3)}\right )}{(\log (x)+3) \log ^2(\log (x)+3)} \, dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {3 \exp \left (5-\frac {3 e^5 \left (x+\frac {(5-x) \log (\log (x)+3)}{e^5}\right )}{\log (\log (x)+3)}\right ) \left (\frac {(\log (x)+3) \log ^2(\log (x)+3)}{e^5}-(\log (x)+3) \log (\log (x)+3)+1\right )}{(\log (x)+3) \log ^2(\log (x)+3)}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 3 \int \frac {\exp \left (5-\frac {3 e^5 \left (x+\frac {(5-x) \log (\log (x)+3)}{e^5}\right )}{\log (\log (x)+3)}\right ) \left (\frac {(\log (x)+3) \log ^2(\log (x)+3)}{e^5}-(\log (x)+3) \log (\log (x)+3)+1\right )}{(\log (x)+3) \log ^2(\log (x)+3)}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle 3 \int \frac {\exp \left (\frac {3 \log (\log (x)+3) x-3 e^5 x-10 \log (\log (x)+3)}{\log (\log (x)+3)}\right ) \left (\frac {(\log (x)+3) \log ^2(\log (x)+3)}{e^5}-(\log (x)+3) \log (\log (x)+3)+1\right )}{(\log (x)+3) \log ^2(\log (x)+3)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 3 \int \left (\exp \left (\frac {3 \log (\log (x)+3) x-3 e^5 x-10 \log (\log (x)+3)}{\log (\log (x)+3)}-5\right )-\frac {\exp \left (\frac {3 \log (\log (x)+3) x-3 e^5 x-10 \log (\log (x)+3)}{\log (\log (x)+3)}\right )}{\log (\log (x)+3)}+\frac {\exp \left (\frac {3 \log (\log (x)+3) x-3 e^5 x-10 \log (\log (x)+3)}{\log (\log (x)+3)}\right )}{\log ^2(\log (x)+3) (\log (x)+3)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \left (\int \frac {\exp \left (\frac {3 \log (\log (x)+3) x-3 e^5 x-10 \log (\log (x)+3)}{\log (\log (x)+3)}\right )}{(\log (x)+3) \log ^2(\log (x)+3)}dx-\int \frac {\exp \left (\frac {3 \log (\log (x)+3) x-3 e^5 x-10 \log (\log (x)+3)}{\log (\log (x)+3)}\right )}{\log (\log (x)+3)}dx+\int e^{3 \left (-\frac {e^5 x}{\log (\log (x)+3)}+x-5\right )}dx\right )\) |
Input:
Int[-((E^(5 - (E^5*(3*x - ((-15 + 3*x)*Log[3 + Log[x]])/E^5))/Log[3 + Log[ x]])*(-3 + (9 + 3*Log[x])*Log[3 + Log[x]] - ((9 + 3*Log[x])*Log[3 + Log[x] ]^2)/E^5))/((3 + Log[x])*Log[3 + Log[x]]^2)),x]
Output:
$Aborted
Time = 39.48 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75
method | result | size |
risch | \({\mathrm e}^{\frac {3 \left (\ln \left (3+\ln \left (x \right )\right ) {\mathrm e}^{-5} x -5 \ln \left (3+\ln \left (x \right )\right ) {\mathrm e}^{-5}-x \right ) {\mathrm e}^{5}}{\ln \left (3+\ln \left (x \right )\right )}}\) | \(35\) |
parallelrisch | \({\mathrm e}^{-\frac {\left (\left (3 x -15\right ) {\mathrm e}^{\ln \left (-\ln \left (3+\ln \left (x \right )\right )\right )-5}+3 x \right ) {\mathrm e}^{5}}{\ln \left (3+\ln \left (x \right )\right )}}\) | \(37\) |
Input:
int(((3*ln(x)+9)*ln(3+ln(x))*exp(ln(-ln(3+ln(x)))-5)+(3*ln(x)+9)*ln(3+ln(x ))-3)*exp(((3*x-15)*exp(ln(-ln(3+ln(x)))-5)+3*x)/exp(ln(-ln(3+ln(x)))-5))/ (3+ln(x))/ln(3+ln(x))/exp(ln(-ln(3+ln(x)))-5),x,method=_RETURNVERBOSE)
Output:
exp(3*(ln(3+ln(x))*exp(-5)*x-5*ln(3+ln(x))*exp(-5)-x)*exp(5)/ln(3+ln(x)))
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.55 \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx=-e^{\left (-\frac {3 \, x e^{5} - {\left (3 \, x - 10\right )} \log \left (\log \left (x\right ) + 3\right ) + \log \left (-\log \left (\log \left (x\right ) + 3\right )\right ) \log \left (\log \left (x\right ) + 3\right )}{\log \left (\log \left (x\right ) + 3\right )} - 5\right )} \log \left (\log \left (x\right ) + 3\right ) \] Input:
integrate(((3*log(x)+9)*log(3+log(x))*exp(log(-log(3+log(x)))-5)+(3*log(x) +9)*log(3+log(x))-3)*exp(((3*x-15)*exp(log(-log(3+log(x)))-5)+3*x)/exp(log (-log(3+log(x)))-5))/(3+log(x))/log(3+log(x))/exp(log(-log(3+log(x)))-5),x , algorithm="fricas")
Output:
-e^(-(3*x*e^5 - (3*x - 10)*log(log(x) + 3) + log(-log(log(x) + 3))*log(log (x) + 3))/log(log(x) + 3) - 5)*log(log(x) + 3)
Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.55 \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx=e^{- \frac {\left (3 x - \frac {\left (3 x - 15\right ) \log {\left (\log {\left (x \right )} + 3 \right )}}{e^{5}}\right ) e^{5}}{\log {\left (\log {\left (x \right )} + 3 \right )}}} \] Input:
integrate(((3*ln(x)+9)*ln(3+ln(x))*exp(ln(-ln(3+ln(x)))-5)+(3*ln(x)+9)*ln( 3+ln(x))-3)*exp(((3*x-15)*exp(ln(-ln(3+ln(x)))-5)+3*x)/exp(ln(-ln(3+ln(x)) )-5))/(3+ln(x))/ln(3+ln(x))/exp(ln(-ln(3+ln(x)))-5),x)
Output:
exp(-(3*x - (3*x - 15)*exp(-5)*log(log(x) + 3))*exp(5)/log(log(x) + 3))
Exception generated. \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((3*log(x)+9)*log(3+log(x))*exp(log(-log(3+log(x)))-5)+(3*log(x) +9)*log(3+log(x))-3)*exp(((3*x-15)*exp(log(-log(3+log(x)))-5)+3*x)/exp(log (-log(3+log(x)))-5))/(3+log(x))/log(3+log(x))/exp(log(-log(3+log(x)))-5),x , algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
Exception generated. \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((3*log(x)+9)*log(3+log(x))*exp(log(-log(3+log(x)))-5)+(3*log(x) +9)*log(3+log(x))-3)*exp(((3*x-15)*exp(log(-log(3+log(x)))-5)+3*x)/exp(log (-log(3+log(x)))-5))/(3+log(x))/log(3+log(x))/exp(log(-log(3+log(x)))-5),x , algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{27,[ 0,1,0,3]%%%}+%%%{81,[0,0,0,3]%%%} / %%%{27,[0,2,0,3]%%%}+%%%{162,[0,1,0,3] %%%}+%%%{243
Time = 4.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx={\mathrm {e}}^{3\,x}\,{\mathrm {e}}^{-15}\,{\mathrm {e}}^{-\frac {3\,x\,{\mathrm {e}}^5}{\ln \left (\ln \left (x\right )+3\right )}} \] Input:
int((exp(exp(5 - log(-log(log(x) + 3)))*(3*x + exp(log(-log(log(x) + 3)) - 5)*(3*x - 15)))*exp(5 - log(-log(log(x) + 3)))*(log(log(x) + 3)*(3*log(x) + 9) + exp(log(-log(log(x) + 3)) - 5)*log(log(x) + 3)*(3*log(x) + 9) - 3) )/(log(log(x) + 3)*(log(x) + 3)),x)
Output:
exp(3*x)*exp(-15)*exp(-(3*x*exp(5))/log(log(x) + 3))
Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.30 \[ \int -\frac {e^{5-\frac {e^5 \left (3 x-\frac {(-15+3 x) \log (3+\log (x))}{e^5}\right )}{\log (3+\log (x))}} \left (-3+(9+3 \log (x)) \log (3+\log (x))-\frac {(9+3 \log (x)) \log ^2(3+\log (x))}{e^5}\right )}{(3+\log (x)) \log ^2(3+\log (x))} \, dx=\frac {e^{3 x}}{e^{\frac {3 e^{5} x}{\mathrm {log}\left (\mathrm {log}\left (x \right )+3\right )}} e^{15}} \] Input:
int(((3*log(x)+9)*log(3+log(x))*exp(log(-log(3+log(x)))-5)+(3*log(x)+9)*lo g(3+log(x))-3)*exp(((3*x-15)*exp(log(-log(3+log(x)))-5)+3*x)/exp(log(-log( 3+log(x)))-5))/(3+log(x))/log(3+log(x))/exp(log(-log(3+log(x)))-5),x)
Output:
e**(3*x)/(e**((3*e**5*x)/log(log(x) + 3))*e**15)