Integrand size = 130, antiderivative size = 26 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\frac {1}{2} \left (-x+\log \left (x \left (e^x+x\right )\right )\right ) \left (-1-x+\log ^2(\log (x))\right ) \] Output:
(ln((exp(x)+x)*x)-x)*(1/2*ln(ln(x))^2-1/2*x-1/2)
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\frac {1}{2} \left (x+x^2-\log (x)-\log \left (e^x+x\right )-x \log \left (x \left (e^x+x\right )\right )-\left (x-\log \left (x \left (e^x+x\right )\right )\right ) \log ^2(\log (x))\right ) \] Input:
Integrate[((-2*x - x^2 + 2*x^3 + E^x*(-1 - x + x^2))*Log[x] + (-(E^x*x) - x^2)*Log[x]*Log[E^x*x + x^2] + (-2*E^x*x - 2*x^2 + (2*E^x + 2*x)*Log[E^x*x + x^2])*Log[Log[x]] + (E^x + 2*x - x^2)*Log[x]*Log[Log[x]]^2)/((2*E^x*x + 2*x^2)*Log[x]),x]
Output:
(x + x^2 - Log[x] - Log[E^x + x] - x*Log[x*(E^x + x)] - (x - Log[x*(E^x + x)])*Log[Log[x]]^2)/2
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.
| \(\displaystyle \int \frac {\left (-x^2+2 x+e^x\right ) \log (x) \log ^2(\log (x))+\left (-2 x^2+\left (2 x+2 e^x\right ) \log \left (x^2+e^x x\right )-2 e^x x\right ) \log (\log (x))+\left (-x^2-e^x x\right ) \log (x) \log \left (x^2+e^x x\right )+\left (2 x^3-x^2+e^x \left (x^2-x-1\right )-2 x\right ) \log (x)}{\left (2 x^2+2 e^x x\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-x^2+2 x+e^x\right ) \log (x) \log ^2(\log (x))+\left (-2 x^2+\left (2 x+2 e^x\right ) \log \left (x^2+e^x x\right )-2 e^x x\right ) \log (\log (x))+\left (-x^2-e^x x\right ) \log (x) \log \left (x^2+e^x x\right )+\left (2 x^3-x^2+e^x \left (x^2-x-1\right )-2 x\right ) \log (x)}{2 x \left (x+e^x\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {-\left (\left (-x^2+2 x+e^x\right ) \log (x) \log ^2(\log (x))\right )+2 \left (x^2+e^x x-\left (x+e^x\right ) \log \left (x^2+e^x x\right )\right ) \log (\log (x))+\left (-2 x^3+x^2+2 x+e^x \left (-x^2+x+1\right )\right ) \log (x)+\left (x^2+e^x x\right ) \log (x) \log \left (x^2+e^x x\right )}{x \left (x+e^x\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {-\left (\left (-x^2+2 x+e^x\right ) \log (x) \log ^2(\log (x))\right )+2 \left (x^2+e^x x-\left (x+e^x\right ) \log \left (x^2+e^x x\right )\right ) \log (\log (x))+\left (-2 x^3+x^2+2 x+e^x \left (-x^2+x+1\right )\right ) \log (x)+\left (x^2+e^x x\right ) \log (x) \log \left (x^2+e^x x\right )}{x \left (x+e^x\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (\frac {-\log (x) x^2+\log (x) x+\log (x) \log \left (x \left (x+e^x\right )\right ) x+2 \log (\log (x)) x-\log (x) \log ^2(\log (x))+\log (x)-2 \log \left (x \left (x+e^x\right )\right ) \log (\log (x))}{x \log (x)}-\frac {(x-1) \left (-\log ^2(\log (x))+x+1\right )}{x+e^x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2 \int \frac {\log \left (x^2+e^x x\right ) \log (\log (x))}{x \log (x)}dx-\int \frac {1}{x+e^x}dx+\int \frac {x}{x+e^x}dx+\int \frac {\log ^2(\log (x))}{x+e^x}dx-\int \frac {x \log ^2(\log (x))}{x+e^x}dx-2 \int \frac {\log (\log (x))}{\log (x)}dx+x^2-x \log \left (x^2+e^x x\right )+\log (x) \log ^2(\log (x))+\log (x)-2 \log (x) \log (\log (x))\right )\) |
Input:
Int[((-2*x - x^2 + 2*x^3 + E^x*(-1 - x + x^2))*Log[x] + (-(E^x*x) - x^2)*L og[x]*Log[E^x*x + x^2] + (-2*E^x*x - 2*x^2 + (2*E^x + 2*x)*Log[E^x*x + x^2 ])*Log[Log[x]] + (E^x + 2*x - x^2)*Log[x]*Log[Log[x]]^2)/((2*E^x*x + 2*x^2 )*Log[x]),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(50\) vs. \(2(24)=48\).
Time = 6.40 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96
| method | result | size |
| parallelrisch | \(-\frac {\ln \left (\ln \left (x \right )\right )^{2} x}{2}+\frac {\ln \left (\left ({\mathrm e}^{x}+x \right ) x \right ) \ln \left (\ln \left (x \right )\right )^{2}}{2}+\frac {x^{2}}{2}-\frac {\ln \left (\left ({\mathrm e}^{x}+x \right ) x \right ) x}{2}+\frac {x}{2}-\frac {\ln \left (\left ({\mathrm e}^{x}+x \right ) x \right )}{2}\) | \(51\) |
| risch | \(\left (\frac {\ln \left (\ln \left (x \right )\right )^{2}}{2}-\frac {x}{2}\right ) \ln \left ({\mathrm e}^{x}+x \right )-\frac {\ln \left (\ln \left (x \right )\right )^{2} x}{2}+\frac {\ln \left (x \right ) \ln \left (\ln \left (x \right )\right )^{2}}{2}-\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{3}}{4}-\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}{4}+\frac {i x \pi {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{3}}{4}+\frac {i x \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) \operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}{4}-\frac {x \ln \left (x \right )}{2}+\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2}}{4}+\frac {i \ln \left (\ln \left (x \right )\right )^{2} \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2}}{4}-\frac {i x \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right )\right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2}}{4}-\frac {i x \pi \,\operatorname {csgn}\left (i x \right ) {\operatorname {csgn}\left (i \left ({\mathrm e}^{x}+x \right ) x \right )}^{2}}{4}+\frac {x^{2}}{2}-\frac {\ln \left (x \right )}{2}+\frac {x}{2}-\frac {\ln \left ({\mathrm e}^{x}+x \right )}{2}\) | \(252\) |
Input:
int(((exp(x)-x^2+2*x)*ln(x)*ln(ln(x))^2+((2*exp(x)+2*x)*ln(exp(x)*x+x^2)-2 *exp(x)*x-2*x^2)*ln(ln(x))+(-exp(x)*x-x^2)*ln(x)*ln(exp(x)*x+x^2)+((x^2-x- 1)*exp(x)+2*x^3-x^2-2*x)*ln(x))/(2*exp(x)*x+2*x^2)/ln(x),x,method=_RETURNV ERBOSE)
Output:
-1/2*ln(ln(x))^2*x+1/2*ln((exp(x)+x)*x)*ln(ln(x))^2+1/2*x^2-1/2*ln((exp(x) +x)*x)*x+1/2*x-1/2*ln((exp(x)+x)*x)
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=-\frac {1}{2} \, {\left (x - \log \left (x^{2} + x e^{x}\right )\right )} \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, {\left (x + 1\right )} \log \left (x^{2} + x e^{x}\right ) + \frac {1}{2} \, x \] Input:
integrate(((exp(x)-x^2+2*x)*log(x)*log(log(x))^2+((2*exp(x)+2*x)*log(exp(x )*x+x^2)-2*exp(x)*x-2*x^2)*log(log(x))+(-exp(x)*x-x^2)*log(x)*log(exp(x)*x +x^2)+((x^2-x-1)*exp(x)+2*x^3-x^2-2*x)*log(x))/(2*exp(x)*x+2*x^2)/log(x),x , algorithm="fricas")
Output:
-1/2*(x - log(x^2 + x*e^x))*log(log(x))^2 + 1/2*x^2 - 1/2*(x + 1)*log(x^2 + x*e^x) + 1/2*x
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (24) = 48\).
Time = 0.89 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\frac {x^{2}}{2} - \frac {x \log {\left (\log {\left (x \right )} \right )}^{2}}{2} + \frac {x}{2} + \left (- \frac {x}{2} + \frac {\log {\left (\log {\left (x \right )} \right )}^{2}}{2}\right ) \log {\left (x^{2} + x e^{x} \right )} - \frac {\log {\left (x \right )}}{2} - \frac {\log {\left (x + e^{x} \right )}}{2} \] Input:
integrate(((exp(x)-x**2+2*x)*ln(x)*ln(ln(x))**2+((2*exp(x)+2*x)*ln(exp(x)* x+x**2)-2*exp(x)*x-2*x**2)*ln(ln(x))+(-exp(x)*x-x**2)*ln(x)*ln(exp(x)*x+x* *2)+((x**2-x-1)*exp(x)+2*x**3-x**2-2*x)*ln(x))/(2*exp(x)*x+2*x**2)/ln(x),x )
Output:
x**2/2 - x*log(log(x))**2/2 + x/2 + (-x/2 + log(log(x))**2/2)*log(x**2 + x *exp(x)) - log(x)/2 - log(x + exp(x))/2
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.77 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=-\frac {1}{2} \, {\left (x - \log \left (x\right )\right )} \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, x^{2} + \frac {1}{2} \, {\left (\log \left (\log \left (x\right )\right )^{2} - x - 1\right )} \log \left (x + e^{x}\right ) - \frac {1}{2} \, {\left (x + 1\right )} \log \left (x\right ) + \frac {1}{2} \, x \] Input:
integrate(((exp(x)-x^2+2*x)*log(x)*log(log(x))^2+((2*exp(x)+2*x)*log(exp(x )*x+x^2)-2*exp(x)*x-2*x^2)*log(log(x))+(-exp(x)*x-x^2)*log(x)*log(exp(x)*x +x^2)+((x^2-x-1)*exp(x)+2*x^3-x^2-2*x)*log(x))/(2*exp(x)*x+2*x^2)/log(x),x , algorithm="maxima")
Output:
-1/2*(x - log(x))*log(log(x))^2 + 1/2*x^2 + 1/2*(log(log(x))^2 - x - 1)*lo g(x + e^x) - 1/2*(x + 1)*log(x) + 1/2*x
Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (23) = 46\).
Time = 0.17 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.38 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=-\frac {1}{2} \, x \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, \log \left (x + e^{x}\right ) \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, \log \left (x\right ) \log \left (\log \left (x\right )\right )^{2} + \frac {1}{2} \, x^{2} - \frac {1}{2} \, x \log \left (x + e^{x}\right ) - \frac {1}{2} \, x \log \left (x\right ) + \frac {1}{2} \, x - \frac {1}{2} \, \log \left (x + e^{x}\right ) - \frac {1}{2} \, \log \left (x\right ) \] Input:
integrate(((exp(x)-x^2+2*x)*log(x)*log(log(x))^2+((2*exp(x)+2*x)*log(exp(x )*x+x^2)-2*exp(x)*x-2*x^2)*log(log(x))+(-exp(x)*x-x^2)*log(x)*log(exp(x)*x +x^2)+((x^2-x-1)*exp(x)+2*x^3-x^2-2*x)*log(x))/(2*exp(x)*x+2*x^2)/log(x),x , algorithm="giac")
Output:
-1/2*x*log(log(x))^2 + 1/2*log(x + e^x)*log(log(x))^2 + 1/2*log(x)*log(log (x))^2 + 1/2*x^2 - 1/2*x*log(x + e^x) - 1/2*x*log(x) + 1/2*x - 1/2*log(x + e^x) - 1/2*log(x)
Timed out. \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\int -\frac {-\ln \left (x\right )\,\left (2\,x+{\mathrm {e}}^x-x^2\right )\,{\ln \left (\ln \left (x\right )\right )}^2+\left (2\,x\,{\mathrm {e}}^x-\ln \left (x\,{\mathrm {e}}^x+x^2\right )\,\left (2\,x+2\,{\mathrm {e}}^x\right )+2\,x^2\right )\,\ln \left (\ln \left (x\right )\right )+\ln \left (x\right )\,\left (2\,x+{\mathrm {e}}^x\,\left (-x^2+x+1\right )+x^2-2\,x^3\right )+\ln \left (x\,{\mathrm {e}}^x+x^2\right )\,\ln \left (x\right )\,\left (x\,{\mathrm {e}}^x+x^2\right )}{\ln \left (x\right )\,\left (2\,x\,{\mathrm {e}}^x+2\,x^2\right )} \,d x \] Input:
int(-(log(log(x))*(2*x*exp(x) - log(x*exp(x) + x^2)*(2*x + 2*exp(x)) + 2*x ^2) + log(x)*(2*x + exp(x)*(x - x^2 + 1) + x^2 - 2*x^3) - log(log(x))^2*lo g(x)*(2*x + exp(x) - x^2) + log(x*exp(x) + x^2)*log(x)*(x*exp(x) + x^2))/( log(x)*(2*x*exp(x) + 2*x^2)),x)
Output:
int(-(log(log(x))*(2*x*exp(x) - log(x*exp(x) + x^2)*(2*x + 2*exp(x)) + 2*x ^2) + log(x)*(2*x + exp(x)*(x - x^2 + 1) + x^2 - 2*x^3) - log(log(x))^2*lo g(x)*(2*x + exp(x) - x^2) + log(x*exp(x) + x^2)*log(x)*(x*exp(x) + x^2))/( log(x)*(2*x*exp(x) + 2*x^2)), x)
Time = 0.57 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {\left (-2 x-x^2+2 x^3+e^x \left (-1-x+x^2\right )\right ) \log (x)+\left (-e^x x-x^2\right ) \log (x) \log \left (e^x x+x^2\right )+\left (-2 e^x x-2 x^2+\left (2 e^x+2 x\right ) \log \left (e^x x+x^2\right )\right ) \log (\log (x))+\left (e^x+2 x-x^2\right ) \log (x) \log ^2(\log (x))}{\left (2 e^x x+2 x^2\right ) \log (x)} \, dx=\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} \mathrm {log}\left (e^{x} x +x^{2}\right )}{2}-\frac {\mathrm {log}\left (\mathrm {log}\left (x \right )\right )^{2} x}{2}-\frac {\mathrm {log}\left (e^{x} x +x^{2}\right ) x}{2}-\frac {\mathrm {log}\left (e^{x} x +x^{2}\right )}{2}+\frac {x^{2}}{2}+\frac {x}{2} \] Input:
int(((exp(x)-x^2+2*x)*log(x)*log(log(x))^2+((2*exp(x)+2*x)*log(exp(x)*x+x^ 2)-2*exp(x)*x-2*x^2)*log(log(x))+(-exp(x)*x-x^2)*log(x)*log(exp(x)*x+x^2)+ ((x^2-x-1)*exp(x)+2*x^3-x^2-2*x)*log(x))/(2*exp(x)*x+2*x^2)/log(x),x)
Output:
(log(log(x))**2*log(e**x*x + x**2) - log(log(x))**2*x - log(e**x*x + x**2) *x - log(e**x*x + x**2) + x**2 + x)/2