Integrand size = 143, antiderivative size = 30 \[ \int \frac {-8-e^4+x^2-\log (x)+\left (9+e^4+8 x+x^2+\log (x)\right ) \log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )+\left (9+e^4+8 x+x^2+e^x \left (-9-e^4-8 x-x^2\right )+\left (1-e^x\right ) \log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )}{\left (9+e^4+8 x+x^2+\log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \, dx=-e^x+x+\frac {x}{\log \left (\frac {x}{e^4+2 x+(3+x)^2+\log (x)}\right )} \]
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {-8-e^4+x^2-\log (x)+\left (9+e^4+8 x+x^2+\log (x)\right ) \log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )+\left (9+e^4+8 x+x^2+e^x \left (-9-e^4-8 x-x^2\right )+\left (1-e^x\right ) \log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )}{\left (9+e^4+8 x+x^2+\log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \, dx=-e^x+x+\frac {x}{\log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \]
Integrate[(-8 - E^4 + x^2 - Log[x] + (9 + E^4 + 8*x + x^2 + Log[x])*Log[x/ (9 + E^4 + 8*x + x^2 + Log[x])] + (9 + E^4 + 8*x + x^2 + E^x*(-9 - E^4 - 8 *x - x^2) + (1 - E^x)*Log[x])*Log[x/(9 + E^4 + 8*x + x^2 + Log[x])]^2)/((9 + E^4 + 8*x + x^2 + Log[x])*Log[x/(9 + E^4 + 8*x + x^2 + Log[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.
| \(\displaystyle \int \frac {x^2+\left (x^2+e^x \left (-x^2-8 x-e^4-9\right )+8 x+\left (1-e^x\right ) \log (x)+e^4+9\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+e^4+9}\right )+\left (x^2+8 x+\log (x)+e^4+9\right ) \log \left (\frac {x}{x^2+8 x+\log (x)+e^4+9}\right )-\log (x)-e^4-8}{\left (x^2+8 x+\log (x)+e^4+9\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+e^4+9}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {x^2+\left (x^2+e^x \left (-x^2-8 x-e^4-9\right )+8 x+\left (1-e^x\right ) \log (x)+e^4+9\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+e^4+9}\right )+\left (x^2+8 x+\log (x)+e^4+9\right ) \log \left (\frac {x}{x^2+8 x+\log (x)+e^4+9}\right )-\log (x)-8 \left (1+\frac {e^4}{8}\right )}{\left (x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2}{\left (x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}+\frac {\log (x)}{\left (-x^2-8 x-\log (x)-9 \left (1+\frac {e^4}{9}\right )\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}+\frac {-8-e^4}{\left (x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}+\frac {x^2}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}+\frac {8 x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}+\frac {\log (x)}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}+\frac {9 \left (1+\frac {e^4}{9}\right )}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}+\frac {1}{\log \left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}-e^x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {\log (x)}{\left (-x^2-8 x-\log (x)-9 \left (1+\frac {e^4}{9}\right )\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}dx-\left (8+e^4\right ) \int \frac {1}{\left (x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}dx+\int \frac {x^2}{\left (x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )\right ) \log ^2\left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}dx+\left (9+e^4\right ) \int \frac {1}{-x^2-8 x-\log (x)-9 \left (1+\frac {e^4}{9}\right )}dx+8 \int \frac {x}{-x^2-8 x-\log (x)-9 \left (1+\frac {e^4}{9}\right )}dx+\int \frac {x^2}{-x^2-8 x-\log (x)-9 \left (1+\frac {e^4}{9}\right )}dx+\left (9+e^4\right ) \int \frac {1}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}dx+8 \int \frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}dx+\int \frac {x^2}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}dx+\int \frac {1}{\log \left (\frac {x}{x^2+8 x+\log (x)+9 \left (1+\frac {e^4}{9}\right )}\right )}dx+x-e^x\) |
Int[(-8 - E^4 + x^2 - Log[x] + (9 + E^4 + 8*x + x^2 + Log[x])*Log[x/(9 + E ^4 + 8*x + x^2 + Log[x])] + (9 + E^4 + 8*x + x^2 + E^x*(-9 - E^4 - 8*x - x ^2) + (1 - E^x)*Log[x])*Log[x/(9 + E^4 + 8*x + x^2 + Log[x])]^2)/((9 + E^4 + 8*x + x^2 + Log[x])*Log[x/(9 + E^4 + 8*x + x^2 + Log[x])]^2),x]
3.13.77.3.1 Defintions of rubi rules used
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 36.96 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.77
\[x -{\mathrm e}^{x}+\frac {2 i x}{\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{4}+x^{2}+8 x +9}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )+{\mathrm e}^{4}+x^{2}+8 x +9}\right )-\pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )+{\mathrm e}^{4}+x^{2}+8 x +9}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )+{\mathrm e}^{4}+x^{2}+8 x +9}\right )^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )+{\mathrm e}^{4}+x^{2}+8 x +9}\right )^{2}+\pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )+{\mathrm e}^{4}+x^{2}+8 x +9}\right )^{3}-2 i \ln \left (\ln \left (x \right )+{\mathrm e}^{4}+x^{2}+8 x +9\right )+2 i \ln \left (x \right )}\]
int((((1-exp(x))*ln(x)+(-exp(4)-x^2-8*x-9)*exp(x)+exp(4)+x^2+8*x+9)*ln(x/( ln(x)+exp(4)+x^2+8*x+9))^2+(ln(x)+exp(4)+x^2+8*x+9)*ln(x/(ln(x)+exp(4)+x^2 +8*x+9))-ln(x)-exp(4)+x^2-8)/(ln(x)+exp(4)+x^2+8*x+9)/ln(x/(ln(x)+exp(4)+x ^2+8*x+9))^2,x)
x-exp(x)+2*I*x/(Pi*csgn(I/(ln(x)+exp(4)+x^2+8*x+9))*csgn(I*x)*csgn(I*x/(ln (x)+exp(4)+x^2+8*x+9))-Pi*csgn(I/(ln(x)+exp(4)+x^2+8*x+9))*csgn(I*x/(ln(x) +exp(4)+x^2+8*x+9))^2-Pi*csgn(I*x)*csgn(I*x/(ln(x)+exp(4)+x^2+8*x+9))^2+Pi *csgn(I*x/(ln(x)+exp(4)+x^2+8*x+9))^3-2*I*ln(ln(x)+exp(4)+x^2+8*x+9)+2*I*l n(x))
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {-8-e^4+x^2-\log (x)+\left (9+e^4+8 x+x^2+\log (x)\right ) \log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )+\left (9+e^4+8 x+x^2+e^x \left (-9-e^4-8 x-x^2\right )+\left (1-e^x\right ) \log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )}{\left (9+e^4+8 x+x^2+\log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \, dx=\frac {{\left (x - e^{x}\right )} \log \left (\frac {x}{x^{2} + 8 \, x + e^{4} + \log \left (x\right ) + 9}\right ) + x}{\log \left (\frac {x}{x^{2} + 8 \, x + e^{4} + \log \left (x\right ) + 9}\right )} \]
integrate((((1-exp(x))*log(x)+(-exp(4)-x^2-8*x-9)*exp(x)+exp(4)+x^2+8*x+9) *log(x/(log(x)+exp(4)+x^2+8*x+9))^2+(log(x)+exp(4)+x^2+8*x+9)*log(x/(log(x )+exp(4)+x^2+8*x+9))-log(x)-exp(4)+x^2-8)/(log(x)+exp(4)+x^2+8*x+9)/log(x/ (log(x)+exp(4)+x^2+8*x+9))^2,x, algorithm=\
Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {-8-e^4+x^2-\log (x)+\left (9+e^4+8 x+x^2+\log (x)\right ) \log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )+\left (9+e^4+8 x+x^2+e^x \left (-9-e^4-8 x-x^2\right )+\left (1-e^x\right ) \log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )}{\left (9+e^4+8 x+x^2+\log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \, dx=x + \frac {x}{\log {\left (\frac {x}{x^{2} + 8 x + \log {\left (x \right )} + 9 + e^{4}} \right )}} - e^{x} \]
integrate((((1-exp(x))*ln(x)+(-exp(4)-x**2-8*x-9)*exp(x)+exp(4)+x**2+8*x+9 )*ln(x/(ln(x)+exp(4)+x**2+8*x+9))**2+(ln(x)+exp(4)+x**2+8*x+9)*ln(x/(ln(x) +exp(4)+x**2+8*x+9))-ln(x)-exp(4)+x**2-8)/(ln(x)+exp(4)+x**2+8*x+9)/ln(x/( ln(x)+exp(4)+x**2+8*x+9))**2,x)
Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83 \[ \int \frac {-8-e^4+x^2-\log (x)+\left (9+e^4+8 x+x^2+\log (x)\right ) \log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )+\left (9+e^4+8 x+x^2+e^x \left (-9-e^4-8 x-x^2\right )+\left (1-e^x\right ) \log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )}{\left (9+e^4+8 x+x^2+\log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \, dx=\frac {{\left (x - e^{x}\right )} \log \left (x^{2} + 8 \, x + e^{4} + \log \left (x\right ) + 9\right ) - x \log \left (x\right ) + e^{x} \log \left (x\right ) - x}{\log \left (x^{2} + 8 \, x + e^{4} + \log \left (x\right ) + 9\right ) - \log \left (x\right )} \]
integrate((((1-exp(x))*log(x)+(-exp(4)-x^2-8*x-9)*exp(x)+exp(4)+x^2+8*x+9) *log(x/(log(x)+exp(4)+x^2+8*x+9))^2+(log(x)+exp(4)+x^2+8*x+9)*log(x/(log(x )+exp(4)+x^2+8*x+9))-log(x)-exp(4)+x^2-8)/(log(x)+exp(4)+x^2+8*x+9)/log(x/ (log(x)+exp(4)+x^2+8*x+9))^2,x, algorithm=\
((x - e^x)*log(x^2 + 8*x + e^4 + log(x) + 9) - x*log(x) + e^x*log(x) - x)/ (log(x^2 + 8*x + e^4 + log(x) + 9) - log(x))
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (28) = 56\).
Time = 0.48 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.23 \[ \int \frac {-8-e^4+x^2-\log (x)+\left (9+e^4+8 x+x^2+\log (x)\right ) \log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )+\left (9+e^4+8 x+x^2+e^x \left (-9-e^4-8 x-x^2\right )+\left (1-e^x\right ) \log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )}{\left (9+e^4+8 x+x^2+\log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \, dx=\frac {x \log \left (x^{2} + 8 \, x + e^{4} + \log \left (x\right ) + 9\right ) - e^{x} \log \left (x^{2} + 8 \, x + e^{4} + \log \left (x\right ) + 9\right ) - x \log \left (x\right ) + e^{x} \log \left (x\right ) - x}{\log \left (x^{2} + 8 \, x + e^{4} + \log \left (x\right ) + 9\right ) - \log \left (x\right )} \]
integrate((((1-exp(x))*log(x)+(-exp(4)-x^2-8*x-9)*exp(x)+exp(4)+x^2+8*x+9) *log(x/(log(x)+exp(4)+x^2+8*x+9))^2+(log(x)+exp(4)+x^2+8*x+9)*log(x/(log(x )+exp(4)+x^2+8*x+9))-log(x)-exp(4)+x^2-8)/(log(x)+exp(4)+x^2+8*x+9)/log(x/ (log(x)+exp(4)+x^2+8*x+9))^2,x, algorithm=\
(x*log(x^2 + 8*x + e^4 + log(x) + 9) - e^x*log(x^2 + 8*x + e^4 + log(x) + 9) - x*log(x) + e^x*log(x) - x)/(log(x^2 + 8*x + e^4 + log(x) + 9) - log(x ))
Time = 13.80 (sec) , antiderivative size = 167, normalized size of antiderivative = 5.57 \[ \int \frac {-8-e^4+x^2-\log (x)+\left (9+e^4+8 x+x^2+\log (x)\right ) \log \left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )+\left (9+e^4+8 x+x^2+e^x \left (-9-e^4-8 x-x^2\right )+\left (1-e^x\right ) \log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )}{\left (9+e^4+8 x+x^2+\log (x)\right ) \log ^2\left (\frac {x}{9+e^4+8 x+x^2+\log (x)}\right )} \, dx=\frac {\frac {x\,\left (120\,x+{\mathrm {e}}^4+16\,x\,{\mathrm {e}}^4+6\,x^2\,{\mathrm {e}}^4+47\,x^2-2\,x^4+7\right )}{2\,x^2-1}+\frac {x\,\ln \left (x\right )\,\left (6\,x^2+16\,x+1\right )}{2\,x^2-1}}{{\mathrm {e}}^4+\ln \left (x\right )-x^2+8}-{\mathrm {e}}^x-\frac {2\,x+4}{x^2-\frac {1}{2}}-x+\frac {x-\frac {x\,\ln \left (\frac {x}{8\,x+{\mathrm {e}}^4+\ln \left (x\right )+x^2+9}\right )\,\left (8\,x+{\mathrm {e}}^4+\ln \left (x\right )+x^2+9\right )}{{\mathrm {e}}^4+\ln \left (x\right )-x^2+8}}{\ln \left (\frac {x}{8\,x+{\mathrm {e}}^4+\ln \left (x\right )+x^2+9}\right )} \]
int(-(exp(4) + log(x) - log(x/(8*x + exp(4) + log(x) + x^2 + 9))^2*(8*x + exp(4) - log(x)*(exp(x) - 1) - exp(x)*(8*x + exp(4) + x^2 + 9) + x^2 + 9) - log(x/(8*x + exp(4) + log(x) + x^2 + 9))*(8*x + exp(4) + log(x) + x^2 + 9) - x^2 + 8)/(log(x/(8*x + exp(4) + log(x) + x^2 + 9))^2*(8*x + exp(4) + log(x) + x^2 + 9)),x)
((x*(120*x + exp(4) + 16*x*exp(4) + 6*x^2*exp(4) + 47*x^2 - 2*x^4 + 7))/(2 *x^2 - 1) + (x*log(x)*(16*x + 6*x^2 + 1))/(2*x^2 - 1))/(exp(4) + log(x) - x^2 + 8) - exp(x) - (2*x + 4)/(x^2 - 1/2) - x + (x - (x*log(x/(8*x + exp(4 ) + log(x) + x^2 + 9))*(8*x + exp(4) + log(x) + x^2 + 9))/(exp(4) + log(x) - x^2 + 8))/log(x/(8*x + exp(4) + log(x) + x^2 + 9))