Integrand size = 149, antiderivative size = 25 \[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\frac {\log \left (\log \left (7-e^{-\frac {1}{x}+\frac {6+x}{9}}\right )\right )}{x} \] Output:
ln(ln(7-exp(2/3-1/x+1/9*x)))/x
Time = 1.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\frac {\log \left (\log \left (7-e^{\frac {2}{3}-\frac {1}{x}+\frac {x}{9}}\right )\right )}{x} \] Input:
Integrate[(E^((-9 + 6*x + x^2)/(9*x))*(9 + x^2) + (63*x - 9*E^((-9 + 6*x + x^2)/(9*x))*x)*Log[7 - E^((-9 + 6*x + x^2)/(9*x))]*Log[Log[7 - E^((-9 + 6 *x + x^2)/(9*x))]])/((-63*x^3 + 9*E^((-9 + 6*x + x^2)/(9*x))*x^3)*Log[7 - E^((-9 + 6*x + x^2)/(9*x))]),x]
Output:
Log[Log[7 - E^(2/3 - x^(-1) + x/9)]]/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.
| \(\displaystyle \int \frac {e^{\frac {x^2+6 x-9}{9 x}} \left (x^2+9\right )+\left (63 x-9 e^{\frac {x^2+6 x-9}{9 x}} x\right ) \log \left (7-e^{\frac {x^2+6 x-9}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {x^2+6 x-9}{9 x}}\right )\right )}{\left (9 e^{\frac {x^2+6 x-9}{9 x}} x^3-63 x^3\right ) \log \left (7-e^{\frac {x^2+6 x-9}{9 x}}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \left (\frac {e^{\frac {x}{9}+\frac {2}{3}} \left (x^2+9\right )}{9 \left (e^{\frac {x+6}{9}}-7 e^{\frac {1}{x}}\right ) x^3 \log \left (7-e^{\frac {1}{9} \left (x-\frac {9}{x}+6\right )}\right )}-\frac {\log \left (\log \left (7-e^{\frac {1}{9} \left (x-\frac {9}{x}+6\right )}\right )\right )}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {e^{\frac {x}{9}+\frac {2}{3}}}{\left (e^{\frac {x}{9}+\frac {2}{3}}-7 e^{\frac {1}{x}}\right ) x^3 \log \left (7-e^{\frac {1}{9} \left (x+6-\frac {9}{x}\right )}\right )}dx-\int \frac {\log \left (\log \left (7-e^{\frac {1}{9} \left (x+6-\frac {9}{x}\right )}\right )\right )}{x^2}dx+\frac {1}{9} \int \frac {e^{\frac {x}{9}+\frac {2}{3}}}{\left (e^{\frac {x}{9}+\frac {2}{3}}-7 e^{\frac {1}{x}}\right ) x \log \left (7-e^{\frac {1}{9} \left (x+6-\frac {9}{x}\right )}\right )}dx\) |
Input:
Int[(E^((-9 + 6*x + x^2)/(9*x))*(9 + x^2) + (63*x - 9*E^((-9 + 6*x + x^2)/ (9*x))*x)*Log[7 - E^((-9 + 6*x + x^2)/(9*x))]*Log[Log[7 - E^((-9 + 6*x + x ^2)/(9*x))]])/((-63*x^3 + 9*E^((-9 + 6*x + x^2)/(9*x))*x^3)*Log[7 - E^((-9 + 6*x + x^2)/(9*x))]),x]
Output:
$Aborted
Time = 1.80 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(\frac {\ln \left (\ln \left (-{\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}+7\right )\right )}{x}\) | \(25\) |
| parallelrisch | \(\frac {\ln \left (\ln \left (-{\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}+7\right )\right )}{x}\) | \(25\) |
Input:
int(((-9*x*exp(1/9*(x^2+6*x-9)/x)+63*x)*ln(-exp(1/9*(x^2+6*x-9)/x)+7)*ln(l n(-exp(1/9*(x^2+6*x-9)/x)+7))+(x^2+9)*exp(1/9*(x^2+6*x-9)/x))/(9*x^3*exp(1 /9*(x^2+6*x-9)/x)-63*x^3)/ln(-exp(1/9*(x^2+6*x-9)/x)+7),x,method=_RETURNVE RBOSE)
Output:
1/x*ln(ln(-exp(1/9*(x^2+6*x-9)/x)+7))
Time = 0.11 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\frac {\log \left (\log \left (-e^{\left (\frac {x^{2} + 6 \, x - 9}{9 \, x}\right )} + 7\right )\right )}{x} \] Input:
integrate(((-9*x*exp(1/9*(x^2+6*x-9)/x)+63*x)*log(-exp(1/9*(x^2+6*x-9)/x)+ 7)*log(log(-exp(1/9*(x^2+6*x-9)/x)+7))+(x^2+9)*exp(1/9*(x^2+6*x-9)/x))/(9* x^3*exp(1/9*(x^2+6*x-9)/x)-63*x^3)/log(-exp(1/9*(x^2+6*x-9)/x)+7),x, algor ithm="fricas")
Output:
log(log(-e^(1/9*(x^2 + 6*x - 9)/x) + 7))/x
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\frac {\log {\left (\log {\left (7 - e^{\frac {\frac {x^{2}}{9} + \frac {2 x}{3} - 1}{x}} \right )} \right )}}{x} \] Input:
integrate(((-9*x*exp(1/9*(x**2+6*x-9)/x)+63*x)*ln(-exp(1/9*(x**2+6*x-9)/x) +7)*ln(ln(-exp(1/9*(x**2+6*x-9)/x)+7))+(x**2+9)*exp(1/9*(x**2+6*x-9)/x))/( 9*x**3*exp(1/9*(x**2+6*x-9)/x)-63*x**3)/ln(-exp(1/9*(x**2+6*x-9)/x)+7),x)
Output:
log(log(7 - exp((x**2/9 + 2*x/3 - 1)/x)))/x
Time = 0.13 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\frac {\log \left (x \log \left (-e^{\left (\frac {1}{9} \, x + \frac {2}{3}\right )} + 7 \, e^{\frac {1}{x}}\right ) - 1\right ) - \log \left (x\right )}{x} \] Input:
integrate(((-9*x*exp(1/9*(x^2+6*x-9)/x)+63*x)*log(-exp(1/9*(x^2+6*x-9)/x)+ 7)*log(log(-exp(1/9*(x^2+6*x-9)/x)+7))+(x^2+9)*exp(1/9*(x^2+6*x-9)/x))/(9* x^3*exp(1/9*(x^2+6*x-9)/x)-63*x^3)/log(-exp(1/9*(x^2+6*x-9)/x)+7),x, algor ithm="maxima")
Output:
(log(x*log(-e^(1/9*x + 2/3) + 7*e^(1/x)) - 1) - log(x))/x
\[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\int { -\frac {9 \, {\left (x e^{\left (\frac {x^{2} + 6 \, x - 9}{9 \, x}\right )} - 7 \, x\right )} \log \left (-e^{\left (\frac {x^{2} + 6 \, x - 9}{9 \, x}\right )} + 7\right ) \log \left (\log \left (-e^{\left (\frac {x^{2} + 6 \, x - 9}{9 \, x}\right )} + 7\right )\right ) - {\left (x^{2} + 9\right )} e^{\left (\frac {x^{2} + 6 \, x - 9}{9 \, x}\right )}}{9 \, {\left (x^{3} e^{\left (\frac {x^{2} + 6 \, x - 9}{9 \, x}\right )} - 7 \, x^{3}\right )} \log \left (-e^{\left (\frac {x^{2} + 6 \, x - 9}{9 \, x}\right )} + 7\right )} \,d x } \] Input:
integrate(((-9*x*exp(1/9*(x^2+6*x-9)/x)+63*x)*log(-exp(1/9*(x^2+6*x-9)/x)+ 7)*log(log(-exp(1/9*(x^2+6*x-9)/x)+7))+(x^2+9)*exp(1/9*(x^2+6*x-9)/x))/(9* x^3*exp(1/9*(x^2+6*x-9)/x)-63*x^3)/log(-exp(1/9*(x^2+6*x-9)/x)+7),x, algor ithm="giac")
Output:
integrate(-1/9*(9*(x*e^(1/9*(x^2 + 6*x - 9)/x) - 7*x)*log(-e^(1/9*(x^2 + 6 *x - 9)/x) + 7)*log(log(-e^(1/9*(x^2 + 6*x - 9)/x) + 7)) - (x^2 + 9)*e^(1/ 9*(x^2 + 6*x - 9)/x))/((x^3*e^(1/9*(x^2 + 6*x - 9)/x) - 7*x^3)*log(-e^(1/9 *(x^2 + 6*x - 9)/x) + 7)), x)
Time = 4.68 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\frac {\ln \left (\ln \left (7-{\mathrm {e}}^{x/9}\,{\mathrm {e}}^{2/3}\,{\mathrm {e}}^{-\frac {1}{x}}\right )\right )}{x} \] Input:
int((exp(((2*x)/3 + x^2/9 - 1)/x)*(x^2 + 9) + log(7 - exp(((2*x)/3 + x^2/9 - 1)/x))*log(log(7 - exp(((2*x)/3 + x^2/9 - 1)/x)))*(63*x - 9*x*exp(((2*x )/3 + x^2/9 - 1)/x)))/(log(7 - exp(((2*x)/3 + x^2/9 - 1)/x))*(9*x^3*exp((( 2*x)/3 + x^2/9 - 1)/x) - 63*x^3)),x)
Output:
log(log(7 - exp(x/9)*exp(2/3)*exp(-1/x)))/x
\[ \int \frac {e^{\frac {-9+6 x+x^2}{9 x}} \left (9+x^2\right )+\left (63 x-9 e^{\frac {-9+6 x+x^2}{9 x}} x\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right ) \log \left (\log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )\right )}{\left (-63 x^3+9 e^{\frac {-9+6 x+x^2}{9 x}} x^3\right ) \log \left (7-e^{\frac {-9+6 x+x^2}{9 x}}\right )} \, dx=\int \frac {\left (-9 x \,{\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}+63 x \right ) \mathrm {log}\left (-{\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}+7\right ) \mathrm {log}\left (\mathrm {log}\left (-{\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}+7\right )\right )+\left (x^{2}+9\right ) {\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}}{\left (9 x^{3} {\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}-63 x^{3}\right ) \mathrm {log}\left (-{\mathrm e}^{\frac {x^{2}+6 x -9}{9 x}}+7\right )}d x \] Input:
int(((-9*x*exp(1/9*(x^2+6*x-9)/x)+63*x)*log(-exp(1/9*(x^2+6*x-9)/x)+7)*log (log(-exp(1/9*(x^2+6*x-9)/x)+7))+(x^2+9)*exp(1/9*(x^2+6*x-9)/x))/(9*x^3*ex p(1/9*(x^2+6*x-9)/x)-63*x^3)/log(-exp(1/9*(x^2+6*x-9)/x)+7),x)
Output:
int(((-9*x*exp(1/9*(x^2+6*x-9)/x)+63*x)*log(-exp(1/9*(x^2+6*x-9)/x)+7)*log (log(-exp(1/9*(x^2+6*x-9)/x)+7))+(x^2+9)*exp(1/9*(x^2+6*x-9)/x))/(9*x^3*ex p(1/9*(x^2+6*x-9)/x)-63*x^3)/log(-exp(1/9*(x^2+6*x-9)/x)+7),x)