Integrand size = 128, antiderivative size = 24 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=2+\frac {x}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \]
Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (-x^2+\frac {x^5 (9+x)}{e}\right )\right )} \]
Integrate[(-2*E + 45*x^3 + 6*x^4 + (E - 9*x^3 - x^4)*Log[(-(E*x^2) + 9*x^5 + x^6)/E]*Log[Log[(-(E*x^2) + 9*x^5 + x^6)/E]])/((E - 9*x^3 - x^4)*Log[(- (E*x^2) + 9*x^5 + x^6)/E]*Log[Log[(-(E*x^2) + 9*x^5 + x^6)/E]]^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.
\(\displaystyle \int \frac {6 x^4+45 x^3+\left (-x^4-9 x^3+e\right ) \log \left (\frac {x^6+9 x^5-e x^2}{e}\right ) \log \left (\log \left (\frac {x^6+9 x^5-e x^2}{e}\right )\right )-2 e}{\left (-x^4-9 x^3+e\right ) \log \left (\frac {x^6+9 x^5-e x^2}{e}\right ) \log ^2\left (\log \left (\frac {x^6+9 x^5-e x^2}{e}\right )\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {6 x^4+45 x^3+\left (-x^4-9 x^3+e\right ) \log \left (\frac {x^6+9 x^5-e x^2}{e}\right ) \log \left (\log \left (\frac {x^6+9 x^5-e x^2}{e}\right )\right )-2 e}{\left (-x^4-9 x^3+e\right ) \log \left (\frac {x^2 \left (x^4+9 x^3-e\right )}{e}\right ) \log ^2\left (\log \left (\frac {x^2 \left (x^4+9 x^3-e\right )}{e}\right )\right )}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{\log \left (\log \left (\frac {x^5 (x+9)}{e}-x^2\right )\right )}+\frac {6 x^4+45 x^3-2 e}{\left (-x^4-9 x^3+e\right ) \log \left (\frac {x^5 (x+9)}{e}-x^2\right ) \log ^2\left (\log \left (\frac {x^5 (x+9)}{e}-x^2\right )\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -6 \int \frac {1}{\log \left (\frac {x^5 (x+9)}{e}-x^2\right ) \log ^2\left (\log \left (\frac {x^5 (x+9)}{e}-x^2\right )\right )}dx+\int \frac {1}{\log \left (\log \left (\frac {x^5 (x+9)}{e}-x^2\right )\right )}dx+4 e \int \frac {1}{\left (-x^4-9 x^3+e\right ) \log \left (\frac {x^5 (x+9)}{e}-x^2\right ) \log ^2\left (\log \left (\frac {x^5 (x+9)}{e}-x^2\right )\right )}dx-9 \int \frac {x^3}{\left (-x^4-9 x^3+e\right ) \log \left (\frac {x^5 (x+9)}{e}-x^2\right ) \log ^2\left (\log \left (\frac {x^5 (x+9)}{e}-x^2\right )\right )}dx\) |
Int[(-2*E + 45*x^3 + 6*x^4 + (E - 9*x^3 - x^4)*Log[(-(E*x^2) + 9*x^5 + x^6 )/E]*Log[Log[(-(E*x^2) + 9*x^5 + x^6)/E]])/((E - 9*x^3 - x^4)*Log[(-(E*x^2 ) + 9*x^5 + x^6)/E]*Log[Log[(-(E*x^2) + 9*x^5 + x^6)/E]]^2),x]
3.10.56.3.1 Defintions of rubi rules used
\[\int \frac {\left ({\mathrm e}-x^{4}-9 x^{3}\right ) \ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right ) \ln \left (\ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right )\right )-2 \,{\mathrm e}+6 x^{4}+45 x^{3}}{\left ({\mathrm e}-x^{4}-9 x^{3}\right ) \ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right ) {\ln \left (\ln \left (\left (-x^{2} {\mathrm e}+x^{6}+9 x^{5}\right ) {\mathrm e}^{-1}\right )\right )}^{2}}d x\]
int(((exp(1)-x^4-9*x^3)*ln((-x^2*exp(1)+x^6+9*x^5)/exp(1))*ln(ln((-x^2*exp (1)+x^6+9*x^5)/exp(1)))-2*exp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3)/ln((-x^2 *exp(1)+x^6+9*x^5)/exp(1))/ln(ln((-x^2*exp(1)+x^6+9*x^5)/exp(1)))^2,x)
int(((exp(1)-x^4-9*x^3)*ln((-x^2*exp(1)+x^6+9*x^5)/exp(1))*ln(ln((-x^2*exp (1)+x^6+9*x^5)/exp(1)))-2*exp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3)/ln((-x^2 *exp(1)+x^6+9*x^5)/exp(1))/ln(ln((-x^2*exp(1)+x^6+9*x^5)/exp(1)))^2,x)
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left ({\left (x^{6} + 9 \, x^{5} - x^{2} e\right )} e^{\left (-1\right )}\right )\right )} \]
integrate(((exp(1)-x^4-9*x^3)*log((-x^2*exp(1)+x^6+9*x^5)/exp(1))*log(log( (-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*exp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3) /log((-x^2*exp(1)+x^6+9*x^5)/exp(1))/log(log((-x^2*exp(1)+x^6+9*x^5)/exp(1 )))^2,x, algorithm=\
Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log {\left (\log {\left (\frac {x^{6} + 9 x^{5} - e x^{2}}{e} \right )} \right )}} \]
integrate(((exp(1)-x**4-9*x**3)*ln((-x**2*exp(1)+x**6+9*x**5)/exp(1))*ln(l n((-x**2*exp(1)+x**6+9*x**5)/exp(1)))-2*exp(1)+6*x**4+45*x**3)/(exp(1)-x** 4-9*x**3)/ln((-x**2*exp(1)+x**6+9*x**5)/exp(1))/ln(ln((-x**2*exp(1)+x**6+9 *x**5)/exp(1)))**2,x)
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (x^{4} + 9 \, x^{3} - e\right ) + 2 \, \log \left (x\right ) - 1\right )} \]
integrate(((exp(1)-x^4-9*x^3)*log((-x^2*exp(1)+x^6+9*x^5)/exp(1))*log(log( (-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*exp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3) /log((-x^2*exp(1)+x^6+9*x^5)/exp(1))/log(log((-x^2*exp(1)+x^6+9*x^5)/exp(1 )))^2,x, algorithm=\
Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {x}{\log \left (\log \left (x^{6} + 9 \, x^{5} - x^{2} e\right ) - 1\right )} \]
integrate(((exp(1)-x^4-9*x^3)*log((-x^2*exp(1)+x^6+9*x^5)/exp(1))*log(log( (-x^2*exp(1)+x^6+9*x^5)/exp(1)))-2*exp(1)+6*x^4+45*x^3)/(exp(1)-x^4-9*x^3) /log((-x^2*exp(1)+x^6+9*x^5)/exp(1))/log(log((-x^2*exp(1)+x^6+9*x^5)/exp(1 )))^2,x, algorithm=\
Time = 10.95 (sec) , antiderivative size = 170, normalized size of antiderivative = 7.08 \[ \int \frac {-2 e+45 x^3+6 x^4+\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log \left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )}{\left (e-9 x^3-x^4\right ) \log \left (\frac {-e x^2+9 x^5+x^6}{e}\right ) \log ^2\left (\log \left (\frac {-e x^2+9 x^5+x^6}{e}\right )\right )} \, dx=\frac {\ln \left (x^4+9\,x^3-\mathrm {e}\right )}{4}+\frac {\ln \left (x\right )}{2}+\frac {x}{\ln \left (\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )\right )}-\frac {45\,x^3\,\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )}{4\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )}-\frac {3\,x^4\,\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )}{2\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )}+\frac {\ln \left ({\mathrm {e}}^{-1}\,x^6+9\,{\mathrm {e}}^{-1}\,x^5-x^2\right )\,\mathrm {e}}{2\,\left (6\,x^4+45\,x^3-2\,\mathrm {e}\right )} \]
int((2*exp(1) - 45*x^3 - 6*x^4 + log(exp(-1)*(9*x^5 - x^2*exp(1) + x^6))*l og(log(exp(-1)*(9*x^5 - x^2*exp(1) + x^6)))*(9*x^3 - exp(1) + x^4))/(log(e xp(-1)*(9*x^5 - x^2*exp(1) + x^6))*log(log(exp(-1)*(9*x^5 - x^2*exp(1) + x ^6)))^2*(9*x^3 - exp(1) + x^4)),x)
log(9*x^3 - exp(1) + x^4)/4 + log(x)/2 + x/log(log(9*x^5*exp(-1) + x^6*exp (-1) - x^2)) - (45*x^3*log(9*x^5*exp(-1) + x^6*exp(-1) - x^2))/(4*(45*x^3 - 2*exp(1) + 6*x^4)) - (3*x^4*log(9*x^5*exp(-1) + x^6*exp(-1) - x^2))/(2*( 45*x^3 - 2*exp(1) + 6*x^4)) + (log(9*x^5*exp(-1) + x^6*exp(-1) - x^2)*exp( 1))/(2*(45*x^3 - 2*exp(1) + 6*x^4))